![Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Book Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Conventional Optimization Techniques . . . . . . . . . . . . . . . . . . . 14
2.2.1 Linear Programming (LP) [5, 6] . . . . . . . . . . . . . . . . . . . 14
2.2.2 Quadratic Programming (QP) . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Integer Programming (IP) . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.4 Dynamic Programming (DP) . . . . . . . . . . . . . . . . . . . . . 17
2.3 Artificial Intelligence (AI) Techniques . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Artificial Neural Network (ANN) . . . . . . . . . . . . . . . . . . 17
2.3.2 Fuzzy Linear Programming (FLP) [22, 23] . . . . . . . . . . . 18
2.3.3 Expert Systems (ES) . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Modern Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Evolutionary Optimization Techniques . . . . . . . . . . . . . . . . . . . . 23
2.5.1 Genetic Algorithm (GA) [25, 26] . . . . . . . . . . . . . . . . . . 23
2.6 Differential Evolution (DE) Algorithm . . . . . . . . . . . . . . . . . . . . 24
2.6.1 Standard DE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Particle Swarm Optimization (PSO) Technique . . . . . . . . . . . . . . 25
2.7.1 PSO Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 Seeker Optimization Algorithm (SOA) . . . . . . . . . . . . . . . . . . . . 28
2.8.1 SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.9 Ant Colony Optimization (ACO) Algorithm . . . . . . . . . . . . . . . . 30
2.9.1 Description of Real Ants . . . . . . . . . . . . . . . . . . . . . . . . 32
ix](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-12-320.jpg)
![ð Þ
List of Symbols
Symbol Meaning
μ The membership functions of the decision variables in all of
objectives and constraints within range of [0–1].
λ The satisfaction parameter that to be maximize within range of
[0–1] for all objectives and constraints.
c1, c2 The weight coefficients for each term of the modified velocity.
r1, r2 The random numbers between 0 and 1.
pbesti The personal best of particle i.
gbest The global best of the group (populations).
μi The step length in seeker optimization algorithm (SOA).
P
k
ð Þ
ij t The probabilistic transition rule of ant k to go from city i to city
j at iteration t.
α and β Two parameters that influence the relative weight of
pheromone trail and heuristic guide function, respectively.
τij The pheromone trail deposited between city i and j.
ηij The visibility or sight and equal to the inverse of the distance
between city i and j.
Nr
k
The tabu list in memory of ant that recodes the cities are visited
to avoid stagnations.
ρ The pheromone evaporation constant.
ε The elite path weighting constant.
dbest The shortest tour distance found as in traveling salesman
problem (TSP).
Aij The connectivity matrix.
Nch/V The total number of voltage-measuring channels in phasor
measurement units (PMUs).
Nch/I The total number of current measuring channels in PMUs.
Nb The total number of system buses.
NZIBs The total number of zero injection buses (ZIBs) in the system.
xiii](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-16-320.jpg)
![Chapter 1
Introduction
1.1 General
The term “smart grid (SG)” defines a self-healing network equipped with dynamic
optimization techniques that use real-time measurements to minimize network
losses, maintain voltage levels, increase reliability, and improve asset management.
The operational data collected by the SG and its subsystems will allow system
operators to rapidly identify the best strategy to secure against attacks, vulnerability,
and so on, caused by various contingencies. However, the SG first depends upon
identifying and researching key performance measures, designing and testing appro-
priate tools, and developing the proper education curriculum to equip current and
future personnel with the knowledge and skills for deployment of this highly
advanced system [1].
The transmission system that interconnects all major substation and load centers
is the backbone of an integrated power system. Efficiency and reliability at an
affordable cost continue to be the ultimate aims of transmission planners and
operators. Transmission lines must tolerate dynamic changes in load and contin-
gency without service disruptions. Strategies to achieve SG performance at the
transmission level include the design of analytical tools and advanced technology
with intelligence for performance analysis such as dynamic optimal power flow,
robust state estimation, real-time stability assessment, and reliability and market
simulation tools. Real-time monitoring based on phasor measurement unit (PMU),
state estimators sensors, and communication technologies are the transmission sub-
system’s intelligent enabling tools for developing smart transmission functionality.
The distribution system is at the heart of the emerging smart electrical system, and
the role of distribution system operators is on the verge of a real revolution.
Operators are in fact the ones who will have to master technical complexity, limit
the sudden rise in costs, and ensure the quality of service expected by customers. The
distribution grid, which ensures continuity of supply, is a vital infrastructure for our
economies to function effectively. This has been true for a long time and is
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. A. A. El-Ela et al., Modern Optimization Techniques for Smart Grids,
https://doi.org/10.1007/978-3-030-96025-4_1
1](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-21-320.jpg)
![increasingly becoming the case. The distribution network is one of the most complex
industrial facilities currently in use [2].
2 1 Introduction
In fact, until now, electricity was very predictable, going from large power plants
to large electricity transmission grids, then to the distribution grids to supply
customers. The direction of the flow was predictable. With the development of
distributed generation (DG), a significant proportion of generation is connected to
the distribution grid. Furthermore, the means of DG are mainly wind and solar power
stations, whose generation varies depending on the wind and sun. It is therefore
possible, for example, to have a generation greater than the local consumption in the
middle of the afternoon in a housing development where there are many solar panels,
at a time when people are at work or school, and when the panels produce at their
maximum rate. Therefore, the electricity temporarily flows that will return a higher
voltage to the grids. At night, the flow returns to its usual direction. There are
therefore varying directions of flow [2].
The term “optimization” refers to a choice that must be made from several
possible solutions while respecting a finite number of constraints. A human desire
for perfection finds its expression in the optimization theory, which teaches how to
describe and fulfill an optimum. The optimization tries to improve system perfor-
mances in the direction of the optimal point or points. The optimization can be
defined as a part of the applied or numerical mathematics or a method for system
design by computer in accordance with either one stress theoretical aspect (existence
of the optimum solution conditions) or the practical aspect (procedures for obtaining
the optimum solution) [3].
The analytical solution [4] used to solve an optimization problem depends on the
form of the criterion and constraint functions. The simplest situation to be considered
is the unconstrained optimization problem. In such a problem, no constraints are
imposed on the decision variable, and different calculus can be used to analyze them.
Another relatively simple form of the general optimization problem is the case in
which all the constraints of the problem can be expressed as equality relationships.
However, conventional optimization techniques have been developed that can
efficiently solve several classes of problems with these inequality restrictions.
Linear programming (LP) [5, 6] problems constitute the most important class for
which efficient solution techniques have been developed. LP problem is specified by
a linear, multivariable function which is to be optimized (maximized or minimized)
subject to a number of linear constraints. In 1963, the mathematician Dantzig
developed an algorithm called the simplex method to solve problems of this type.
The original simplex method has been modified into an efficient algorithm to solve
large LP problems by computer. Other classes of problems include integer program-
ming (IP) [7, 8] problems, in which at least some of the variables must assume only
integer values, and quadratic programming (QP) problems [9–17], in which the
objective relationship is a quadratic function of the decision variables, as well as
dynamic programming (DP) [18, 19], in which decision variables are to be made in
sequential or multistage decision-making problems.](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-22-320.jpg)
![1.1 General 3
The artificial intelligence (AI) is the science and engineering of making intelli-
gent machines, especially intelligent computer programs. It is related to the similar
task of using computers to understand human intelligence. Intelligence is the com-
putational part of the ability to achieve goals in the world. The artificial neural
network (ANN), fuzzy logic, and expert systems (ES) represent the AI techniques.
The ANN is composed of many nonlinear computational elements called neurons
operating in parallel and linked with each other through connections characterized
by weighting factors. The ANN has been applied to a wide range of problems in
different disciplines in power systems such as dynamic security assessment, identi-
fication of harmonic loads, fault diagnosis, electric load forecasting, and online
determination of ready power reserve [20, 21].
Fuzzy linear programming (FLP) is a traditional multi-objective approach to
combine the objectives under some weighting, which creates a problem of selecting
the proper weighting. In an FLP approach, the objectives are reformulated as fuzzy
constraints, and membership functions are viewed as normalizing factors. The fuzzy
set of feasible solutions is obtained as an intersection of all fuzzy regions of
acceptability defined through fuzzy constraints. The fuzzy optimization-based tech-
niques have been developed to solve some of the power system problems in SG such
as optimal PMU placement and optimal placement of DGs and capacitor banks
[22, 23].
Expert systems (ES) are systems that are based on expert knowledge, on any
subject, in order to emulate human expertise in the specific field. To obtain this
knowledge, the knowledge engineers, also called software engineers, need to
develop methodologies for intelligent systems. Despite its earlier high hopes, ES
technology has found application only in areas where information can be reduced to
a set of computational rules, such as insurance underwriting or some aspects of
securities trading [24].
Many difficulties such as multi-modality, dimensionality, and differentiability are
associated with the optimization of large-scale problems. Traditional techniques
such as steepest decent, LP, and DP generally fail to solve nondeterministic
polynomial-time hard (NP-hard) problems such large-scale problems especially
with nonlinear objective functions. Most of the traditional techniques require gradi-
ent information, and hence it is not possible to solve non-differentiable functions
with the help of such traditional techniques. Moreover, such techniques often fail to
solve optimization problems that have many local optima. To overcome these
problems, there is a need to develop more powerful optimization techniques, and
research is going on to find effective optimization techniques since the last three
decades. Some of the well-known population-based optimization techniques devel-
oped during the last three decades are:
• Genetic algorithm (GA), which works on the principle of the Darwinian theory of
the survival of the fittest and the theory of evolution of the living beings
• Differential evolution (DE) algorithm, which is similar to GA works with spe-
cialized crossover and selection method
• Particle swarm optimization (PSO) algorithm, which works on the principle of
foraging behavior of the swarm of birds](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-23-320.jpg)
![4 1 Introduction
• Seeker optimization algorithm (SOA), which works on the principle of simulating
the act of human search to obtain the optimal solution by a seeker population
• Ant colony optimization (ACO) algorithm, which works on the principle of
foraging behavior of the ant for the food
Genetic algorithm (GA) is an adaptive method which may be used to solve
complex search and optimization problems. They are based on principles inspired
from the genetic and evaluation mechanisms observed in natural systems and
populations of living beings, i.e., natural selection, mutation, and recombination
(crossover) [25]. Individuals are represented by bit strings, i.e., the strings of 0 s and
1 s. Each individual is assigned the merit function, which represents how close to the
solution every individual is. The process of selection of individuals for the repro-
duction is the following: the better (the greater for the maximization or the smaller
for the minimization) merit function an individual has, the more times it is likely to
be selected to reproduce. The crossover operator randomly chooses a position in the
individual and exchanges the subsequences before and after that position between
two individuals to create two new offspring. The mutation operator randomly flips
some of the bits in the individual. The GA has been applied to various areas of SG
such as optimal PMU placement and optimal placement of DGs and capacitor banks.
The most interesting aspect of GAs is that they do not require any prior knowledge or
space limitations, such as smoothness or convexity of the function to be optimized
[26]. However, it becomes time consuming and needs great effort to adjust the many
parameters it has.
Differential evolution (DE) algorithm is an optimization approach that addresses
these requirements. Since its invention by Storn and Price in 1995, DE algorithm,
together with evolution strategies, GA, and evolutionary programming (EP), can be
categorized into a class of population-based, derivative-free methods known as
evolutionary algorithms (EAs). Like nearly all EAs, DE algorithm is a population-
based optimizer that starts the optimization process by sampling the search space at
multiple, randomly chosen initial points (i.e., a population of individual vectors). DE
algorithm is in nature a derivative-free continuous function optimizer, as it encodes
parameters as floating-point numbers and manipulates them with simple arithmetic
operations such as addition, subtraction, and multiplication. DE algorithm generates
new points that are the perturbations/mutations of existing points [27–29]. The DE
algorithm has been applied to various areas of SG such as optimal PMU placement
and optimal placement of DGs and optimal capacitor banks. Like GA, DE algorithm
becomes time consuming.
Particle swarm optimization (PSO) was developed by Kennedy and Eberhart in
1995, based on swarm behavior such as fish and bird schooling in nature. PSO may
have some similarities with GAs, but it is simpler because it does not use mutation/
crossover operators. Instead, it uses the real-number randomness and the global
communication among the swarm particles. In this sense, it is also easier to imple-
ment, as there is no encoding or decoding of the parameters into binary strings as
those in GAs, which can also use real-number strings. This algorithm searches the
space of an objective function by adjusting the trajectories of individual agents,](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-24-320.jpg)
![1.1 General 5
called particles, as these trajectories form piecewise paths in a quasi-stochastic
manner. In addition, the PSO has few parameters to be adjusted [30–32]. Recently,
the PSO has been applied in various fields in SG optimization problems, such as
optimal PMU placement and optimal placement of DGs and capacitor banks.
Seeker optimization algorithm (SOA) is a novel swarm intelligence paradigm for
the real-parameter optimization. The SOA is based on the concept of simulating the
act of humans’ intelligent search with their memory, experience, and uncertainty
reasoning. SOA operates on a set of solutions called search population. The indi-
vidual of this population is called seeker. In order to add a social component for
social sharing of information, a neighborhood is defined for each seeker. After given
start point, search direction, search radius, and trust degree, every seeker moves to a
new position (next solution) based on its social learning, cognitive learning, and
uncertainty reasoning [33–37].
Ant colony optimization (ACO) is a meta-heuristic inspired by the foraging
behavior of ants, in order to find the shortest path from the nest to a food source,
which is first introduced by Marco Dorigo and co-workers in 1991. Ant colonies
exploit a positive feedback mechanism, where they use a form of indirect commu-
nication called stigmergy, which is based on the laying and detection of pheromone
trials. In the ACO, a set of software agents called artificial ants search for good
solutions to a given optimization problem. To apply the ACO algorithm, the
optimization problem is transformed into the problem of finding the best path that
corresponds to the objective function of a problem on a weighted graph. The
artificial ants incrementally build solutions by moving on the graph. The solution
construction process is stochastic and is biased by a pheromone model, that is, a set
of parameters associated with graph components (either nodes or edges) whose
values are modified at runtime by ants [38–41]. Recently, the ACO has been applied
in various areas in SG optimization problems, such as optimal PMU placement and
optimal placement of DGs and capacitor banks.
The SG environment requires the upgrade of tools for sensing, metering, and
measurements at all levels of the grid. These components will provide the data
necessary for monitoring the grid and the power market. Sensing provides outage
detection and response, evaluates the health of equipment and the integrity of the
grid, eliminates meter estimations, provides energy theft protection, and enables
consumer choice, demand-side management (DSM), and various grid monitoring
functions. In this regard, new digital technologies use two-way communications, a
variety of inputs and outputs, the ability to connect and disconnect, and interfaces
with generators, grid operators, and customer portals to enhance power measure-
ment. The integrated sensors will interface with the communication network. Phasor
measurements are a current technology that is a component of most SG designs [42–
48].
Phasor measurement units (PMUs) are designed to measure in real time both
magnitudes and phase angles of the bus voltages and/or the branch currents with
high accuracy, through numerical algorithms implemented in the unit. This device
enables the long-term desire of performing local computations in real time and
solves the problem of measurement time skews through sampling clock](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-25-320.jpg)
![6 1 Introduction
synchronization. PMUs are able to provide immediate state of the buses they are
installed and, if the system parameters are accurately known, calculate in real time
the state of neighboring buses through one simple linear step. As a new class of
measurement, synchronized phasor measurements greatly elevate the availability as
well as the quality of data and information useful to improve system monitoring,
protection, control, and operation of the increasingly stressed power systems.
Though currently the number of PMUs installed in the system is not large enough
to make significant differences, the power industry has acknowledged the impor-
tance of rapidly propagating the use of PMUs into systems threatened by more
frequent blackouts. Many utilities and consultants are endeavoring with research
institutions to search for the practical PMU installation strategies [49–52].
PMU placement algorithms developed for specific applications are not optimal
for other applications. Ultimately, only a PMU placement algorithm for full observ-
ability of the system would be beneficial to most monitoring and control applica-
tions. Though widely deploying PMUs on every bus will allow any possible
application to be implemented, this installation strategy will require a major eco-
nomic undertaking.
The optimal placement of PMUs [53–99] is a discrete optimization scheme to
determine the optimal number and locations of PMUs taking into account the
minimum availability of PMU measuring channels and maximize the measurement
redundancy (MR) while satisfying complete observability constraint in smart power
systems. The optimal placement of PMUs is assessed under normal operating
condition as well as emergency conditions such as any single line or PMU outage.
The PMU measuring channels are reduced using the proposed reduction strategy
(RS) with keeping the complete observability.
Capacitor banks are commonly used in a distribution system to provide reactive
power locally, resulting in reduced maximum kVA demand, improved voltage
profile, reduced line/feeder losses, and decreased payments for the energy. Capacitor
banks are installed close to the load, on the poles, or at the substations. Maximum
benefit can be obtained by installing the shunt capacitor banks at the load. This is not
always practical due to the size of the load, distribution of the load, and voltage level.
In distribution and certain industrial loads, the reactive power requirement to meet
the required power factor is constant. In such applications, fixed capacitor banks are
used. Sometimes such fixed capacitor banks can be switched along with the load. If
the load is constant for the 24-hour period, the capacitor banks can be on without the
need for switching on and off. In high-voltage and feeder applications, the reactive
power support is required during peak load conditions. Therefore, the capacitor
banks are switched on during the peak load and switched off during off-peak load.
The switching schemes keep the reactive power levels more or less constant,
maintain the desired power factor, reduce overvoltage during light load conditions,
and reduce losses at the transformers and feeders [100, 101].
The optimal capacitor placement problem [102–123] is considered as an optimi-
zation problem to determine the optimal locations and sizes of capacitors with an
objective of power loss reduction and voltage profile improvement in distribution](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-26-320.jpg)
![1.1 General 7
systems, while security and operational constraints are satisfied. The fixed and
practical switched capacitors are the available choices to find the optimal solution.
The increasing dependency on the use of electricity has compelled utility suppliers to
maintain a high standard of system reliability. This has increased the importance of
analyzing and preventing voltage deviation that is likely to take place in heavily
loaded systems. Sensitivity analysis using loss sensitivity indices (LSIs) is used to
select the candidate locations for the capacitors to reduce the search space in the
optimization procedure of the optimization algorithm. LSIs are derived depending
on the load flow laws.
The distribution feeder is nonlinear because most loads are assumed to be
constant complex power loads. The approach of the linear system can be modified
to take into account the nonlinear characteristics of the distribution feeder. In this
approach, the backward/forward sweep (BFS) algorithm is one of the most common
ways used for load flow in a distribution system. The BFS algorithm involves mainly
iterative three basic steps based on Kirchhoff’s current law (KCL) and Kirchhoff’s
voltage law (KVL). The three steps are named as the nodal current calculation, the
backward sweep, and the forward sweep, and they are repeated until the convergence
is achieved. In the nodal current calculation, all the current injections at different
buses are determined. The backward sweep is primarily a current or a power flow
summation with possible voltage updates. The forward sweep is primarily a voltage
drop calculation with possible current or power flow updates. This algorithm is
based on the fact that the current at the end of the sublateral is zero, whereas the
voltage at the source node is specified. Therefore, by the application of the three
steps in iterative scheme, a radial distribution feeder can be solved.
Distributed generation (DG), also called on-site generation, dispersed generation,
embedded generation, decentralized generation, decentralized energy, or distributed
energy, generates electricity from many small energy sources. A DG is the use of
small-scale power generation technologies located close to the load being served.
DG stakeholders include energy companies, equipment suppliers, regulators, energy
users, and financial and supporting companies. For some customers, DG can lower
costs, enhance efficiency, improve reliability, reduce emissions, or expand their
energy options. DG may add redundancy that increases grid security even while
powering emergency lighting or other critical systems [124–131].
DG reduces the amount of energy lost in transmitting electricity because the
electricity is generated around the place where it is used, perhaps even in the same
building. This also reduces the size and the number of power lines that must be
constructed. Typical distributed power sources in a feed-in tariff (FIT) scheme have
low maintenance, low pollution, and high efficiencies. In the past, these traits
required dedicated operating engineers and large complex plants to reduce pollution.
However, modern embedded systems can provide these traits with automated
operation and renewables, such as sunlight, wind, and geothermal. This reduces
the size of the power plant that can show a profit. Distributed energy resource (DER)
systems are small-scale power generation technologies used to provide an alternative
to or an enhancement of the traditional electric power system. The usual problem
with distributed generators is their high costs. Combined heating plants and power](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-27-320.jpg)
![generators (cogenerators) are also more expensive per watt than central generators.
They find favor because most buildings already burn fuels, and the cogeneration can
extract more value from the fuel [124–131].
The optimal combination of DGs and capacitor banks [132–148] is considered as
a single- and/or multi-objective optimization problem in distribution systems to
determine the optimal locations and sizes of DGs and capacitor banks in different
combination cases of them to achieve the performance enhancement of distribution
systems. The considered objective functions are reducing the power loss, minimiz-
ing the voltage deviation (VD), and maximizing the voltage stability index (VSI),
while the multi-objective function is simultaneously optimizing all the objectives. In
order to solve these objectives, the optimization algorithm should be used under
security and operational equality and inequality constraints.
1.2 Book Contributions
The main contributions of the book can be summarized as follows:
8 1 Introduction
1. An efficient optimization algorithm, called ACO algorithm, is used to find the
optimal number and locations of PMUs considering measuring channels to make
the power system complete observability at normal and emergency conditions.
2. Proposed RS is used to reduce the PMU measuring channels obtained from the
ACO algorithm with keeping the complete observability at normal and emer-
gency conditions. Therefore, the total cost of PMUs will be reduced.
3. A new developed index is proposed besides other LSIs to select the candidate
locations for the DGs and capacitor banks in the distribution systems to reduce the
search space in the optimization procedure.
4. The ACO algorithm is utilized to find the optimal locations and sizes of DGs and
capacitor banks according to single- and multi-objective functions to enhance the
performance of distribution systems.
5. An efficient BFS algorithm is used for the load flow calculations in distribution
systems.
6. A comparison between the proposed procedure using the ACO algorithm and
other optimization techniques such as DP, fuzzy, GA, and PSO to find the optimal
combination of DGs and capacitor banks is presented.
1.3 Scope of the Book
The book contains seven chapters followed by two appendices for IEEE standard
transmission and distribution test system data as well as two real systems that are the
West Delta Network (WDN) transmission system and East Delta Network (EDN)
radial distribution system. In addition, published papers and references list are
presented. These contents can be summarized as follows:](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-28-320.jpg)
![Chapter 2
Optimization Techniques
2.1 Introduction
Optimization is a scientific discipline that deals with the detection of optimal
solutions for a problem, among alternatives. The optimality of solutions is based
on one or several criteria that are usually problem and user dependent. For example,
a structural engineering problem can admit solutions that primarily adhere to fun-
damental engineering specifications, as well as to the aesthetic and operational
expectations of the designer. Constraints can be posed by the user or the problem
itself, thereby reducing the number of prospective solutions. If a solution fulfills all
constraints, it is called a feasible solution. Among all feasible solutions, the global
optimization problem concerns the detection of the optimal one. However, this is not
always possible or necessary. Indeed, there are cases where suboptimal solutions are
acceptable, depending on their quality compared to the optimal one. This is usually
described as local optimization, although the same term has been also used to
describe local search in a strict vicinity of the search space. The different nature
and mathematical characteristics of optimization problems necessitated the special-
ization of algorithms to specific problem categories that share common properties,
such as nonlinearity, convexity, differentiability, continuity, function evaluation
accuracy, etc. Moreover, the inherent characteristics of each algorithm may render
it more suitable either for local or global optimization problems [30].
Optimization problems can be classified as unconstrained and constrained
optimization problems. Constrained optimization is much harder than unconstrained
optimization, because it is required to find the best point of the objective function. In
addition, the constraints of the problem must be verified. There is no single method
available for solving all optimization problems efficiently. Hence, a number of
optimization methods have been developed for solving different types of optimiza-
tion problems. The optimum-seeking methods are also known as mathematical
programming techniques and are generally studied as a part of operations research.
Operations research is a branch of mathematics concerned with the application of
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
A. A. A. El-Ela et al., Modern Optimization Techniques for Smart Grids,
https://doi.org/10.1007/978-3-030-96025-4_2
11](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-30-320.jpg)
![scientific methods and techniques to decision-making problems and with
establishing the best or optimal solutions [149]. Mathematical programming tech-
niques are useful in finding the minimum of function of several variables under a
prescribed set of constraints. In recent years, the modern optimization methods have
emerged as powerful and popular methods for solving complex engineering optimi-
zation problems. The objective function is accompanied by a domain, i.e., a set of
feasible candidate solutions. The domain is delimited by problem constraints, which
need to be quantified properly and described mathematically using equality and
inequality relations. In the simplest cases, constraints are limited to bounding boxes
of the variables. In harder problems, complex relations among the variables must
hold in the final solution, rendering the minimization procedure rather complicated.
12 2 Optimization Techniques
The main elements of any constrained optimization problem are:
• Variables (also called decision variables): The values of the variables are not
known when you start the problem. The variables usually represent things that
you can adjust or control, where the goal is to find values of the variables that
provide the best value of the objective function.
• Objective function: This mathematical expression combines the variables to
express your goal, which will be required to either maximize or minimize the
objective function.
• Constraints: These mathematical expressions combine the variables to express
limits on the possible solutions.
• Variable bounds: Most of the variables in an optimization problem are permitted
to take on any value between minimum limit and maximum limit.
Optimization techniques consist of static and dynamic techniques for optimiza-
tion, such as linear programming (LP), mixed-integer programming (MIP), dynamic
programming, and so on, for the development of smart grid (SG) optimization and
planning activities. LP, MIP, DP, and Lagrangian relaxation methods are used for
power system and operation issues, but they are limited for use in the SG due to the
static network of the programs they can solve. They work better when computed in
conjunction with decision support (DS) tools and computational tool techniques.
Therefore, the modern optimization techniques such as genetic algorithm (GA),
particle swarm optimization (PSO), and ant colony optimization (ACO) algorithm
are suitable for the applications of SG. Figure 2.1 shows the classification of
optimization techniques [1]. Major optimization subfields can be further distin-
guished based on properties of the objective function, its domain, as well as the
form of the constraints. A simple categorization of optimization problems is sum-
marized in Table 2.1 [30].
There are many techniques for solving optimization problems such as conven-
tional and modern optimization techniques.](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-31-320.jpg)
![9
Classification criterion Special characteristics
14 2 Optimization Techniques
Table 2.1 A categorization of optimization problems based on different criteria
Type of
optimization
problem
Form of the objective
function and/or
constraints
Linear Linear objective function and constraints
Nonlinear Nonlinear objective function and/or constraints
Convex Convex objective function and feasible set
Quadratic Quadratic objective function and linear
constraints
Stochastic Noisy evaluation of the objective function or
probabilistically determined problem variables
and/or parameters
Nature of the search space Discrete Discrete variables of the objective function
Continuous Real variables of the objective function
Mixed integer Both real and integer variables
Nature of the problem Dynamic Time-varying objective function
Multi-objective Multitude of objective functions
2.2 Conventional Optimization Techniques
Many of the existing conventional types of optimization techniques such as linear
programming (LP) and quadratic programming (QP) accept only changes that yield
immediate improvement, which are effective for the optimization problems with
deterministic quadratic objective function that has only one minimum. However, it
often induces local optima rather than global optima and sometimes results in
divergence when minimizing both objective functions at the same time. As a result,
these techniques usually achieve local optima rather than global optima.
2.2.1 Linear Programming (LP) [5, 6]
LP is the most common optimization technique applied for constrained optimization
problems, where the entire objective functions and equality and inequality con-
straints are linear. It is widely used due to many practical problems that can be
formulated as LP problems; there are noniterative, simplex, and efficient methods for
solving optimization problems. The classical LP problem is to obtain the optimal
decision variables that maximize the linear objective function under linear equality
and inequality constraints. Specifically, the general structure of problems solved by
LP is:
Maximize cT
x
Subject to A x b
x 0
=
;
ð2:1Þ](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-33-320.jpg)
![T n
2.2 Conventional Optimization Techniques 15
where x ¼ [x1, . . .. . ., xn ] 2 R is the vector of decision variables to be determined,
c ¼ [c1, . . .. . ., cn] 2 Rn
is the vector of the objective function coefficients,
A ¼ [aij ] 2 Rm n
is the matrix of the constraint coefficients with its elements,
and b ¼ [b1, b2, .., bm]T
2 Rm
is the bounding vector.
Assume that the model is statically linear. The process to achieve the global
optimum uses simplex like techniques, variants of the interior point method, or
integer programming (IP). These methods are applicable to problems involving
linear objective functions and linear constraints. The solution procedure includes
[5, 6]:
1. Initialization step: Introduce slack variables (if needed) and determine initial
point as a corner point solution of the equality constraints.
2. At each iteration, move from the current basic feasible solution to a better,
adjacent basic feasible solution.
3. Determine the entering basic variable: Select the nonbasic variable that, when
increased, would increase the objective at the fastest rate. Determine the leaving
basic variable: Select the basic variable that reaches zero first as the entering basic
variable is increased.
4. Determine the new basic feasible solution.
5. Optimality test and termination criteria: Check if the objective can be increased
by increasing any nonbasic variable by rewriting the objective function in terms
of the nonbasic variables only and then checking the sign of the coefficient of
each nonbasic variable. If all coefficients are nonpositive, this solution is optimal
and stop. Otherwise, go to the iterative.
Applying the LP method to the SG requires improving it to accommodate the
grid’s productivity, adaptivity, and randomness. The traditional linear method is
confined to static problems and thus is insufficient for SG implementation.
2.2.2 Quadratic Programming (QP)
Quadratic programming (QP) [9–17] is a linearly constrained optimization problem
with a quadratic objective function to be minimized subject to linear equality and
inequality constraints. The general quadratic program can be written as:
Maximize C:x þ
1
2
xT
:Q:x
Subject to A x b
x 0
9
=
;
ð2:2Þ
where C is an n-dimensional row vector describing the coefficients of the objective
function and Q 2 Rn n
is a symmetric matrix describing the coefficients of the
quadratic terms. QP problem can be solved by different optimization methods such](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-34-320.jpg)
![as the simplex method [11], interior point method [12], active set method [13], a
branch-and-cut algorithm [14], objective separable method [15], Wolfe’s modified
simplex method [16], and the Karush-Kuhn-Tucker (KKT) conditions [17]. The
KKT conditions extend the ideas of Lagrange multipliers to handle constraints in
addition to equality constraints. These conditions provide a first-order optimality
condition for the problem.
2.2.3 Integer Programming (IP)
Integer programming (IP) is a special case of LP where all or some of the decision
variables are restricted to discrete integer values, for example, where the discrete
values are restricted to zero and one only, that is, yes or no decisions, or binary
decision variables. The general structure of the MIP problem is [7, 8]:
Maximize P x
ð Þ ¼
Pn
j¼1cjxi
Subject to
Pm
i¼1
Pn
j¼1aij xj bi
xj 0, j 2 1, n
f g, and xj is an integer
9
=
;
ð2:3Þ
The branch-and-bound procedure features:
16 2 Optimization Techniques
1. Initialization: Set P*
¼ 1, where P*
is the optimal value of P.
2. Branching: This step involves developing subproblems by fixing the binary
variables at 0 or 1, or choosing the first element in the natural ordering of the
variables as the branching variable.
3. Bounding: For each of the subproblems, a bound can be obtained to determine the
goodness of its best feasible solution. For each new subproblem, obtain its bound
by applying the simplex method to its LP relaxation and use the value of the P for
the resulting optimal solution.
4. Fathoming: If a subproblem has a feasible solution, it should be stored as the first
incumbent (the best feasible solution found so far) for the whole problem along
with its value of P. This value is denoted P*
, which is the current incumbent for P.
5. Optimality test: The iterative procedure stops when no subproblems remain. At
this stage, the current incumbent for P is the optimal solution.
Pure integer or MIP problems pose a great computational challenge. While highly
efficient LP techniques can enumerate the basic LP problem at each possible
combination of the discrete variables (nodes), the problem lies in the astronomically
large number of combinations to be enumerated. If there are N discrete variables, the
total number of combinations becomes 2N
. The branch-and-bound technique for
binary integer or reformulated MIP problems overcomes this challenge by dividing
the overall problem into smaller and smaller subproblems and enumerating them in a
logical sequence. Note that the direct use of these techniques for solving the smart](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-35-320.jpg)
![grid optimization problem will be limited, because they are generally static and are
not designed for handling real-time and dynamic optimization problems.
2.3 Artificial Intelligence (AI) Techniques 17
2.2.4 Dynamic Programming (DP)
The abstract framework for dynamic programming (DP) was first introduced in
[18]. This approach was developed to solve sequential or multistage decision-
making problems. Basically, it solves a multivariable problem by solving a series
of single-variable problems. This is achieved by tandem projection onto the space of
each of the variables. In other words, it projects first onto a subset of the variables,
then onto a subset of these, and so on. It is a candidate optimization technique for
handling time variability and noise in the objective and constrained optimization
problem [18]. The general structure of the DP problem is [19]:
Problem P : p
∶ ¼ opt p x
ð Þ, x 2 X ð2:4Þ
where p is a real-valued function or the objective function on some set X,¼ denotes
definition, opt denotes optimality function, X is the decision space, and x is the
decision variables. The solution procedure for Problem P, defined by Eq. (2.4),
would run as follows [19]:
1. For every x 2 X, solve problem P(x) and determine the value of p(x).
2. Determine the value of p
by optimizing p(x) over x 2 X.
2.3 Artificial Intelligence (AI) Techniques
2.3.1 Artificial Neural Network (ANN)
There are a wide variety of artificial neural networks (ANNs) that are used to model
real neural networks, study behavior, and control in animals and machines, but also
there are ANNs which are used for engineering purposes, such as economic dispatch
and reactive power dispatch [20, 21].
The ANNs were inspired by biological findings relating to the behavior of the
brain as a network of units called neurons. Each neuron receives signals through
synapses that control the effects of the signal on the neuron. The fundamental
building block in an ANN is the mathematical model of a neuron as shown in
Fig. 2.2. The three basic components of the artificial neuron are:
• The synapses or connecting links that provide weights, wi, to the input values, xi,
for i ¼ 1,. . .,m.](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-36-320.jpg)
![X
18 2 Optimization Techniques
Φ(v)
i
y
Output
Activation
Function
i
v
Synaptic
Weights
Input
Signals
m
X
2
X
1
X
1
=
o
X
∑
φ
o
W
1
W
2
W
m
W
Fig. 2.2 An artificial neuron
• An adder that sums the weighted input values to compute the input to the
activation function:
vi ¼ wo þ
m
i¼1
wi xi ð2:5Þ
where wo is called the bias that is a numerical value associated with the neuron. It is
convenient to think of the bias, as the weight for an input xo whose value is always
equal to one, so that:
vi ¼
Xm
i¼0
wi xi ð2:6Þ
• An activation function φ (also called a squashing function) that maps vi to φ(v),
the output value of the neuron.
2.3.2 Fuzzy Linear Programming (FLP) [22, 23]
Fuzzy logic is a superset of conventional logic that has been extended to handle the
concept of partial truth, which are the truth values between “completely true” and
“completely false.” Special membership functions transform uncertainty in data to a
crisp form for analysis. Fuzzy logic control is often used to determine whether the
process variables are within acceptable tolerances. A fuzzy set is a generalization of
ordinary sets that allows assigning a degree of membership for each element to range
from (0, 1) interval. Figure 2.3 shows the simplified block diagram of the fuzzy logic
approach. For optimization problems, it is quite restrictive to require the objective
and constraints to be specified in precise, crisp terms to apply fuzziness in the LP](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-37-320.jpg)
![9
problem. Fuzzy LP (FLP) is applied to solve optimization problems such as eco-
nomic dispatch [22, 23]. Specifically, the standard FLP problem is:
Maximize λ
Subject to λ μ
ð2:7Þ
The fuzzy objective function for maximization or minimization purposes takes
the linear membership function. There are three types of FLP problems, which are:
2.3 Artificial Intelligence (AI) Techniques 19
Fuzzyfication Decision-making Defuzzyfication
Start End
Fig. 2.3 Simplified block diagram of a fuzzy logic approach
μi(CX)
1
0
Zo Z1
(CXλ)i
1
0
bi bi+ti
(AXλ)i
a. Minimization b. Maximization
μi(AX)
λ
λ
AX Z
Fig. 2.4 Membership function with fuzzy resources
(a) Linear programming with fuzzy resources: It can be formulated in LP with LP
objective and constraints with fuzzy resources and can be written as follows:
Maximize e
CX
Subject to A X e
b
X 0
=
;
ð2:8Þ
y e
where the fuzzy inequalit is characterized by the membership functions. The
fuzzy objective and inequality constraint for maximization or minimization purposes
takes the linear membership function as shown in Fig. 2.4. In this figure, the
relationship between the value of an objective function and constraint CXλ and
AXλ and the degree of satisfaction λ is illustrated for the linear membership function,
where Zo and bi + ti represent an aspiration level of the decision-maker.
The fuzzy membership function for minimizing an inequality constraint can be
defined as follows:](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-38-320.jpg)
![22 2 Optimization Techniques
2.3.3 Expert Systems (ES)
ES are used as a method of optimization that relies on heuristic or rule-driven
decision-making. They are sometimes used for fault diagnosis with prescription
for corrective actions. While the expert system/computer application performs a
task that would otherwise be performed by a human, the method is only as reliable as
the designed engineering rule base. Figure 2.7 shows the components of ES [24].
Expert systems have several advantages over human experts, including increased
availability and reliability, lower cost and response time, and increased confidence in
decision-making ability via provision of clear reason for a given answer. Its uses for
the power system include optimal load shedding, resource allocation such as VAR,
discrete control such as series capacitors, and economic dispatch. The operator-
assisted functions in the management of the SG will benefit from the use of this
knowledge-based system or its hybrid with ANN or other variations of computa-
tional/heuristic optimization techniques. Some of the problems can be reformulated
using expert systems, real-time data, and hybrid ANN.
2.4 Modern Optimization Methods
The modern optimization methods (also called nontraditional optimization methods)
have emerged as powerful and popular methods for solving complex engineering
optimization problems in recent years. These methods are simulated annealing,
evolutionary programming (EP), Tabu search (TS), neural network-based methods,
genetic algorithm (GA), differential evolution (DE) algorithm, particle swarm opti-
mization (PSO) technique, seeker optimization algorithm (SOA), and ant colony
optimization (ACO) algorithm. Most of these methods are labeled on certain char-
acteristics and behavior of biological, molecular, swarm of insects, and neurobio-
logical systems. These new heuristic tools have been combined among themselves
and with knowledge elements, as well as with more traditional approaches such as
statistical analysis to solve extremely challenging problems. Five of such modern
techniques will be covered in some details, namely, GA, DE algorithm, PSO
technique, SOA, and ACO algorithm.
User
Expert system
Knowledge base
Inference engine
Facts
Expertise
Fig. 2.7 Fundamental components of an expert system](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-41-320.jpg)
![2.5 Evolutionary Optimization Techniques
2.5.1 Genetic Algorithm (GA) [25, 26]
Genetic algorithm (GA) is an optimization and search technique based on the
principles of genetics and natural selection. GA was invented by John Holland
who developed this idea in his book Adaptation in Natural and Artificial Systems
in the year 1975. Holland proposed GA as a heuristic method based on “survival of
the fittest.” GA was discovered as a useful tool for search and optimization problems.
GA handles a population of possible solutions. Each solution is represented through
a chromosome, which is just an abstract representation. Coding of all the possible
solutions into a chromosome is the first part, but certainly not the most straightfor-
ward one of a GA. A set of reproduction operators has to be determined. Also,
reproduction operators are applied directly on the chromosomes and are used to
perform mutations and recombinations over solutions of the problem. GA is very
different from classical optimization algorithms in:
2.5 Evolutionary Optimization Techniques 23
• Use of the encoding of the parameters, not the parameters themselves.
• Work on a population of points, not a unique one.
• Use the only values of the function to optimize, not their derived function or other
auxiliary knowledge.
• Use probabilistic transition function, not the determinist ones.
The steps of the GA can be represented as the flowchart shown in Fig. 2.8.
Yes
Initialization
Generate Initial Population
Roulette wheel selection [25,26]
Mutation [25,26]
Check the problem constraints
Crossover [25,26]
Stopping criterion
Get the optimal solution
Evaluation Function
Offspring chromosome evaluation [25,26]
Elitism [25,26]
No
Fig. 2.8 Flowchart for a simple GA](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-42-320.jpg)
![24 2 Optimization Techniques
2.6 Differential Evolution (DE) Algorithm
In 1995, Storn and Price proposed a new floating-point [27] encoded evolutionary
algorithm for global optimization and named it differential evolution (DE) owing to
a special kind of differential operator, which they invoked to create new offspring
from parent chromosomes instead of classical crossover or mutation. The DE
algorithm differs from GA with respect to the mechanics of mutation, crossover,
and selection performed. GA relies on crossover, while DE relies on mutation
operation. In GA, the mutation takes place randomly, whereas in DE, it takes
place by some rule.
2.6.1 Standard DE Algorithm
In DE algorithm, solutions are represented as chromosomes based on floating-point
numbers. In the mutation process of this algorithm, the weighted difference between
two randomly selected population members is added to a third member to generate a
mutated solution followed by a crossover operator to combine the mutated solution
with the target solution so as to generate a trial solution. Then, a selection operator is
applied to compare the fitness function value of both competing solutions, namely,
target and trial solutions, to determine who can survive for the next generation. The
basic DE algorithm consists of four steps, namely, initialization of population,
mutation, crossover, and selection, as in Fig. 2.9 [28, 29].
Fig. 2.9 DE cycle of stages
Initialization of Chromosomes
Mutation Differential Operator
Crossover
Selection](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-43-320.jpg)
![•
2.7 Particle Swarm Optimization (PSO) Technique
Particle swarm optimization (PSO) is a population-based stochastic optimization
technique developed by Kennedy and Eberhart in 1995 [30, 31], discovered through
simplified social model simulation. It stimulates the behaviors of bird flocking or fish
schooling involving the scenario of a group of birds or fishes randomly looking for
food in an area. PSO is motivated from this scenario and is developed to solve
complex optimization problems. PSO shares many similarities with evolutionary
computation techniques such as GA. The system is initialized with a population of
random solutions and searches for optima by updating generations. However, unlike
GA, PSO has no evolution operators such as crossover and mutation [25]. In PSO,
the potential solutions called particles fly through the problem space by following
the current optimum particles.
This is performed by particles in multidimensional space that have a position and
a velocity. These particles are flying through hyperspace and have two essential
reasoning capabilities: their memory of their own best position and knowledge of the
swarm’s best, “best” simply meaning the position with the smallest objective value.
Members of a swarm communicate good positions to each other and adjust their own
position and velocity based on these good positions. There are two main ways to
do this:
2.7 Particle Swarm Optimization (PSO) Technique 25
• A global best that is known to all and immediately updated when a new best
position is found by any particle in the swarm
“Neighborhood” bests where each particle only immediately communicates with
a subset of the swarm about positions
2.7.1 PSO Mathematical Model
PSO technique is based on the social behavior of bird flocking or fish schooling for
searching the food. Each particle keeps track of its coordinates in hyperspace, which
are associated with the best solution (fitness) it has achieved so far. The value of the
fitness is called personal best “pbest” and stored in the particle memory. Another
best value is also tracked, the global version of the particle swarm optimizer keeps
track of the overall best value, and its location is obtained thus far by any particle in
the population; this is called global best “gbest.”
The modified velocity of each agent can be calculated using the current velocity,
and the distance between pbest and gbest positions with respect to current position is
shown below:
vkþ1
id ¼ wi:vk
id þ c1:r1: pbestid xk
id
þ c2:r2: gbest xk
id
i
¼ 1, 2, . . . , n d ¼ 1, 2, . . . , m ð2:15Þ](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-44-320.jpg)
![where n is the number of particles in a population and m is the number of members in
26 2 Optimization Techniques
a particle (number of variables). Using the above equation, a certain velocity that
gradually gets close to pbest and gbest can be calculated. The current position
(searching point in the search space) can be modified by the following equation:
xkþ1
id ¼ xk
id þ vkþ1
id ð2:16Þ
Figure 2.10 shows the above concept of modification of searching points. Dis-
crete variables c1 and c2 can be handled in Eqs. (2.15) and (2.16) with little
modification. Discrete numbers can be used to express the current position and
velocity. If a discrete random number is used in Eq. (2.15) and the whole calculation
of the right-hand side (RHS) of Eq. (2.15) is discretized to the existing discrete
number, both continuous and discrete numbers can be handled in the algorithm with
no inconsistency.
Suitable selection of inertia weight w in Eq. (2.15) provides a balance between
global and local explorations, thus requiring less iteration on average to find a
sufficiently optimal solution. As originally developed, the inertia weight w often
decreases linearly from wmax to wmin during a run. In general, the inertia weight w is
set according to the following equation:
w ¼ wmax
wmax wmin
kmax
k ð2:17Þ
where kmax is the maximum number of iterations (generations) and k is the current
number of iterations.
Despite its features, a general problem with the PSO, similar to other optimization
algorithms that are not exhaustive methods, such as the brute-force search, is that of
becoming trapped in a local optimum, or suboptimal solution, such that it may work
well on one problem but yet fail on another problem. In general, the main drawbacks
of PSO can be summarized as follows [32]:
Fig. 2.10 Concept of
modification of searching
point in two dimensions
xi
k
xi
k+1
vi
k
vpbest
vgbest
X-Axis
Y-Axis](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-45-320.jpg)
![2.8 Seeker Optimization Algorithm (SOA)
Seeker optimization algorithm (SOA) was originally proposed by Dai et al.
[33]. SOA is a relatively new intelligent algorithm that may be used to find optimal
(or near optimal) solutions to numerical and qualitative problems. This method is a
population-based heuristic stochastic search algorithm based on simulating the act of
human search to obtain the optimal solution by a seeker population. In this algo-
rithm, the next position of seekers is obtained from their current positions based on
their historical and social experiences. The seekers exchange their information with
those defined in their neighbors to negotiate the quickest way to reach the best
solution [34, 35].
2.8.1 SOA
28 2 Optimization Techniques
(i) Implementation of SOA
In SOA, for each seeker i, the position update on each variable j is given by the
following equation [35, 36]:
xij t þ 1
ð Þ ¼ xij t
ð Þ þ αij t
ð Þ dij t
ð Þ ð2:18Þ
where xij (t + 1) and xij (t) are the positions of seeker i on the variable j at time steps
t + 1 and t, respectively. dij (t) and αij (t) are search direction and step length of seeker
i on the variable j at time step t, where αij (t) 0 and dij (t) 2 {21, 0, 1}. Here,
i represents the population number and j the variable number to be optimized. In
Eq. (2.18), if dij(t) ¼ 1, the ith seeker goes to the positive orientation of the
coordinate axis on dimension j. If dij(t) ¼ 21, the seeker goes to the negative
orientation, and if dij(t) ¼ 0, the seeker stays at the current position.
Moreover, seekers in the same subpopulation are searching for the optimal
solution using their own information. In order to avoid the convergence of sub-
populations trapping into local optima, the position of the worst seeker of each
subpopulation is combined with the best one in each of the other subpopulations
using the binomial crossover operator as follows:
xknj,worst ¼
xlj,best if Rj 0:5
xknj,worst otherwise
ð2:19Þ
xknj,worst i
where Rj is a uniformly random real number within [0, 1], s denoted as the
jth variable of the nth worst position in the kth subpopulation, and xlj, best is the jth
variable of the best position in the lth subpopulation. Here, n,k,l ¼ 1, 2,. . ., K 2
1 and k 6¼ l. Thus, the diversity of population is increased by sharing good
information among subpopulations.](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-47-320.jpg)
![d t sign g t x t 2 21
d t sign l t x t ð2 22
ð
2.8 Seeker Optimization Algorithm (SOA) 29
(ii) Search Direction
In SOA, each seeker selects his search direction based on several empirical
gradients (EGs) by comparing the current or historical positions of himself or his
neighbors. For seeker i, the empirical directions involved are [35, 36]:
di,ego t
ð Þ ¼ sign pi,best t
ð Þ xi t
ð Þ
ð2:20Þ
i,alt1ð Þ ¼ i,bestð Þ ið Þ ð : Þ
i,alt2ð Þ ¼ i,bestð Þ ið Þ
ð Þ : Þ
where di,ego(t) is the egotistic direction, di,alt1(t) and di,alt2(t) are the altruistic direc-
tions, and pi,best(t), gi,best(t), and li,best(t) represent the personal historical best posi-
tion, neighbors’ historical position, and current best position, respectively. The
function sign() is a signum function on each variable of the input vector. In addition,
each seeker i as an agent enjoys the properties of pro-activeness and exhibits goal-
directed behavior, which means he may change his search direction in advance
according to his past behavior. This behavior is modeled as an empirical direction
called pro-activeness direction given as:
di,pro t
ð Þ ¼ sign xi t1
ð Þ xi t2
ð Þ
ð Þ 2:23Þ
where t1,t2 2 {t, t-1, t-2} and xi(t1) is better than xi(t2). Every variable j of di(t) is
selected applying the following proportional selection rule as:
dij ¼
0 if rj p
0
ð Þ
j
þ1 if p
0
ð Þ
j rj p
0
ð Þ
j þ p
þ1
ð Þ
j
1 if p
0
ð Þ
j þ p
þ1
ð Þ
j rj 1
8
:
ð2:24Þ
where rj is a uniform random number in [0, 1] and pj
(m)
(m 2{0,+1,-1}) is the
percentage of the number of m from the set {dij,ego(t), dij,alt1(t), dij,alt2(t), dij,pro(t)} on
each dimension j of all the four empirical directions, that is:
pm
j ¼
the number of m
4
ð2:25Þ
(iii) Step Length
In SOA, a fuzzy system is adopted to represent the understanding and linguistic
behavioral pattern of human searching tendency. The fitness values of all the seekers
are sorted in descending manner and turned into the sequence numbers from 1 to S as
the inputs of fuzzy reasoning, where S is the size of population. This design renders a
fuzzy system to be applicable to a wide range of optimization problems. The
calculation expression of step length is presented as [37]:](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-48-320.jpg)
![S I
ð
ð
30 2 Optimization Techniques
μmin
μ
ij
μ
α
3δ
2δ
δ
0
-δ
-2δ
-3δ ij
α
Bell function curve
Fig. 2.12 The action part of the fuzzy reasoning
μi ¼ μmax
i
S 1
μmax μmin
ð Þ 2:26Þ
where Ii is the sequence number of xi(t) after sorting the fitness values and μmax is the
maximum membership degree value which is equal to or a little less than 1. The Bell
membership function μ x
ð Þ ¼ ex2
=2δ2
is well utilized in the action part of fuzzy
reasoning as shown in Fig. 2.12 [37], and the parameter δ in the function is
determined as follows:
δ ¼ ω abs xbest xrand
ð Þ 2:27Þ
where the symbol abs() produces an absolute value in response to input vector and ω
is used to decrease the step length with time step increasing so as to gradually
improve the search precision. The xbest and xrand are the best seeker and a randomly
selected seeker, respectively, from the same subpopulation to which the ith seeker
belongs, and xrand is different from xbest. The step length αij(t) for every variable j is
given as:
αij t
ð Þ ¼ δj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ln rand μi, 1
ð Þ
ð Þ
p
ð2:28Þ
where rand(μi,1) returns a uniformly random real number within [μi,1].
The steps of the SOA can be represented as in the flowchart shown in Fig. 2.13.
2.9 Ant Colony Optimization (ACO) Algorithm
Ant colony optimization (ACO) algorithm is used for solving the combinatorial
optimization problems, which can be regarded as a paradigm for all ant colony
search algorithms that are inspired by the foraging behavior of the social insects,
especially the ants. The first member of the ACO class of algorithms, called the ant
system, was initially presented by Colorni, Dorigo, and Maniezzo [38, 39]. The main
underlying idea, loosely inspired by the behavior of real ants, is that of a parallel](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-49-320.jpg)
![search over several constructive computational threads based on local problem data
and on a dynamic memory structure containing information on the quality of the
previously obtained result. The collective behavior emerging from the interaction of
the different search threads has proved effective in solving combinatorial optimiza-
tion (CO) problems. The main elements of a biological stigmergic system are:
32 2 Optimization Techniques
• The insect as the acting individual
• The pheromone as an information carrier, used to create a dissipation field
• The environment as a display and distribution mechanism for information
Real ants are capable of finding the shortest path from a food source to the nest
without using visual cues. Also, they are capable of adapting to changes in the
environment, for example, finding a new shortest path once the old one is no longer
feasible due to a new obstacle. The characteristics of ant colony optimization are as
follows [38, 39]:
1. Natural algorithm: Since it is based on the behavior of real ants in establishing
paths from their colony to the source of food and back.
2. Parallel and distributed: Since it concerns a population of agents moving simul-
taneously, independently, and without a supervisor.
3. Cooperative: Since each agent chooses a path on the basis of the information,
pheromone trails laid by the other agents, which have previously selected the
same path. This cooperative behavior is also autocatalytic, i.e., it provides a
positive feedback, since the probability of choosing a path increases with the
number of agents that previously chose that path.
4. Versatile: That it can be applied to similar versions of the same problem; e.g.,
there is a straightforward extension from the traveling salesman problem (TSP) to
the asymmetric traveling salesman problem (ATSP).
5. Robust: That it can be applied with minimal changes to other combinatorial
optimization problems such as quadratic assignment problem (QAP) and the
jobshop scheduling problem (JSP).
2.9.1 Description of Real Ants
The ACO algorithm is based on the behavior of real ants that are members of a
family of social insects. However, a group of explorer ants leave the colony for
finding the food source in random directions where they marked their routes by
laying a chemical substance on the ground. Other ants are attracted to the route that
has the largest amount of pheromone. Hence, they are found the shortest route
between the nest and food source by indirect communication media called phero-
mone that laid on the ground as a guide for another ants. Figure 2.14 shows how the
real ants can find the shortest path between nest or colony and food source. In
Fig. 2.14a, there are no obstacles between nest and food source. However, the
shortest route is the straight line. If an obstacle is located on the route of ants to](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-51-320.jpg)
![become two routes around the obstacle, some of the ants choose the left side around
the obstacle and the other will choose the right side as shown in Fig. 2.14b. The
pheromone laid on the left side will be concentrated than the right side of the obstacle
because ants in the shortest path take minimum time in leaving and returning for nest
where they move in the same speed. Therefore, they will be laid a largest amount of
pheromone than other ants on the other route, while the other ants are attracted to the
shortest route. Hence, all ants in the colony will take the shortest route around the
obstacle as shown in Fig. 2.14c.
2.9.2 Comparison Between Artificial and Real Ant Systems
Artificial ants are characterized as agents that imitate the behavior of real ants for
obtaining the optimal solutions according to the problem’s constraints. However, it
should be noted that an artificial ant colony system (ACS) has some differences in
comparison with a real ACS [40], as follows:
2.9 Ant Colony Optimization (ACO) Algorithm 33
Fig. 2.14 Illustration of real ants behavior. (a) Real ants follow a path between the nest and a food
source. (b) An obstacle appears on the path; the pheromone is deposited more quickly on the shorter
path. (c) All ants have chosen the shorter path
1. Artificial ants have memory.
2. They are not completely blind.
3. They live in an environment where time is discrete.
On the other hand, an artificial ACS has several characteristics adapted from
real ACS:
• Artificial ants have a probabilistic preference for paths with a larger amount of
pheromone.
• Shorter paths tend to have larger rates of growth in their amount of pheromone.
• The ants use an indirect communication system based on the amount of phero-
mone deposited in each path.](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-52-320.jpg)
![τ t 1 1 ρ τ t εΔτ t 2:31
34 2 Optimization Techniques
2.9.3 ACO Mathematical Model
A random amount of pheromone is deposited in each route after each ant completes
its tour. Other ants attract to the shortest route according to the probabilistic
transition rule that depends on the amount of pheromone deposited and a heuristic
guide function. The heuristic guide function is the inverse of the distance between
the beginning and ending of each route. The probabilistic transition rule of ant k to go
from city i to city j at iteration t can be expressed as in traveling salesman problem
(TSP) [41] as:
Pk
ij t
ð Þ ¼
τij t
ð Þ
α
ηij t
ð Þ
β
P
q τiq t
ð Þ
α
ηiq t
ð Þ
β
; j, q 2 Nk
i ð2:29Þ
If α ¼ 0, the closest cities are more likely to be selected that corresponds to a
classical greedy algorithm. On the contrary, if β ¼ 0, the probability will depend on
the pheromone trial only. These two parameters should be tuned with each other.
After each tour is completed, a local pheromone update is determined by each ant
depending on the route of each ant as in Eq. (2.30). After all ants are attracted to the
shortest route, a global pheromone update is considered to show the influence of the
new addition deposits by other ants that are attracted to the best tour as shown in
Eq. (2.31):
τij t þ 1
ð Þ ¼ 1 ρ
ð Þτij t
ð Þ þ ρτo ð2:30Þ
ij þ
ð Þ ¼
ð Þ ijð Þ þ ijð Þ ð Þ
where τij(t + 1) is the pheromone after one tour or iteration. τo ¼ 1/dij is the
incremental value of pheromone of each ant, while Δτij is the amount of pheromone
for the elite path as:
Δτij t
ð Þ ¼ Q
=dbest
ð2:32Þ
where Q is a large positive constant and dbest is the shortest tour distance found as
in TSP.
2.9.4 ACO Algorithm
The process of ACO algorithm for solving the optimization problems can be
summarized as follows:
Step 1: Initialization
Specify the lower and upper boundaries of each control variable [xi
min
, xi
max
]. Then,
insert the initial values of control variables randomly within their permissible limits
inside the d-search space that has a matrix of dimension d(m n), where n refers to](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-53-320.jpg)
![the number of stages, while m refers to the number of states (ants) in each stage
shown in Fig. 2.15. So, the control variable xi at iteration 0 can be represented as the
vector of:
2.9 Ant Colony Optimization (ACO) Algorithm 35
Destination
Colony
State
1
State
1
State
1
State
1
State
2
State
2
State
2
State
2
State
m-1
State
m-1
State
m-1
State
m-1
State
m
State
m
State
m
State
m
Stage 1 Stage 2 Stage n-1 Stage n
Fig. 2.15 Search space for the optimization problem
x0
i ¼ x0
i1, x0
i2, x0
i3, . . . , x0
in
where xi
min
xi xi
max
. Also, insert the ACO parameters, which are n, m that is
equal to the number of ants, α, β, ρ, and ε. These parameters must be selected
carefully by trial and error in order to obtain the best global solution of the problem.
In addition, initialize the ant’s tours with random values limited by [1, m] with the
same dimension of d-space.
Step 2: Provide first position
Each ant is positioned on the initial state randomly within the reasonable range of
each control variable in a search space with one ant in each control variable in the
length of randomly distributed values.
Step 3: Evaluation
The merit of each initial control variable in the d-search space is found using the
corresponding objective function of each ant in each stage of the problem called
individual evaluation function with the same dimension of d-space. The whole initial
objective functions for each ant are obtained by summing the individual objective
functions with dimension (m 1) as:
F0
i ¼
Pm
i¼1f i þ
Pm
i¼2f i þ
Pm
i¼3f i þ . . .
¼ F0
1, F0
2, F0
3, . . . , F0
m
T
The initial global objective function is the minimum value among the values of
the whole initial objective functions as:](https://image.slidesharecdn.com/23078748-250509202557-ec3f934c/85/Modern-Optimization-Techniques-For-Smart-Grids-Manjusha-Pandey-54-320.jpg)
Modern Optimization Techniques For Smart Grids Manjusha Pandey
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- 5. Adel Ali Abou El-Ela MohamedT. Mouwafi Adel A. Elbaset Modern Optimization Techniques for Smart Grids
- 6. Modern Optimization Techniques for Smart Grids
- 7. Adel Ali Abou El-Ela • Mohamed T. Mouwafi Adel A. Elbaset Modern Optimization Techniques for Smart Grids
- 8. Adel Ali Abou El-Ela Electrical Engineering Department, Faculty of Engineering Menofia University Shebin El-Kom, Egypt Mohamed T. Mouwafi Electrical Engineering Department, Faculty of Engineering Menofia University Shebin El-Kom, Egypt Adel A. Elbaset Electrical Engineering Department Minia University El-Minia, Egypt Department of Electromechanics Engineering Heliopolis University Cairo, Egypt ISBN 978-3-030-96024-7 ISBN 978-3-030-96025-4 (eBook) https://doi.org/10.1007/978-3-030-96025-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 9. Dedicated to our families and students
- 10. Preface Smart grid (SG) is a type of electrical grid that attempts to predict and intelligently respond to the behavior and actions of all electric power users such as suppliers, consumers, and those that do both to efficiently deliver reliable, economic, and sustainable electricity services. At the heart of the future SG, lie two related challenging optimization problems: monitoring of a power system and enhancement of distribution system performance. SG technologies combine power generation and delivery systems with advanced communication systems to help save energy, reduce energy costs, and improve reliability. A few benefits are connected with the con- sumption of renewable energy technologies, including quite low or no greenhouse- gas emissions, making them a key segment in any environmental change moderation methodology. In this book, a multi-stage method is proposed to make the power system complete observability by the optimal placement of phasor measurement units (PMUs) taking into account the minimum availability of PMUs measuring channels. To solve the optimization problem, a two-stage optimal method is introduced with and without considering zero injection buses (ZIBs). In stage 1, the ant colony optimization (ACO) algorithm is used to find the optimal number and locations of PMUs considering measuring channels and maximize the measurement redundancy (MR) at normal operating conditions as well as emergency conditions such as any single line or PMU outage. In stage 2, the reduction strategy (RS) is proposed to reduce the number of PMUs measuring channels with keeping complete observabil- ity. To prove the robustness and capability of the proposed method, the results are compared with other optimization techniques. Simulation results show the capability of the proposed method to find the optimal PMU placement for significant saving in the total cost with more accuracy and efficiency, especially with increase in the power system sizing. This book presents a two-stage procedure to determine the optimal locations and sizes of capacitors with an objective of power loss reduction for improving the voltage profile in radial distribution systems. In the first stage, the loss sensitivity analysis using two loss sensitivity indices (LSIs) is employed to select candidate vii
- 11. locations for the capacitors to reduce the search space in the optimization procedure. The suggested LSIs are based on the following physical quantities: the variation of the active power losses with respect to the load bus voltage at variant nodes, and the variation of the active power losses with respect to the level of reactive power at variant nodes. In the second stage, the ACO algorithm is used to find the optimal locations and sizes of capacitors considering the minimization of total energy loss and total costs of capacitors as objective functions, while the security and operational constraints are fully achieved. The fixed and practical switched capacitors are considered to find the optimal solution. The backward/forward sweep (BFS) algo- rithm is introduced for the load flow calculations. The numerical results are com- pared with other methods to show the capability of the proposed procedure to find the optimal locations and sizes of capacitors for significant saving in the total cost with more accuracy and efficiency, especially with increase in the distribution system sizing. viii Preface In this book, a proposed procedure which consists of a two-stage methodology is proposed to determine the optimal combination of distributed generations (DGs) and capacitor banks with different single and multi-objective functions in radial distri- bution systems. In the first stage, two LSIs are used to select the candidate locations for the DGs and capacitor banks. The suggested LSIs are based on the following physical quantities: the variation of the active power losses with respect to the level of active power at variant nodes, and the variation of the active power losses with respect to the level of reactive power at variant nodes. In second stage, the ACO algorithm is introduced to find the optimal locations and sizes of DGs and capacitor banks according to single and multi-objective functions. The different single- objective functions are power loss reduction, minimization of voltage deviation (VD), and maximization of voltage stability index (VSI), while the multi-objective function is simultaneously optimizing all the objectives. The obtained results are compared with other methods. Simulation results show the capability of the pro- posed procedure to find the optimal solution for significant minimization in the objective functions with more accuracy and efficiency. Shebin El-Kom, Egypt Adel Ali Abou El-Ela Cairo, Egypt Mohamed T. Mouwafi Adel A. Elbaset
- 12. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Book Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Scope of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Optimization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Conventional Optimization Techniques . . . . . . . . . . . . . . . . . . . 14 2.2.1 Linear Programming (LP) [5, 6] . . . . . . . . . . . . . . . . . . . 14 2.2.2 Quadratic Programming (QP) . . . . . . . . . . . . . . . . . . . . . 15 2.2.3 Integer Programming (IP) . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.4 Dynamic Programming (DP) . . . . . . . . . . . . . . . . . . . . . 17 2.3 Artificial Intelligence (AI) Techniques . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Artificial Neural Network (ANN) . . . . . . . . . . . . . . . . . . 17 2.3.2 Fuzzy Linear Programming (FLP) [22, 23] . . . . . . . . . . . 18 2.3.3 Expert Systems (ES) . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Modern Optimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Evolutionary Optimization Techniques . . . . . . . . . . . . . . . . . . . . 23 2.5.1 Genetic Algorithm (GA) [25, 26] . . . . . . . . . . . . . . . . . . 23 2.6 Differential Evolution (DE) Algorithm . . . . . . . . . . . . . . . . . . . . 24 2.6.1 Standard DE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7 Particle Swarm Optimization (PSO) Technique . . . . . . . . . . . . . . 25 2.7.1 PSO Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . 25 2.8 Seeker Optimization Algorithm (SOA) . . . . . . . . . . . . . . . . . . . . 28 2.8.1 SOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 Ant Colony Optimization (ACO) Algorithm . . . . . . . . . . . . . . . . 30 2.9.1 Description of Real Ants . . . . . . . . . . . . . . . . . . . . . . . . 32 ix
- 13. x Contents 2.9.2 Comparison Between Artificial and Real Ant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.9.3 ACO Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . 34 2.9.4 ACO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Smart Grid Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Smart Grid (SG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Definitions of SG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.2 Traditional Grid Versus SG . . . . . . . . . . . . . . . . . . . . . . 41 3.2.3 Benefits of SG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Smart Metering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Evolution of Electricity Metering . . . . . . . . . . . . . . . . . . 44 3.3.2 Comparison Between Conventional and Smart Metering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.3 Examples of Smart Metering . . . . . . . . . . . . . . . . . . . . . 46 3.4 Phasor Measurement Units (PMUs) . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 PMU Hardware Components . . . . . . . . . . . . . . . . . . . . . 47 3.4.2 Applications of PMUs in Power Systems . . . . . . . . . . . . . 49 3.5 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.6 Capacitor Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6.1 Fixed Versus Switched Capacitor Banks . . . . . . . . . . . . . 51 3.6.2 Benefits of Capacitor Banks . . . . . . . . . . . . . . . . . . . . . . 51 3.7 Distributed Generations (DGs) . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.7.1 Definition of DG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.7.2 Types of DGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.7.3 Applications of DGs . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4 Optimal Placement of PMUs in Smart Power Systems . . . . . . . . . . . 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Rules of Observability Based on PMUs . . . . . . . . . . . . . . . . . . . 62 4.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.1 Formulation of Optimal PMU Placement Problem . . . . . . 62 4.3.2 Installation Cost of PMUs . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 Optimal Solution Using Proposed Multistage Method . . . . . . . . . 64 4.4.1 Optimal PMU Placement Using ACO Algorithm . . . . . . . 64 4.4.2 Proposed Reduction Strategy (RS) Rules for PMU Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.1 Test Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.2 Results and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
- 14. Contents xi 5 Optimal Capacitor Placement for Power Loss Reduction and Voltage Profile Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3 Sensitivity Analysis and Loss Sensitivity Indices . . . . . . . . . . . . 112 5.4 Optimal Capacitor Placement Using ACO Algorithm . . . . . . . . . 117 5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.5.1 Test Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.5.2 Results and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6 Optimal Combination of DGs and Capacitor Banks for Performance Enhancement of Distribution Systems . . . . . . . . . . 141 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.2.1 Objective Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.2.2 Multi-objective Function . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2.3 System Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.3 Two Loss Sensitivity Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.4 Placement of Optimal DGs and Capacitor Banks Using ACO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.5.1 Test Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.5.2 Results and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Appendix A. Test Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 A.1. Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 A.1.1. IEEE 14-Bus Test System . . . . . . . . . . . . . . . . . . . . . . . . 183 A.1.2. IEEE 24-Bus Test System . . . . . . . . . . . . . . . . . . . . . . . . 183 A.1.3. IEEE 30-Bus Test System . . . . . . . . . . . . . . . . . . . . . . . . 184 A.1.4. New England (NE) 39-Bus Test System . . . . . . . . . . . . . . 184 A.1.5. IEEE 57-Bus Test System . . . . . . . . . . . . . . . . . . . . . . . . 184 A.1.6. IEEE 118-Bus Test System . . . . . . . . . . . . . . . . . . . . . . . 185 A.1.7. WDN 52-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 A.2. Distribution Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 A.2.1. 10-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 A.2.2. 34-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
- 15. xii Contents A.2.3. 85-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 A.2.4. East Delta Network (EDN) System . . . . . . . . . . . . . . . . . 192 Appendix B. Backward/Forward Sweep (BFS) Algorithm . . . . . . . . . . . 211 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
- 16. ð Þ List of Symbols Symbol Meaning μ The membership functions of the decision variables in all of objectives and constraints within range of [0–1]. λ The satisfaction parameter that to be maximize within range of [0–1] for all objectives and constraints. c1, c2 The weight coefficients for each term of the modified velocity. r1, r2 The random numbers between 0 and 1. pbesti The personal best of particle i. gbest The global best of the group (populations). μi The step length in seeker optimization algorithm (SOA). P k ð Þ ij t The probabilistic transition rule of ant k to go from city i to city j at iteration t. α and β Two parameters that influence the relative weight of pheromone trail and heuristic guide function, respectively. τij The pheromone trail deposited between city i and j. ηij The visibility or sight and equal to the inverse of the distance between city i and j. Nr k The tabu list in memory of ant that recodes the cities are visited to avoid stagnations. ρ The pheromone evaporation constant. ε The elite path weighting constant. dbest The shortest tour distance found as in traveling salesman problem (TSP). Aij The connectivity matrix. Nch/V The total number of voltage-measuring channels in phasor measurement units (PMUs). Nch/I The total number of current measuring channels in PMUs. Nb The total number of system buses. NZIBs The total number of zero injection buses (ZIBs) in the system. xiii
- 17. xiv List of Symbols KP The annual cost per unit of power loss ($/kW-year). KC The total capacitor banks purchase and installation cost. PLoss Total The total power loss. QC Total The total capacitor banks reactive power. PLossi The power loss in line i. NC The optimal number of capacitor banks placement. NC max The maximum number of possible locations of capacitor banks. VL The voltage magnitude at each load bus. VL min and VL max The minimum and maximum limits of load bus voltage, respectively. PFk The power flow in line k. PFk max The maximum power flow in line k. pfoverall The overall power factor at substation. pf min The minimum limit of overall power factor at substation. QCj The reactive power injection at location j. QCj min and QCj max The minimum and maximum limits of reactive power injection at location j, respectively. PL Total The total active power load in kW. QL Total The total reactive power load in kvar. Peff/q and Qeff/q The total effective active and reactive power loads beyond the node q, respectively. Vp The voltage magnitude at nodes p. δp The voltage angle at nodes p. Rk and Xk The resistance and reactance of branch k, respectively. Si The specified power injection at node i. IL (k) The current flow in branch L at iteration k. ZL The impedance of branch L. ΔVi (k) The voltage mismatch at bus i at iteration k. Vrated The rated or flat voltage at each bus that equals 1 p.u. NDG The optimal number of distributed generations (DGs). NDG max The maximum number of possible locations of DGs. PDGj and QDGj The active and reactive power injections by DGs at bus j, respectively. QCj The reactive power injection by capacitor banks at bus j. PDG Total The total active power injection by DGs. QDG Total The total reactive power injection by DGs.
- 18. Abbreviations ACO Ant colony optimization AI Artificial intelligence AMR Automatic meter reading ANN Artificial neural network APSO Accelerated particle swarm optimization APSO Advanced particle swarm optimization B and B Branch and bound BFA Bacteria foraging algorithm BFOA Bacterial foraging optimization algorithm BFS Backward/forward sweep BPSO Binary particle swarm optimization CBs Circuit breakers CGA Cellular genetic algorithm CHP Combined heat and power CLA Cellular learning automata CSA Cuckoo search algorithm DA Differential evolution DeFS Depth first search DE-PS Differential evolution and pattern search DER Distributed energy resources DG Distributed generations DNOs Distribution network operators DOE Department of energy DP Dynamic programming DPS Dynamic programming search DS Decision support DSM Demand side management DSP Digital signal processing DSTATCOM Static compensator EAs Evolutionary algorithms xv
- 19. xvi Abbreviations EDN East Delta Network EHV Extra-high voltage EILPM Equivalent integer linear programming method EPRI Electric Power Research Institute ES Expert systems FA Firefly algorithm FIT Feed-in tariff FLP Fuzzy linear programming Fuzzy-EP Fuzzy adaptation of evolutionary programming GA Genetic algorithm GEM Grenade explosion method GPS Global Positioning System GUI Graphical user interface HAN Home area network HBMO Honey-bee mating optimization HDPSO Hybrid discrete particle swarm optimization HV High voltage IA Improved analytical ICA Imperialist competitive algorithm IGA Immunity genetic algorithm ILP Integer linear programming ILS Iterated local search INSGA-II Improved nondominated sorting genetic algorithm–II IP Integer programming IPP Independent power producer IQP Integer quadratic programming KCL Kirchhoff’s current law KKT Karush-Kuhn-Tucker KVL Kirchhoff’s voltage law LP Linear programming LSFs Loss sensitivity factors LSIs Loss sensitivity indices MBFO Modified bacterial foraging optimization MI Mutual information MINLP Mixed integer nonlinear programming MIP Mixed integer-programming MLI Maximum loadability index MO-BBO Multi-objective biogeography based optimization MOI Multi-objective index MR Measurement redundancy MST Minimum spanning tree NE New-England NIST National Institute of Standards and Technology NPV Net present value
- 20. Abbreviations xvii NSDE Non-dominated sorting differential evolution OPF Optimal power flow OPP Optimal phasor measurement units placement ORPD Optimal reactive power dispatch PC Personal computer PDCs Phasor data concentrators PGSA Plant growth simulation algorithm PLC Power line carrier PLI Power loss index PMU Phasor measurement unit PPA Pagerank placement algorithm PSO Particle swarm optimization PURPA Public utility regulatory policies act PV Photovoltaic QP Quadratic programming RCOs Random component outages RER Renewable energy resources RS Reduction strategy RTUs Remote terminal units SA Simulated annealing SAHSA Self-adaptive harmony search algorithm SCADA Supervisory control and data acquisition SCIG Squirrel cage induction generator SCRO Simplified chemical reaction optimization SE State estimation SEA Sequential elimination algorithm SG Smart grid SOA Seeker optimization algorithm TLBO Teaching learning-based optimization UEN Unified Egyptian Network VCI Voltage collapse index VD Voltage deviation VSI Voltage stability index WAMS Wide area measurement system WDN West Delta Network WLS Weighted least squares ZIBs Zero injection buses
- 21. Chapter 1 Introduction 1.1 General The term “smart grid (SG)” defines a self-healing network equipped with dynamic optimization techniques that use real-time measurements to minimize network losses, maintain voltage levels, increase reliability, and improve asset management. The operational data collected by the SG and its subsystems will allow system operators to rapidly identify the best strategy to secure against attacks, vulnerability, and so on, caused by various contingencies. However, the SG first depends upon identifying and researching key performance measures, designing and testing appro- priate tools, and developing the proper education curriculum to equip current and future personnel with the knowledge and skills for deployment of this highly advanced system [1]. The transmission system that interconnects all major substation and load centers is the backbone of an integrated power system. Efficiency and reliability at an affordable cost continue to be the ultimate aims of transmission planners and operators. Transmission lines must tolerate dynamic changes in load and contin- gency without service disruptions. Strategies to achieve SG performance at the transmission level include the design of analytical tools and advanced technology with intelligence for performance analysis such as dynamic optimal power flow, robust state estimation, real-time stability assessment, and reliability and market simulation tools. Real-time monitoring based on phasor measurement unit (PMU), state estimators sensors, and communication technologies are the transmission sub- system’s intelligent enabling tools for developing smart transmission functionality. The distribution system is at the heart of the emerging smart electrical system, and the role of distribution system operators is on the verge of a real revolution. Operators are in fact the ones who will have to master technical complexity, limit the sudden rise in costs, and ensure the quality of service expected by customers. The distribution grid, which ensures continuity of supply, is a vital infrastructure for our economies to function effectively. This has been true for a long time and is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. A. A. El-Ela et al., Modern Optimization Techniques for Smart Grids, https://doi.org/10.1007/978-3-030-96025-4_1 1
- 22. increasingly becoming the case. The distribution network is one of the most complex industrial facilities currently in use [2]. 2 1 Introduction In fact, until now, electricity was very predictable, going from large power plants to large electricity transmission grids, then to the distribution grids to supply customers. The direction of the flow was predictable. With the development of distributed generation (DG), a significant proportion of generation is connected to the distribution grid. Furthermore, the means of DG are mainly wind and solar power stations, whose generation varies depending on the wind and sun. It is therefore possible, for example, to have a generation greater than the local consumption in the middle of the afternoon in a housing development where there are many solar panels, at a time when people are at work or school, and when the panels produce at their maximum rate. Therefore, the electricity temporarily flows that will return a higher voltage to the grids. At night, the flow returns to its usual direction. There are therefore varying directions of flow [2]. The term “optimization” refers to a choice that must be made from several possible solutions while respecting a finite number of constraints. A human desire for perfection finds its expression in the optimization theory, which teaches how to describe and fulfill an optimum. The optimization tries to improve system perfor- mances in the direction of the optimal point or points. The optimization can be defined as a part of the applied or numerical mathematics or a method for system design by computer in accordance with either one stress theoretical aspect (existence of the optimum solution conditions) or the practical aspect (procedures for obtaining the optimum solution) [3]. The analytical solution [4] used to solve an optimization problem depends on the form of the criterion and constraint functions. The simplest situation to be considered is the unconstrained optimization problem. In such a problem, no constraints are imposed on the decision variable, and different calculus can be used to analyze them. Another relatively simple form of the general optimization problem is the case in which all the constraints of the problem can be expressed as equality relationships. However, conventional optimization techniques have been developed that can efficiently solve several classes of problems with these inequality restrictions. Linear programming (LP) [5, 6] problems constitute the most important class for which efficient solution techniques have been developed. LP problem is specified by a linear, multivariable function which is to be optimized (maximized or minimized) subject to a number of linear constraints. In 1963, the mathematician Dantzig developed an algorithm called the simplex method to solve problems of this type. The original simplex method has been modified into an efficient algorithm to solve large LP problems by computer. Other classes of problems include integer program- ming (IP) [7, 8] problems, in which at least some of the variables must assume only integer values, and quadratic programming (QP) problems [9–17], in which the objective relationship is a quadratic function of the decision variables, as well as dynamic programming (DP) [18, 19], in which decision variables are to be made in sequential or multistage decision-making problems.
- 23. 1.1 General 3 The artificial intelligence (AI) is the science and engineering of making intelli- gent machines, especially intelligent computer programs. It is related to the similar task of using computers to understand human intelligence. Intelligence is the com- putational part of the ability to achieve goals in the world. The artificial neural network (ANN), fuzzy logic, and expert systems (ES) represent the AI techniques. The ANN is composed of many nonlinear computational elements called neurons operating in parallel and linked with each other through connections characterized by weighting factors. The ANN has been applied to a wide range of problems in different disciplines in power systems such as dynamic security assessment, identi- fication of harmonic loads, fault diagnosis, electric load forecasting, and online determination of ready power reserve [20, 21]. Fuzzy linear programming (FLP) is a traditional multi-objective approach to combine the objectives under some weighting, which creates a problem of selecting the proper weighting. In an FLP approach, the objectives are reformulated as fuzzy constraints, and membership functions are viewed as normalizing factors. The fuzzy set of feasible solutions is obtained as an intersection of all fuzzy regions of acceptability defined through fuzzy constraints. The fuzzy optimization-based tech- niques have been developed to solve some of the power system problems in SG such as optimal PMU placement and optimal placement of DGs and capacitor banks [22, 23]. Expert systems (ES) are systems that are based on expert knowledge, on any subject, in order to emulate human expertise in the specific field. To obtain this knowledge, the knowledge engineers, also called software engineers, need to develop methodologies for intelligent systems. Despite its earlier high hopes, ES technology has found application only in areas where information can be reduced to a set of computational rules, such as insurance underwriting or some aspects of securities trading [24]. Many difficulties such as multi-modality, dimensionality, and differentiability are associated with the optimization of large-scale problems. Traditional techniques such as steepest decent, LP, and DP generally fail to solve nondeterministic polynomial-time hard (NP-hard) problems such large-scale problems especially with nonlinear objective functions. Most of the traditional techniques require gradi- ent information, and hence it is not possible to solve non-differentiable functions with the help of such traditional techniques. Moreover, such techniques often fail to solve optimization problems that have many local optima. To overcome these problems, there is a need to develop more powerful optimization techniques, and research is going on to find effective optimization techniques since the last three decades. Some of the well-known population-based optimization techniques devel- oped during the last three decades are: • Genetic algorithm (GA), which works on the principle of the Darwinian theory of the survival of the fittest and the theory of evolution of the living beings • Differential evolution (DE) algorithm, which is similar to GA works with spe- cialized crossover and selection method • Particle swarm optimization (PSO) algorithm, which works on the principle of foraging behavior of the swarm of birds
- 24. 4 1 Introduction • Seeker optimization algorithm (SOA), which works on the principle of simulating the act of human search to obtain the optimal solution by a seeker population • Ant colony optimization (ACO) algorithm, which works on the principle of foraging behavior of the ant for the food Genetic algorithm (GA) is an adaptive method which may be used to solve complex search and optimization problems. They are based on principles inspired from the genetic and evaluation mechanisms observed in natural systems and populations of living beings, i.e., natural selection, mutation, and recombination (crossover) [25]. Individuals are represented by bit strings, i.e., the strings of 0 s and 1 s. Each individual is assigned the merit function, which represents how close to the solution every individual is. The process of selection of individuals for the repro- duction is the following: the better (the greater for the maximization or the smaller for the minimization) merit function an individual has, the more times it is likely to be selected to reproduce. The crossover operator randomly chooses a position in the individual and exchanges the subsequences before and after that position between two individuals to create two new offspring. The mutation operator randomly flips some of the bits in the individual. The GA has been applied to various areas of SG such as optimal PMU placement and optimal placement of DGs and capacitor banks. The most interesting aspect of GAs is that they do not require any prior knowledge or space limitations, such as smoothness or convexity of the function to be optimized [26]. However, it becomes time consuming and needs great effort to adjust the many parameters it has. Differential evolution (DE) algorithm is an optimization approach that addresses these requirements. Since its invention by Storn and Price in 1995, DE algorithm, together with evolution strategies, GA, and evolutionary programming (EP), can be categorized into a class of population-based, derivative-free methods known as evolutionary algorithms (EAs). Like nearly all EAs, DE algorithm is a population- based optimizer that starts the optimization process by sampling the search space at multiple, randomly chosen initial points (i.e., a population of individual vectors). DE algorithm is in nature a derivative-free continuous function optimizer, as it encodes parameters as floating-point numbers and manipulates them with simple arithmetic operations such as addition, subtraction, and multiplication. DE algorithm generates new points that are the perturbations/mutations of existing points [27–29]. The DE algorithm has been applied to various areas of SG such as optimal PMU placement and optimal placement of DGs and optimal capacitor banks. Like GA, DE algorithm becomes time consuming. Particle swarm optimization (PSO) was developed by Kennedy and Eberhart in 1995, based on swarm behavior such as fish and bird schooling in nature. PSO may have some similarities with GAs, but it is simpler because it does not use mutation/ crossover operators. Instead, it uses the real-number randomness and the global communication among the swarm particles. In this sense, it is also easier to imple- ment, as there is no encoding or decoding of the parameters into binary strings as those in GAs, which can also use real-number strings. This algorithm searches the space of an objective function by adjusting the trajectories of individual agents,
- 25. 1.1 General 5 called particles, as these trajectories form piecewise paths in a quasi-stochastic manner. In addition, the PSO has few parameters to be adjusted [30–32]. Recently, the PSO has been applied in various fields in SG optimization problems, such as optimal PMU placement and optimal placement of DGs and capacitor banks. Seeker optimization algorithm (SOA) is a novel swarm intelligence paradigm for the real-parameter optimization. The SOA is based on the concept of simulating the act of humans’ intelligent search with their memory, experience, and uncertainty reasoning. SOA operates on a set of solutions called search population. The indi- vidual of this population is called seeker. In order to add a social component for social sharing of information, a neighborhood is defined for each seeker. After given start point, search direction, search radius, and trust degree, every seeker moves to a new position (next solution) based on its social learning, cognitive learning, and uncertainty reasoning [33–37]. Ant colony optimization (ACO) is a meta-heuristic inspired by the foraging behavior of ants, in order to find the shortest path from the nest to a food source, which is first introduced by Marco Dorigo and co-workers in 1991. Ant colonies exploit a positive feedback mechanism, where they use a form of indirect commu- nication called stigmergy, which is based on the laying and detection of pheromone trials. In the ACO, a set of software agents called artificial ants search for good solutions to a given optimization problem. To apply the ACO algorithm, the optimization problem is transformed into the problem of finding the best path that corresponds to the objective function of a problem on a weighted graph. The artificial ants incrementally build solutions by moving on the graph. The solution construction process is stochastic and is biased by a pheromone model, that is, a set of parameters associated with graph components (either nodes or edges) whose values are modified at runtime by ants [38–41]. Recently, the ACO has been applied in various areas in SG optimization problems, such as optimal PMU placement and optimal placement of DGs and capacitor banks. The SG environment requires the upgrade of tools for sensing, metering, and measurements at all levels of the grid. These components will provide the data necessary for monitoring the grid and the power market. Sensing provides outage detection and response, evaluates the health of equipment and the integrity of the grid, eliminates meter estimations, provides energy theft protection, and enables consumer choice, demand-side management (DSM), and various grid monitoring functions. In this regard, new digital technologies use two-way communications, a variety of inputs and outputs, the ability to connect and disconnect, and interfaces with generators, grid operators, and customer portals to enhance power measure- ment. The integrated sensors will interface with the communication network. Phasor measurements are a current technology that is a component of most SG designs [42– 48]. Phasor measurement units (PMUs) are designed to measure in real time both magnitudes and phase angles of the bus voltages and/or the branch currents with high accuracy, through numerical algorithms implemented in the unit. This device enables the long-term desire of performing local computations in real time and solves the problem of measurement time skews through sampling clock
- 26. 6 1 Introduction synchronization. PMUs are able to provide immediate state of the buses they are installed and, if the system parameters are accurately known, calculate in real time the state of neighboring buses through one simple linear step. As a new class of measurement, synchronized phasor measurements greatly elevate the availability as well as the quality of data and information useful to improve system monitoring, protection, control, and operation of the increasingly stressed power systems. Though currently the number of PMUs installed in the system is not large enough to make significant differences, the power industry has acknowledged the impor- tance of rapidly propagating the use of PMUs into systems threatened by more frequent blackouts. Many utilities and consultants are endeavoring with research institutions to search for the practical PMU installation strategies [49–52]. PMU placement algorithms developed for specific applications are not optimal for other applications. Ultimately, only a PMU placement algorithm for full observ- ability of the system would be beneficial to most monitoring and control applica- tions. Though widely deploying PMUs on every bus will allow any possible application to be implemented, this installation strategy will require a major eco- nomic undertaking. The optimal placement of PMUs [53–99] is a discrete optimization scheme to determine the optimal number and locations of PMUs taking into account the minimum availability of PMU measuring channels and maximize the measurement redundancy (MR) while satisfying complete observability constraint in smart power systems. The optimal placement of PMUs is assessed under normal operating condition as well as emergency conditions such as any single line or PMU outage. The PMU measuring channels are reduced using the proposed reduction strategy (RS) with keeping the complete observability. Capacitor banks are commonly used in a distribution system to provide reactive power locally, resulting in reduced maximum kVA demand, improved voltage profile, reduced line/feeder losses, and decreased payments for the energy. Capacitor banks are installed close to the load, on the poles, or at the substations. Maximum benefit can be obtained by installing the shunt capacitor banks at the load. This is not always practical due to the size of the load, distribution of the load, and voltage level. In distribution and certain industrial loads, the reactive power requirement to meet the required power factor is constant. In such applications, fixed capacitor banks are used. Sometimes such fixed capacitor banks can be switched along with the load. If the load is constant for the 24-hour period, the capacitor banks can be on without the need for switching on and off. In high-voltage and feeder applications, the reactive power support is required during peak load conditions. Therefore, the capacitor banks are switched on during the peak load and switched off during off-peak load. The switching schemes keep the reactive power levels more or less constant, maintain the desired power factor, reduce overvoltage during light load conditions, and reduce losses at the transformers and feeders [100, 101]. The optimal capacitor placement problem [102–123] is considered as an optimi- zation problem to determine the optimal locations and sizes of capacitors with an objective of power loss reduction and voltage profile improvement in distribution
- 27. 1.1 General 7 systems, while security and operational constraints are satisfied. The fixed and practical switched capacitors are the available choices to find the optimal solution. The increasing dependency on the use of electricity has compelled utility suppliers to maintain a high standard of system reliability. This has increased the importance of analyzing and preventing voltage deviation that is likely to take place in heavily loaded systems. Sensitivity analysis using loss sensitivity indices (LSIs) is used to select the candidate locations for the capacitors to reduce the search space in the optimization procedure of the optimization algorithm. LSIs are derived depending on the load flow laws. The distribution feeder is nonlinear because most loads are assumed to be constant complex power loads. The approach of the linear system can be modified to take into account the nonlinear characteristics of the distribution feeder. In this approach, the backward/forward sweep (BFS) algorithm is one of the most common ways used for load flow in a distribution system. The BFS algorithm involves mainly iterative three basic steps based on Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL). The three steps are named as the nodal current calculation, the backward sweep, and the forward sweep, and they are repeated until the convergence is achieved. In the nodal current calculation, all the current injections at different buses are determined. The backward sweep is primarily a current or a power flow summation with possible voltage updates. The forward sweep is primarily a voltage drop calculation with possible current or power flow updates. This algorithm is based on the fact that the current at the end of the sublateral is zero, whereas the voltage at the source node is specified. Therefore, by the application of the three steps in iterative scheme, a radial distribution feeder can be solved. Distributed generation (DG), also called on-site generation, dispersed generation, embedded generation, decentralized generation, decentralized energy, or distributed energy, generates electricity from many small energy sources. A DG is the use of small-scale power generation technologies located close to the load being served. DG stakeholders include energy companies, equipment suppliers, regulators, energy users, and financial and supporting companies. For some customers, DG can lower costs, enhance efficiency, improve reliability, reduce emissions, or expand their energy options. DG may add redundancy that increases grid security even while powering emergency lighting or other critical systems [124–131]. DG reduces the amount of energy lost in transmitting electricity because the electricity is generated around the place where it is used, perhaps even in the same building. This also reduces the size and the number of power lines that must be constructed. Typical distributed power sources in a feed-in tariff (FIT) scheme have low maintenance, low pollution, and high efficiencies. In the past, these traits required dedicated operating engineers and large complex plants to reduce pollution. However, modern embedded systems can provide these traits with automated operation and renewables, such as sunlight, wind, and geothermal. This reduces the size of the power plant that can show a profit. Distributed energy resource (DER) systems are small-scale power generation technologies used to provide an alternative to or an enhancement of the traditional electric power system. The usual problem with distributed generators is their high costs. Combined heating plants and power
- 28. generators (cogenerators) are also more expensive per watt than central generators. They find favor because most buildings already burn fuels, and the cogeneration can extract more value from the fuel [124–131]. The optimal combination of DGs and capacitor banks [132–148] is considered as a single- and/or multi-objective optimization problem in distribution systems to determine the optimal locations and sizes of DGs and capacitor banks in different combination cases of them to achieve the performance enhancement of distribution systems. The considered objective functions are reducing the power loss, minimiz- ing the voltage deviation (VD), and maximizing the voltage stability index (VSI), while the multi-objective function is simultaneously optimizing all the objectives. In order to solve these objectives, the optimization algorithm should be used under security and operational equality and inequality constraints. 1.2 Book Contributions The main contributions of the book can be summarized as follows: 8 1 Introduction 1. An efficient optimization algorithm, called ACO algorithm, is used to find the optimal number and locations of PMUs considering measuring channels to make the power system complete observability at normal and emergency conditions. 2. Proposed RS is used to reduce the PMU measuring channels obtained from the ACO algorithm with keeping the complete observability at normal and emer- gency conditions. Therefore, the total cost of PMUs will be reduced. 3. A new developed index is proposed besides other LSIs to select the candidate locations for the DGs and capacitor banks in the distribution systems to reduce the search space in the optimization procedure. 4. The ACO algorithm is utilized to find the optimal locations and sizes of DGs and capacitor banks according to single- and multi-objective functions to enhance the performance of distribution systems. 5. An efficient BFS algorithm is used for the load flow calculations in distribution systems. 6. A comparison between the proposed procedure using the ACO algorithm and other optimization techniques such as DP, fuzzy, GA, and PSO to find the optimal combination of DGs and capacitor banks is presented. 1.3 Scope of the Book The book contains seven chapters followed by two appendices for IEEE standard transmission and distribution test system data as well as two real systems that are the West Delta Network (WDN) transmission system and East Delta Network (EDN) radial distribution system. In addition, published papers and references list are presented. These contents can be summarized as follows:
- 29. This chapter presents a brief introduction of the power system problems with 1.3 Scope of the Book 9 introducing proposed solutions for some optimization problems and the optimi- zation techniques. Hence, it summarizes the objectives and contributions of the book. Chapter 2 presents a comparison between the mathematics of conventional optimi- zation techniques (LP, QP, IP, and DP), AI techniques (ANN, FLP, and ES), and modern optimization techniques (GA, DE, PSO, SOA, and ACO). Chapter 3 presents a survey on the SG technologies by introducing the different working definitions of SG, a comparison between traditional grid and SG to show the benefits of SG, and the smart metering devices such as PMUs for monitoring the transmission power systems. In addition, the methods used to enhance the performance of distribution networks are presented such as capacitor banks and DGs. Moreover, a comparison between different methods for the enhancement of SG performance is presented in order to select the best method that will be used in the next chapters. Chapter 4 presents a two-stage method using the ACO algorithm and proposed RS to make the power system complete topological observability by the optimal number and locations of PMUs with a limited number of PMU measuring channels. The optimal results are obtained at normal operating condition as well as emergency conditions such as any single line or PMU outage with and without considering zero injection buses (ZIBs). A comparison between the proposed method and other optimization techniques is presented, which shows the capability of the proposed method to solve the optimal PMU placement problem. Chapter 5 presents a two-stage procedure using two LSIs and the ACO algorithm to determine the optimal locations and sizes of capacitors with minimizing the objective of power loss and improving the voltage profile in distribution systems. In addition, the steps of BFS algorithm for the load flow calculations are presented. A comparison between the proposed procedure and other optimization techniques is presented, which shows the capability of the proposed method to solve the optimal capacitor placement problem. Chapter 6 presents a two-stage procedure using two LSIs and the ACO algorithm to determine the optimal combination of DGs and capacitor banks with different single- and multi-objective functions in radial distribution systems. A comparison between the proposed procedure and other optimization techniques is presented to evaluate the capability of the proposed procedure to solve the placement problem of optimal DGs and capacitor banks. Chapter 7 presents the conclusions of the book and the future work that can be carried out related to this book.
- 30. Chapter 2 Optimization Techniques 2.1 Introduction Optimization is a scientific discipline that deals with the detection of optimal solutions for a problem, among alternatives. The optimality of solutions is based on one or several criteria that are usually problem and user dependent. For example, a structural engineering problem can admit solutions that primarily adhere to fun- damental engineering specifications, as well as to the aesthetic and operational expectations of the designer. Constraints can be posed by the user or the problem itself, thereby reducing the number of prospective solutions. If a solution fulfills all constraints, it is called a feasible solution. Among all feasible solutions, the global optimization problem concerns the detection of the optimal one. However, this is not always possible or necessary. Indeed, there are cases where suboptimal solutions are acceptable, depending on their quality compared to the optimal one. This is usually described as local optimization, although the same term has been also used to describe local search in a strict vicinity of the search space. The different nature and mathematical characteristics of optimization problems necessitated the special- ization of algorithms to specific problem categories that share common properties, such as nonlinearity, convexity, differentiability, continuity, function evaluation accuracy, etc. Moreover, the inherent characteristics of each algorithm may render it more suitable either for local or global optimization problems [30]. Optimization problems can be classified as unconstrained and constrained optimization problems. Constrained optimization is much harder than unconstrained optimization, because it is required to find the best point of the objective function. In addition, the constraints of the problem must be verified. There is no single method available for solving all optimization problems efficiently. Hence, a number of optimization methods have been developed for solving different types of optimiza- tion problems. The optimum-seeking methods are also known as mathematical programming techniques and are generally studied as a part of operations research. Operations research is a branch of mathematics concerned with the application of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. A. A. El-Ela et al., Modern Optimization Techniques for Smart Grids, https://doi.org/10.1007/978-3-030-96025-4_2 11
- 31. scientific methods and techniques to decision-making problems and with establishing the best or optimal solutions [149]. Mathematical programming tech- niques are useful in finding the minimum of function of several variables under a prescribed set of constraints. In recent years, the modern optimization methods have emerged as powerful and popular methods for solving complex engineering optimi- zation problems. The objective function is accompanied by a domain, i.e., a set of feasible candidate solutions. The domain is delimited by problem constraints, which need to be quantified properly and described mathematically using equality and inequality relations. In the simplest cases, constraints are limited to bounding boxes of the variables. In harder problems, complex relations among the variables must hold in the final solution, rendering the minimization procedure rather complicated. 12 2 Optimization Techniques The main elements of any constrained optimization problem are: • Variables (also called decision variables): The values of the variables are not known when you start the problem. The variables usually represent things that you can adjust or control, where the goal is to find values of the variables that provide the best value of the objective function. • Objective function: This mathematical expression combines the variables to express your goal, which will be required to either maximize or minimize the objective function. • Constraints: These mathematical expressions combine the variables to express limits on the possible solutions. • Variable bounds: Most of the variables in an optimization problem are permitted to take on any value between minimum limit and maximum limit. Optimization techniques consist of static and dynamic techniques for optimiza- tion, such as linear programming (LP), mixed-integer programming (MIP), dynamic programming, and so on, for the development of smart grid (SG) optimization and planning activities. LP, MIP, DP, and Lagrangian relaxation methods are used for power system and operation issues, but they are limited for use in the SG due to the static network of the programs they can solve. They work better when computed in conjunction with decision support (DS) tools and computational tool techniques. Therefore, the modern optimization techniques such as genetic algorithm (GA), particle swarm optimization (PSO), and ant colony optimization (ACO) algorithm are suitable for the applications of SG. Figure 2.1 shows the classification of optimization techniques [1]. Major optimization subfields can be further distin- guished based on properties of the objective function, its domain, as well as the form of the constraints. A simple categorization of optimization problems is sum- marized in Table 2.1 [30]. There are many techniques for solving optimization problems such as conven- tional and modern optimization techniques.
- 33. 9 Classification criterion Special characteristics 14 2 Optimization Techniques Table 2.1 A categorization of optimization problems based on different criteria Type of optimization problem Form of the objective function and/or constraints Linear Linear objective function and constraints Nonlinear Nonlinear objective function and/or constraints Convex Convex objective function and feasible set Quadratic Quadratic objective function and linear constraints Stochastic Noisy evaluation of the objective function or probabilistically determined problem variables and/or parameters Nature of the search space Discrete Discrete variables of the objective function Continuous Real variables of the objective function Mixed integer Both real and integer variables Nature of the problem Dynamic Time-varying objective function Multi-objective Multitude of objective functions 2.2 Conventional Optimization Techniques Many of the existing conventional types of optimization techniques such as linear programming (LP) and quadratic programming (QP) accept only changes that yield immediate improvement, which are effective for the optimization problems with deterministic quadratic objective function that has only one minimum. However, it often induces local optima rather than global optima and sometimes results in divergence when minimizing both objective functions at the same time. As a result, these techniques usually achieve local optima rather than global optima. 2.2.1 Linear Programming (LP) [5, 6] LP is the most common optimization technique applied for constrained optimization problems, where the entire objective functions and equality and inequality con- straints are linear. It is widely used due to many practical problems that can be formulated as LP problems; there are noniterative, simplex, and efficient methods for solving optimization problems. The classical LP problem is to obtain the optimal decision variables that maximize the linear objective function under linear equality and inequality constraints. Specifically, the general structure of problems solved by LP is: Maximize cT x Subject to A x b x 0 = ; ð2:1Þ
- 34. T n 2.2 Conventional Optimization Techniques 15 where x ¼ [x1, . . .. . ., xn ] 2 R is the vector of decision variables to be determined, c ¼ [c1, . . .. . ., cn] 2 Rn is the vector of the objective function coefficients, A ¼ [aij ] 2 Rm n is the matrix of the constraint coefficients with its elements, and b ¼ [b1, b2, .., bm]T 2 Rm is the bounding vector. Assume that the model is statically linear. The process to achieve the global optimum uses simplex like techniques, variants of the interior point method, or integer programming (IP). These methods are applicable to problems involving linear objective functions and linear constraints. The solution procedure includes [5, 6]: 1. Initialization step: Introduce slack variables (if needed) and determine initial point as a corner point solution of the equality constraints. 2. At each iteration, move from the current basic feasible solution to a better, adjacent basic feasible solution. 3. Determine the entering basic variable: Select the nonbasic variable that, when increased, would increase the objective at the fastest rate. Determine the leaving basic variable: Select the basic variable that reaches zero first as the entering basic variable is increased. 4. Determine the new basic feasible solution. 5. Optimality test and termination criteria: Check if the objective can be increased by increasing any nonbasic variable by rewriting the objective function in terms of the nonbasic variables only and then checking the sign of the coefficient of each nonbasic variable. If all coefficients are nonpositive, this solution is optimal and stop. Otherwise, go to the iterative. Applying the LP method to the SG requires improving it to accommodate the grid’s productivity, adaptivity, and randomness. The traditional linear method is confined to static problems and thus is insufficient for SG implementation. 2.2.2 Quadratic Programming (QP) Quadratic programming (QP) [9–17] is a linearly constrained optimization problem with a quadratic objective function to be minimized subject to linear equality and inequality constraints. The general quadratic program can be written as: Maximize C:x þ 1 2 xT :Q:x Subject to A x b x 0 9 = ; ð2:2Þ where C is an n-dimensional row vector describing the coefficients of the objective function and Q 2 Rn n is a symmetric matrix describing the coefficients of the quadratic terms. QP problem can be solved by different optimization methods such
- 35. as the simplex method [11], interior point method [12], active set method [13], a branch-and-cut algorithm [14], objective separable method [15], Wolfe’s modified simplex method [16], and the Karush-Kuhn-Tucker (KKT) conditions [17]. The KKT conditions extend the ideas of Lagrange multipliers to handle constraints in addition to equality constraints. These conditions provide a first-order optimality condition for the problem. 2.2.3 Integer Programming (IP) Integer programming (IP) is a special case of LP where all or some of the decision variables are restricted to discrete integer values, for example, where the discrete values are restricted to zero and one only, that is, yes or no decisions, or binary decision variables. The general structure of the MIP problem is [7, 8]: Maximize P x ð Þ ¼ Pn j¼1cjxi Subject to Pm i¼1 Pn j¼1aij xj bi xj 0, j 2 1, n f g, and xj is an integer 9 = ; ð2:3Þ The branch-and-bound procedure features: 16 2 Optimization Techniques 1. Initialization: Set P* ¼ 1, where P* is the optimal value of P. 2. Branching: This step involves developing subproblems by fixing the binary variables at 0 or 1, or choosing the first element in the natural ordering of the variables as the branching variable. 3. Bounding: For each of the subproblems, a bound can be obtained to determine the goodness of its best feasible solution. For each new subproblem, obtain its bound by applying the simplex method to its LP relaxation and use the value of the P for the resulting optimal solution. 4. Fathoming: If a subproblem has a feasible solution, it should be stored as the first incumbent (the best feasible solution found so far) for the whole problem along with its value of P. This value is denoted P* , which is the current incumbent for P. 5. Optimality test: The iterative procedure stops when no subproblems remain. At this stage, the current incumbent for P is the optimal solution. Pure integer or MIP problems pose a great computational challenge. While highly efficient LP techniques can enumerate the basic LP problem at each possible combination of the discrete variables (nodes), the problem lies in the astronomically large number of combinations to be enumerated. If there are N discrete variables, the total number of combinations becomes 2N . The branch-and-bound technique for binary integer or reformulated MIP problems overcomes this challenge by dividing the overall problem into smaller and smaller subproblems and enumerating them in a logical sequence. Note that the direct use of these techniques for solving the smart
- 36. grid optimization problem will be limited, because they are generally static and are not designed for handling real-time and dynamic optimization problems. 2.3 Artificial Intelligence (AI) Techniques 17 2.2.4 Dynamic Programming (DP) The abstract framework for dynamic programming (DP) was first introduced in [18]. This approach was developed to solve sequential or multistage decision- making problems. Basically, it solves a multivariable problem by solving a series of single-variable problems. This is achieved by tandem projection onto the space of each of the variables. In other words, it projects first onto a subset of the variables, then onto a subset of these, and so on. It is a candidate optimization technique for handling time variability and noise in the objective and constrained optimization problem [18]. The general structure of the DP problem is [19]: Problem P : p ∶ ¼ opt p x ð Þ, x 2 X ð2:4Þ where p is a real-valued function or the objective function on some set X,¼ denotes definition, opt denotes optimality function, X is the decision space, and x is the decision variables. The solution procedure for Problem P, defined by Eq. (2.4), would run as follows [19]: 1. For every x 2 X, solve problem P(x) and determine the value of p(x). 2. Determine the value of p by optimizing p(x) over x 2 X. 2.3 Artificial Intelligence (AI) Techniques 2.3.1 Artificial Neural Network (ANN) There are a wide variety of artificial neural networks (ANNs) that are used to model real neural networks, study behavior, and control in animals and machines, but also there are ANNs which are used for engineering purposes, such as economic dispatch and reactive power dispatch [20, 21]. The ANNs were inspired by biological findings relating to the behavior of the brain as a network of units called neurons. Each neuron receives signals through synapses that control the effects of the signal on the neuron. The fundamental building block in an ANN is the mathematical model of a neuron as shown in Fig. 2.2. The three basic components of the artificial neuron are: • The synapses or connecting links that provide weights, wi, to the input values, xi, for i ¼ 1,. . .,m.
- 37. X 18 2 Optimization Techniques Φ(v) i y Output Activation Function i v Synaptic Weights Input Signals m X 2 X 1 X 1 = o X ∑ φ o W 1 W 2 W m W Fig. 2.2 An artificial neuron • An adder that sums the weighted input values to compute the input to the activation function: vi ¼ wo þ m i¼1 wi xi ð2:5Þ where wo is called the bias that is a numerical value associated with the neuron. It is convenient to think of the bias, as the weight for an input xo whose value is always equal to one, so that: vi ¼ Xm i¼0 wi xi ð2:6Þ • An activation function φ (also called a squashing function) that maps vi to φ(v), the output value of the neuron. 2.3.2 Fuzzy Linear Programming (FLP) [22, 23] Fuzzy logic is a superset of conventional logic that has been extended to handle the concept of partial truth, which are the truth values between “completely true” and “completely false.” Special membership functions transform uncertainty in data to a crisp form for analysis. Fuzzy logic control is often used to determine whether the process variables are within acceptable tolerances. A fuzzy set is a generalization of ordinary sets that allows assigning a degree of membership for each element to range from (0, 1) interval. Figure 2.3 shows the simplified block diagram of the fuzzy logic approach. For optimization problems, it is quite restrictive to require the objective and constraints to be specified in precise, crisp terms to apply fuzziness in the LP
- 38. 9 problem. Fuzzy LP (FLP) is applied to solve optimization problems such as eco- nomic dispatch [22, 23]. Specifically, the standard FLP problem is: Maximize λ Subject to λ μ ð2:7Þ The fuzzy objective function for maximization or minimization purposes takes the linear membership function. There are three types of FLP problems, which are: 2.3 Artificial Intelligence (AI) Techniques 19 Fuzzyfication Decision-making Defuzzyfication Start End Fig. 2.3 Simplified block diagram of a fuzzy logic approach μi(CX) 1 0 Zo Z1 (CXλ)i 1 0 bi bi+ti (AXλ)i a. Minimization b. Maximization μi(AX) λ λ AX Z Fig. 2.4 Membership function with fuzzy resources (a) Linear programming with fuzzy resources: It can be formulated in LP with LP objective and constraints with fuzzy resources and can be written as follows: Maximize e CX Subject to A X e b X 0 = ; ð2:8Þ y e where the fuzzy inequalit is characterized by the membership functions. The fuzzy objective and inequality constraint for maximization or minimization purposes takes the linear membership function as shown in Fig. 2.4. In this figure, the relationship between the value of an objective function and constraint CXλ and AXλ and the degree of satisfaction λ is illustrated for the linear membership function, where Zo and bi + ti represent an aspiration level of the decision-maker. The fuzzy membership function for minimizing an inequality constraint can be defined as follows:
- 39. 8 9 20 2 Optimization Techniques μi AX ð Þ ¼ 1 if AX bi 1 AX bi ti if bi AX bi þ ti 0 if AX bi þ ti : ð2:9Þ where μi(AX) is the degree to which decision variables satisfy the ith constraint and ti is the tolerance of the ith resource bi. Also, the fuzzy membership function for maximizing the degree of optimality of an objective function with fuzzy resources can be defined as follows: μi CX ð Þ ¼ 1 if CX Z1 1 Z1 cx Z1 Zo if Zo CX Z1 0 if CX Zo 8 : ð2:10Þ where μi(CX) is the degree to which decision variables satisfy the ith objective function. (b) Linear programming with fuzzy objective coefficients: LP objective and con- straints with fuzzy objective coefficients can be written as follows: Maximize e CX Subject to A X b X 0 = ; ð2:11Þ C ~ i where s triangular fuzzy numbers characterized by the membership functions of fuzzy objective coefficients as shown in Fig. 2.5, where C ~ ¼ C i , Co i , Cþ i , C ¼ C 1 , . . . , C n , Co ¼ Co 1, . . . , Co n , and Cþ ¼ Cþ 1 , . . . , Cþ n for a multi-objective linear programming problem. (c) Linear programming with fuzzy constraint coefficients: LP objective and con- straints with fuzzy objective coefficients can be formulated as: Fig. 2.5 Membership function with fuzzy objective coefficients μi(Ci) Ci Ci + Ci o Ci - 1 0
- 40. Maximize CX e 9 = ð 2.3 Artificial Intelligence (AI) Techniques 21 Fig. 2.6 Membership function with fuzzy constraint coefficients μi(Ci) 1 0 A- Ao A+ A Subject to A X b X 0 ; ð2:12Þ A ~ where is triangular fuzzy numbers characterized by the membership functions of fuzzy constraint coefficients as shown in Fig. 2.6, where A ~ ¼ a ~ ij , that is, a ~ ij ¼ a ~ ij , a ~ ij o, a ~ ij þ , A ¼ a ij h i , Ao ¼ ao ij h i , and Aþ ¼ aþ ij h i for a multiple-constraint linear programming problem. There are other types of FLP problems, which are the combination of the three basic problems, and can be written as: Maximize e CX Subject to e A X e b X 0 9 = ; ð2:13Þ There is a max-min method to solve the multi-objective optimization problems. Specifically, the problem becomes: max min μo X ð Þ, μ1 X ð Þ, . . . , μm X ð Þ ½ 2:14Þ Finally, the fuzzy resources LP problems are solved using standard LP after the objectives and constraints are prepared in the form of LP technique. In SG design, many real-time decisions require the attribute of fuzziness. The design of automatic generation control (AGC), static security assessment (SSA), and state estimation (SE) will therefore be used for proposed SG functions and control.
- 41. 22 2 Optimization Techniques 2.3.3 Expert Systems (ES) ES are used as a method of optimization that relies on heuristic or rule-driven decision-making. They are sometimes used for fault diagnosis with prescription for corrective actions. While the expert system/computer application performs a task that would otherwise be performed by a human, the method is only as reliable as the designed engineering rule base. Figure 2.7 shows the components of ES [24]. Expert systems have several advantages over human experts, including increased availability and reliability, lower cost and response time, and increased confidence in decision-making ability via provision of clear reason for a given answer. Its uses for the power system include optimal load shedding, resource allocation such as VAR, discrete control such as series capacitors, and economic dispatch. The operator- assisted functions in the management of the SG will benefit from the use of this knowledge-based system or its hybrid with ANN or other variations of computa- tional/heuristic optimization techniques. Some of the problems can be reformulated using expert systems, real-time data, and hybrid ANN. 2.4 Modern Optimization Methods The modern optimization methods (also called nontraditional optimization methods) have emerged as powerful and popular methods for solving complex engineering optimization problems in recent years. These methods are simulated annealing, evolutionary programming (EP), Tabu search (TS), neural network-based methods, genetic algorithm (GA), differential evolution (DE) algorithm, particle swarm opti- mization (PSO) technique, seeker optimization algorithm (SOA), and ant colony optimization (ACO) algorithm. Most of these methods are labeled on certain char- acteristics and behavior of biological, molecular, swarm of insects, and neurobio- logical systems. These new heuristic tools have been combined among themselves and with knowledge elements, as well as with more traditional approaches such as statistical analysis to solve extremely challenging problems. Five of such modern techniques will be covered in some details, namely, GA, DE algorithm, PSO technique, SOA, and ACO algorithm. User Expert system Knowledge base Inference engine Facts Expertise Fig. 2.7 Fundamental components of an expert system
- 42. 2.5 Evolutionary Optimization Techniques 2.5.1 Genetic Algorithm (GA) [25, 26] Genetic algorithm (GA) is an optimization and search technique based on the principles of genetics and natural selection. GA was invented by John Holland who developed this idea in his book Adaptation in Natural and Artificial Systems in the year 1975. Holland proposed GA as a heuristic method based on “survival of the fittest.” GA was discovered as a useful tool for search and optimization problems. GA handles a population of possible solutions. Each solution is represented through a chromosome, which is just an abstract representation. Coding of all the possible solutions into a chromosome is the first part, but certainly not the most straightfor- ward one of a GA. A set of reproduction operators has to be determined. Also, reproduction operators are applied directly on the chromosomes and are used to perform mutations and recombinations over solutions of the problem. GA is very different from classical optimization algorithms in: 2.5 Evolutionary Optimization Techniques 23 • Use of the encoding of the parameters, not the parameters themselves. • Work on a population of points, not a unique one. • Use the only values of the function to optimize, not their derived function or other auxiliary knowledge. • Use probabilistic transition function, not the determinist ones. The steps of the GA can be represented as the flowchart shown in Fig. 2.8. Yes Initialization Generate Initial Population Roulette wheel selection [25,26] Mutation [25,26] Check the problem constraints Crossover [25,26] Stopping criterion Get the optimal solution Evaluation Function Offspring chromosome evaluation [25,26] Elitism [25,26] No Fig. 2.8 Flowchart for a simple GA
- 43. 24 2 Optimization Techniques 2.6 Differential Evolution (DE) Algorithm In 1995, Storn and Price proposed a new floating-point [27] encoded evolutionary algorithm for global optimization and named it differential evolution (DE) owing to a special kind of differential operator, which they invoked to create new offspring from parent chromosomes instead of classical crossover or mutation. The DE algorithm differs from GA with respect to the mechanics of mutation, crossover, and selection performed. GA relies on crossover, while DE relies on mutation operation. In GA, the mutation takes place randomly, whereas in DE, it takes place by some rule. 2.6.1 Standard DE Algorithm In DE algorithm, solutions are represented as chromosomes based on floating-point numbers. In the mutation process of this algorithm, the weighted difference between two randomly selected population members is added to a third member to generate a mutated solution followed by a crossover operator to combine the mutated solution with the target solution so as to generate a trial solution. Then, a selection operator is applied to compare the fitness function value of both competing solutions, namely, target and trial solutions, to determine who can survive for the next generation. The basic DE algorithm consists of four steps, namely, initialization of population, mutation, crossover, and selection, as in Fig. 2.9 [28, 29]. Fig. 2.9 DE cycle of stages Initialization of Chromosomes Mutation Differential Operator Crossover Selection
- 44. • 2.7 Particle Swarm Optimization (PSO) Technique Particle swarm optimization (PSO) is a population-based stochastic optimization technique developed by Kennedy and Eberhart in 1995 [30, 31], discovered through simplified social model simulation. It stimulates the behaviors of bird flocking or fish schooling involving the scenario of a group of birds or fishes randomly looking for food in an area. PSO is motivated from this scenario and is developed to solve complex optimization problems. PSO shares many similarities with evolutionary computation techniques such as GA. The system is initialized with a population of random solutions and searches for optima by updating generations. However, unlike GA, PSO has no evolution operators such as crossover and mutation [25]. In PSO, the potential solutions called particles fly through the problem space by following the current optimum particles. This is performed by particles in multidimensional space that have a position and a velocity. These particles are flying through hyperspace and have two essential reasoning capabilities: their memory of their own best position and knowledge of the swarm’s best, “best” simply meaning the position with the smallest objective value. Members of a swarm communicate good positions to each other and adjust their own position and velocity based on these good positions. There are two main ways to do this: 2.7 Particle Swarm Optimization (PSO) Technique 25 • A global best that is known to all and immediately updated when a new best position is found by any particle in the swarm “Neighborhood” bests where each particle only immediately communicates with a subset of the swarm about positions 2.7.1 PSO Mathematical Model PSO technique is based on the social behavior of bird flocking or fish schooling for searching the food. Each particle keeps track of its coordinates in hyperspace, which are associated with the best solution (fitness) it has achieved so far. The value of the fitness is called personal best “pbest” and stored in the particle memory. Another best value is also tracked, the global version of the particle swarm optimizer keeps track of the overall best value, and its location is obtained thus far by any particle in the population; this is called global best “gbest.” The modified velocity of each agent can be calculated using the current velocity, and the distance between pbest and gbest positions with respect to current position is shown below: vkþ1 id ¼ wi:vk id þ c1:r1: pbestid xk id þ c2:r2: gbest xk id i ¼ 1, 2, . . . , n d ¼ 1, 2, . . . , m ð2:15Þ
- 45. where n is the number of particles in a population and m is the number of members in 26 2 Optimization Techniques a particle (number of variables). Using the above equation, a certain velocity that gradually gets close to pbest and gbest can be calculated. The current position (searching point in the search space) can be modified by the following equation: xkþ1 id ¼ xk id þ vkþ1 id ð2:16Þ Figure 2.10 shows the above concept of modification of searching points. Dis- crete variables c1 and c2 can be handled in Eqs. (2.15) and (2.16) with little modification. Discrete numbers can be used to express the current position and velocity. If a discrete random number is used in Eq. (2.15) and the whole calculation of the right-hand side (RHS) of Eq. (2.15) is discretized to the existing discrete number, both continuous and discrete numbers can be handled in the algorithm with no inconsistency. Suitable selection of inertia weight w in Eq. (2.15) provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution. As originally developed, the inertia weight w often decreases linearly from wmax to wmin during a run. In general, the inertia weight w is set according to the following equation: w ¼ wmax wmax wmin kmax k ð2:17Þ where kmax is the maximum number of iterations (generations) and k is the current number of iterations. Despite its features, a general problem with the PSO, similar to other optimization algorithms that are not exhaustive methods, such as the brute-force search, is that of becoming trapped in a local optimum, or suboptimal solution, such that it may work well on one problem but yet fail on another problem. In general, the main drawbacks of PSO can be summarized as follows [32]: Fig. 2.10 Concept of modification of searching point in two dimensions xi k xi k+1 vi k vpbest vgbest X-Axis Y-Axis
- 46. 2.7 Particle Swarm Optimization (PSO) Technique 27 • Premature convergence of a swarm: Particles try to converge to a single point, located on a line between the global best and the personal best positions (local best). This point is not guaranteed for a local optimum. Another reason could be the fast rate of information flow between particles, which leads to the creation of similar particles. This results in a loss in diversity, and the possibility of being trapped in local optima is increased. • Parameter setting dependence: This leads to the high performance variances for a stochastic search algorithm. In general, there is not any specific set of parameters for different problems. As an example, and by simple observation of Eq. (2.23), increasing the inertia weight w will increase the speed of the particles v and cause more exploration (global search) and less exploitation (local search). As a result, finding the best set of parameters is not a trivial task, and it might be different from one problem to another. The steps of PSO can be implemented as the flowchart shown in Fig. 2.11. Fig. 2.11 Flowchart of PSO algorithm Yes No Initialization Evaluation function Velocity and position updating pbest and gbest updating Initialization of pbest and gbest Check problem constraints Stopping criterion Get the optimal solution
- 47. 2.8 Seeker Optimization Algorithm (SOA) Seeker optimization algorithm (SOA) was originally proposed by Dai et al. [33]. SOA is a relatively new intelligent algorithm that may be used to find optimal (or near optimal) solutions to numerical and qualitative problems. This method is a population-based heuristic stochastic search algorithm based on simulating the act of human search to obtain the optimal solution by a seeker population. In this algo- rithm, the next position of seekers is obtained from their current positions based on their historical and social experiences. The seekers exchange their information with those defined in their neighbors to negotiate the quickest way to reach the best solution [34, 35]. 2.8.1 SOA 28 2 Optimization Techniques (i) Implementation of SOA In SOA, for each seeker i, the position update on each variable j is given by the following equation [35, 36]: xij t þ 1 ð Þ ¼ xij t ð Þ þ αij t ð Þ dij t ð Þ ð2:18Þ where xij (t + 1) and xij (t) are the positions of seeker i on the variable j at time steps t + 1 and t, respectively. dij (t) and αij (t) are search direction and step length of seeker i on the variable j at time step t, where αij (t) 0 and dij (t) 2 {21, 0, 1}. Here, i represents the population number and j the variable number to be optimized. In Eq. (2.18), if dij(t) ¼ 1, the ith seeker goes to the positive orientation of the coordinate axis on dimension j. If dij(t) ¼ 21, the seeker goes to the negative orientation, and if dij(t) ¼ 0, the seeker stays at the current position. Moreover, seekers in the same subpopulation are searching for the optimal solution using their own information. In order to avoid the convergence of sub- populations trapping into local optima, the position of the worst seeker of each subpopulation is combined with the best one in each of the other subpopulations using the binomial crossover operator as follows: xknj,worst ¼ xlj,best if Rj 0:5 xknj,worst otherwise ð2:19Þ xknj,worst i where Rj is a uniformly random real number within [0, 1], s denoted as the jth variable of the nth worst position in the kth subpopulation, and xlj, best is the jth variable of the best position in the lth subpopulation. Here, n,k,l ¼ 1, 2,. . ., K 2 1 and k 6¼ l. Thus, the diversity of population is increased by sharing good information among subpopulations.
- 48. d t sign g t x t 2 21 d t sign l t x t ð2 22 ð 2.8 Seeker Optimization Algorithm (SOA) 29 (ii) Search Direction In SOA, each seeker selects his search direction based on several empirical gradients (EGs) by comparing the current or historical positions of himself or his neighbors. For seeker i, the empirical directions involved are [35, 36]: di,ego t ð Þ ¼ sign pi,best t ð Þ xi t ð Þ ð2:20Þ i,alt1ð Þ ¼ i,bestð Þ ið Þ ð : Þ i,alt2ð Þ ¼ i,bestð Þ ið Þ ð Þ : Þ where di,ego(t) is the egotistic direction, di,alt1(t) and di,alt2(t) are the altruistic direc- tions, and pi,best(t), gi,best(t), and li,best(t) represent the personal historical best posi- tion, neighbors’ historical position, and current best position, respectively. The function sign() is a signum function on each variable of the input vector. In addition, each seeker i as an agent enjoys the properties of pro-activeness and exhibits goal- directed behavior, which means he may change his search direction in advance according to his past behavior. This behavior is modeled as an empirical direction called pro-activeness direction given as: di,pro t ð Þ ¼ sign xi t1 ð Þ xi t2 ð Þ ð Þ 2:23Þ where t1,t2 2 {t, t-1, t-2} and xi(t1) is better than xi(t2). Every variable j of di(t) is selected applying the following proportional selection rule as: dij ¼ 0 if rj p 0 ð Þ j þ1 if p 0 ð Þ j rj p 0 ð Þ j þ p þ1 ð Þ j 1 if p 0 ð Þ j þ p þ1 ð Þ j rj 1 8 : ð2:24Þ where rj is a uniform random number in [0, 1] and pj (m) (m 2{0,+1,-1}) is the percentage of the number of m from the set {dij,ego(t), dij,alt1(t), dij,alt2(t), dij,pro(t)} on each dimension j of all the four empirical directions, that is: pm j ¼ the number of m 4 ð2:25Þ (iii) Step Length In SOA, a fuzzy system is adopted to represent the understanding and linguistic behavioral pattern of human searching tendency. The fitness values of all the seekers are sorted in descending manner and turned into the sequence numbers from 1 to S as the inputs of fuzzy reasoning, where S is the size of population. This design renders a fuzzy system to be applicable to a wide range of optimization problems. The calculation expression of step length is presented as [37]:
- 49. S I ð ð 30 2 Optimization Techniques μmin μ ij μ α 3δ 2δ δ 0 -δ -2δ -3δ ij α Bell function curve Fig. 2.12 The action part of the fuzzy reasoning μi ¼ μmax i S 1 μmax μmin ð Þ 2:26Þ where Ii is the sequence number of xi(t) after sorting the fitness values and μmax is the maximum membership degree value which is equal to or a little less than 1. The Bell membership function μ x ð Þ ¼ ex2 =2δ2 is well utilized in the action part of fuzzy reasoning as shown in Fig. 2.12 [37], and the parameter δ in the function is determined as follows: δ ¼ ω abs xbest xrand ð Þ 2:27Þ where the symbol abs() produces an absolute value in response to input vector and ω is used to decrease the step length with time step increasing so as to gradually improve the search precision. The xbest and xrand are the best seeker and a randomly selected seeker, respectively, from the same subpopulation to which the ith seeker belongs, and xrand is different from xbest. The step length αij(t) for every variable j is given as: αij t ð Þ ¼ δj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln rand μi, 1 ð Þ ð Þ p ð2:28Þ where rand(μi,1) returns a uniformly random real number within [μi,1]. The steps of the SOA can be represented as in the flowchart shown in Fig. 2.13. 2.9 Ant Colony Optimization (ACO) Algorithm Ant colony optimization (ACO) algorithm is used for solving the combinatorial optimization problems, which can be regarded as a paradigm for all ant colony search algorithms that are inspired by the foraging behavior of the social insects, especially the ants. The first member of the ACO class of algorithms, called the ant system, was initially presented by Colorni, Dorigo, and Maniezzo [38, 39]. The main underlying idea, loosely inspired by the behavior of real ants, is that of a parallel
- 50. 2.9 Ant Colony Optimization (ACO) Algorithm 31 Yes Real coded initialization of S seekers Start Set t = 0 Divide the population into K subpopulations randomly Calculate the objective function value for each seeker Calculate the personal best position, neighborhood best position and population best position Compute search direction for each seeker by using Equation (2.24) Compute step length for each seeker by using Equation (2.28) Update the position of each seeker using Equation (2.18) Calculate the objective function value for each seeker Update the personal best position, neighborhood best position and population best position Subpopulations learn from each other by using Equation (2.19) Increment t = t +1 Meet stopping criterion? Display the optimal fitness value and optimal solution Stop No Fig. 2.13 Flowchart of the SOA
- 51. search over several constructive computational threads based on local problem data and on a dynamic memory structure containing information on the quality of the previously obtained result. The collective behavior emerging from the interaction of the different search threads has proved effective in solving combinatorial optimiza- tion (CO) problems. The main elements of a biological stigmergic system are: 32 2 Optimization Techniques • The insect as the acting individual • The pheromone as an information carrier, used to create a dissipation field • The environment as a display and distribution mechanism for information Real ants are capable of finding the shortest path from a food source to the nest without using visual cues. Also, they are capable of adapting to changes in the environment, for example, finding a new shortest path once the old one is no longer feasible due to a new obstacle. The characteristics of ant colony optimization are as follows [38, 39]: 1. Natural algorithm: Since it is based on the behavior of real ants in establishing paths from their colony to the source of food and back. 2. Parallel and distributed: Since it concerns a population of agents moving simul- taneously, independently, and without a supervisor. 3. Cooperative: Since each agent chooses a path on the basis of the information, pheromone trails laid by the other agents, which have previously selected the same path. This cooperative behavior is also autocatalytic, i.e., it provides a positive feedback, since the probability of choosing a path increases with the number of agents that previously chose that path. 4. Versatile: That it can be applied to similar versions of the same problem; e.g., there is a straightforward extension from the traveling salesman problem (TSP) to the asymmetric traveling salesman problem (ATSP). 5. Robust: That it can be applied with minimal changes to other combinatorial optimization problems such as quadratic assignment problem (QAP) and the jobshop scheduling problem (JSP). 2.9.1 Description of Real Ants The ACO algorithm is based on the behavior of real ants that are members of a family of social insects. However, a group of explorer ants leave the colony for finding the food source in random directions where they marked their routes by laying a chemical substance on the ground. Other ants are attracted to the route that has the largest amount of pheromone. Hence, they are found the shortest route between the nest and food source by indirect communication media called phero- mone that laid on the ground as a guide for another ants. Figure 2.14 shows how the real ants can find the shortest path between nest or colony and food source. In Fig. 2.14a, there are no obstacles between nest and food source. However, the shortest route is the straight line. If an obstacle is located on the route of ants to
- 52. become two routes around the obstacle, some of the ants choose the left side around the obstacle and the other will choose the right side as shown in Fig. 2.14b. The pheromone laid on the left side will be concentrated than the right side of the obstacle because ants in the shortest path take minimum time in leaving and returning for nest where they move in the same speed. Therefore, they will be laid a largest amount of pheromone than other ants on the other route, while the other ants are attracted to the shortest route. Hence, all ants in the colony will take the shortest route around the obstacle as shown in Fig. 2.14c. 2.9.2 Comparison Between Artificial and Real Ant Systems Artificial ants are characterized as agents that imitate the behavior of real ants for obtaining the optimal solutions according to the problem’s constraints. However, it should be noted that an artificial ant colony system (ACS) has some differences in comparison with a real ACS [40], as follows: 2.9 Ant Colony Optimization (ACO) Algorithm 33 Fig. 2.14 Illustration of real ants behavior. (a) Real ants follow a path between the nest and a food source. (b) An obstacle appears on the path; the pheromone is deposited more quickly on the shorter path. (c) All ants have chosen the shorter path 1. Artificial ants have memory. 2. They are not completely blind. 3. They live in an environment where time is discrete. On the other hand, an artificial ACS has several characteristics adapted from real ACS: • Artificial ants have a probabilistic preference for paths with a larger amount of pheromone. • Shorter paths tend to have larger rates of growth in their amount of pheromone. • The ants use an indirect communication system based on the amount of phero- mone deposited in each path.
- 53. τ t 1 1 ρ τ t εΔτ t 2:31 34 2 Optimization Techniques 2.9.3 ACO Mathematical Model A random amount of pheromone is deposited in each route after each ant completes its tour. Other ants attract to the shortest route according to the probabilistic transition rule that depends on the amount of pheromone deposited and a heuristic guide function. The heuristic guide function is the inverse of the distance between the beginning and ending of each route. The probabilistic transition rule of ant k to go from city i to city j at iteration t can be expressed as in traveling salesman problem (TSP) [41] as: Pk ij t ð Þ ¼ τij t ð Þ α ηij t ð Þ β P q τiq t ð Þ α ηiq t ð Þ β ; j, q 2 Nk i ð2:29Þ If α ¼ 0, the closest cities are more likely to be selected that corresponds to a classical greedy algorithm. On the contrary, if β ¼ 0, the probability will depend on the pheromone trial only. These two parameters should be tuned with each other. After each tour is completed, a local pheromone update is determined by each ant depending on the route of each ant as in Eq. (2.30). After all ants are attracted to the shortest route, a global pheromone update is considered to show the influence of the new addition deposits by other ants that are attracted to the best tour as shown in Eq. (2.31): τij t þ 1 ð Þ ¼ 1 ρ ð Þτij t ð Þ þ ρτo ð2:30Þ ij þ ð Þ ¼ ð Þ ijð Þ þ ijð Þ ð Þ where τij(t + 1) is the pheromone after one tour or iteration. τo ¼ 1/dij is the incremental value of pheromone of each ant, while Δτij is the amount of pheromone for the elite path as: Δτij t ð Þ ¼ Q =dbest ð2:32Þ where Q is a large positive constant and dbest is the shortest tour distance found as in TSP. 2.9.4 ACO Algorithm The process of ACO algorithm for solving the optimization problems can be summarized as follows: Step 1: Initialization Specify the lower and upper boundaries of each control variable [xi min , xi max ]. Then, insert the initial values of control variables randomly within their permissible limits inside the d-search space that has a matrix of dimension d(m n), where n refers to
- 54. the number of stages, while m refers to the number of states (ants) in each stage shown in Fig. 2.15. So, the control variable xi at iteration 0 can be represented as the vector of: 2.9 Ant Colony Optimization (ACO) Algorithm 35 Destination Colony State 1 State 1 State 1 State 1 State 2 State 2 State 2 State 2 State m-1 State m-1 State m-1 State m-1 State m State m State m State m Stage 1 Stage 2 Stage n-1 Stage n Fig. 2.15 Search space for the optimization problem x0 i ¼ x0 i1, x0 i2, x0 i3, . . . , x0 in where xi min xi xi max . Also, insert the ACO parameters, which are n, m that is equal to the number of ants, α, β, ρ, and ε. These parameters must be selected carefully by trial and error in order to obtain the best global solution of the problem. In addition, initialize the ant’s tours with random values limited by [1, m] with the same dimension of d-space. Step 2: Provide first position Each ant is positioned on the initial state randomly within the reasonable range of each control variable in a search space with one ant in each control variable in the length of randomly distributed values. Step 3: Evaluation The merit of each initial control variable in the d-search space is found using the corresponding objective function of each ant in each stage of the problem called individual evaluation function with the same dimension of d-space. The whole initial objective functions for each ant are obtained by summing the individual objective functions with dimension (m 1) as: F0 i ¼ Pm i¼1f i þ Pm i¼2f i þ Pm i¼3f i þ . . . ¼ F0 1, F0 2, F0 3, . . . , F0 m T The initial global objective function is the minimum value among the values of the whole initial objective functions as:
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- 56. the decrees promulgated during the first month after the overthrow of the Tsar. We need not fear healthy reaction. No power on earth can deprive the peasant of the land now acquired, in the teeth of landlord and Bolshevik alike, on a basis of private ownership. By a strange irony of fate, the Communist régime has made the Russian peasant still less communistic than he was under the Tsar. And with the assurance of personal possession, there must rapidly develop that sense of responsibility, dignity, and pride which well-tended property always engenders. For the Russian loves the soil with all his heart, with all his soul, and with all his mind. His folk-songs are full of affectionate descriptions of it. His plough and his harrow are to him more than mere wood and iron. He loves to think of them as living things, as personal friends. Barbaric instincts have been aroused by the Revolution, and this simple but exalted mentality will remain in abeyance as long as those continue to rule who despise the peasant’s primitive aspirations and whose world- revolutionary aims are incomprehensible to him. A veiled threat still lies behind ambiguous and inconsistent Bolshevist protestations. When this veiled threat is eliminated and the peasant comes fully into his own I am convinced that he will be found to have developed independent ideas and an unlooked-for capacity for judgment and reflection which will astonish the world, and which with but little practice will thoroughly fit him for all the duties of citizenship. Shortly after the Baltic republic of Lithuania had come to terms with Soviet Russia, one of the members of the Lithuanian delegation who had just returned from Moscow told me the following incident. In discussing with the Bolsheviks, out of official hours, the internal Russian situation, the Lithuanians asked how, in view of the universal misery and lack of liberty, the Communists continued to maintain their dominance. To which a prominent Bolshevist leader laconically replied: “Our power is based on three things: first, on Jewish brains; secondly, on Lettish and Chinese bayonets; and thirdly, on the crass stupidity of the Russian people.”
- 57. This incident betrays the true sentiments of the Bolshevist leaders toward the Russians. They despise the people over whom they rule. They regard themselves as of superior type, a sort of cream of humanity, the “vanguard of the revolutionary proletariat,” as they often call themselves. The Tsarist Government, except in its final degenerate days, was at least Russian in its sympathies. The kernel of the Russian tragedy lies not in the brutality of the Extraordinary Commission, nor even in the suppression of every form of freedom, but in the fact that the Revolution, which dawned so auspiciously and promised so much, has actually given Russia a government utterly alienated from the sympathies, aspirations, and ideals of the nation. The Bolshevist leader would find but few disputants of his admission that Bolshevist power rests to a large extent on Jewish brains and Chinese bayonets. But his gratitude for the stupidity of the Russian people is misplaced. The Russian people have shown not stupidity but eminent wisdom in repudiating both Communism and the alternative to it presented by the landlords and the generals. Their tolerance of the Red in preference to the White is based upon the conviction, universal throughout Russia, that the Red is a merely passing phenomenon. Human nature decrees this, but there was no such guarantee against the Whites with the support of the Allies behind them. A people culturally and politically immature like the Russians may not easily be able to embody in a formula the longings that stir the hidden depths of their souls, but you cannot on this account call them stupid. The Bolsheviks are all formula—empty formula—and no soul. The Russians are all soul with no formula. They possess no developed system of self-expression outside the arts. To the Bolshevik the letter is all in all. He is the slave of his shibboleths. To the Russian the letter is nothing; it is only the spirit that matters. More keenly than is common in the Western world he feels that the kingdom of heaven is to be found not in politics or creeds of any sort or kind, but simply within each one of us as individuals.
- 58. The man who says: “The Russians are a nation of fools,” assumes a prodigious responsibility. You cannot call a people stupid who in a single century have raised themselves from obscurity to a position of pre-eminence in the arts, literature, and philosophy. And whence did this galaxy of geniuses from Glinka to Scriabin and Stravinsky, or such as Dostoievsky, Turgeniev, Tolstoy, and the host of others whose works have so profoundly affected the thought of the last half-century—whence did they derive their inspiration if not from the common people around them? The Russian nation, indeed, is not one of fools, but of potential geniuses. But the trend of their genius is not that of Western races. It lies in the arts and philosophy and rarely descends to the more sordid realms of politics and commerce. Yet, in spite of a reputation for unpracticalness, the Russians have shown the world at least one supreme example of economic organization. It is forgotten nowadays that Russia deserves an equal share in the honours of the Great War. She bore the brunt of the first two years of it and made possible the long defence of the Western front. And it is forgotten (if ever it was fully recognized) that while corruption at Court and treachery in highest military circles were leading Russia to perdition, the provisioning of the army and of the cities was upheld heroically, with chivalrous self-sacrifice, and with astonishing proficiency, by the one great democratic and popularly controlled organization Russia has ever possessed, to wit, the Union of Co-operative Societies. The almost incredible success of the Russian co-operative movement was due, I believe, more than anything else to the spirit of devotion that actuated its leaders. It is futile to point, as some do, to exceptional cases of malpractices. When an organization springs up with mushroom growth, as did the Russian co-operatives, defects are bound to arise. The fact remains that by the time the Revolution came, the Russian co-operative societies were not only supplying the army but also providing for the needs of almost the entire nation with an efficiency unsurpassed in any other country.
- 59. The Bolsheviks waged a ruthless and desperate war against public co-operation. The Co-operative Unions represented an organization independent of the State and could therefore not be tolerated under a Communist régime. But, like religion, co-operation could never be completely uprooted. On the contrary, their own administration being so incompetent, the Bolsheviks have on many occasions been compelled to appeal to what was left of the co-operative societies to help them out, especially in direct dealings with the peasantry. So that, although free co-operation is entirely suppressed, the shell of the former great organization exists in a mutilated form, and offers hope for its resuscitation in the future when all co-operative leaders are released from prison. There are many ways of reducing the Russian problem to simple terms, and not the least apt is a struggle between Co-operation and Coercion. A deeper significance is attached in Russia to the word “Co- operation” than is usual in western countries. The Russian Co- operative Unions up to the time when the Bolsheviks seized power by no means limited their activities to the mere acquisition and distribution of the first necessities of life. They had also their own press organs, independent and well-informed, they were opening scholastic establishments, public libraries and reading-rooms, and they were organizing departments of Public Health and Welfare. Russian Co-operation must be understood in the widest possible sense of mutual aid and the dissemination of mental and moral as well as of physical sustenance. It is a literal application on a wide social scale of the exhortation to do unto others as you would that they should do to you. This comprehensive and idealistic movement was the nearest expression yet manifested of the Russian social ideal, and I believe that, whatever the outward form of the future constitution of Russia may be, in essence it will resolve itself into a Co-operative Commonwealth. There is one factor in the Russian problem which is bound to play a large part in its solution, although it is the most indefinite. I mean the power of emotionalism. Emotionalism is the strongest trait of the
- 60. Russian character and it manifests itself most often, especially in the peasantry, in religion. The calculated efforts of the Bolsheviks to suppress religion were shattered on the rocks of popular belief. Their categorical prohibition to participate in or attend any religious rites was ultimately confined solely to Communists, who when convicted of attending divine services are liable to expulsion from the privileged ranks for “tarnishing the reputation of the party.” As regards the general populace, to proclaim that Christianity is “the opium of the people” is as far as the Communists now dare go in their dissuasions. But the people flock to church more than ever they did before, and this applies not only to the peasants and factory hands but also to the bourgeoisie, who it was thought were growing indifferent to religion. This is not the first time that under national affliction the Russian people have sought solace in higher things. Under the Tartar yoke they did the same, forgetting their material woes in the creation of many of those architectural monuments, often quaint and fantastic but always impressive, in which they now worship. I will not venture to predict what precisely may be the outcome of the religious revival which undoubtedly is slowly developing, but will content myself with quoting the words of a Moscow workman, just arrived from the Red capital, whom I met in the northern Ukraine in November, 1920. “There is only one man in the whole of Russia,” said this workman, “whom the Bolsheviks fear from the bottom of their hearts, and that is Tihon, the Patriarch of the Russian Church.” A story runs of a Russian peasant, who dreamt that he was presented with a huge bowl of delicious gruel. But, alas, he was given no spoon to eat it with. And he awoke. And his mortification at having been unable to enjoy the gruel was so great that on the following night, in anticipation of a recurrence of the same dream, he was careful to take with him to bed a large wooden spoon to eat the gruel with when next it should appear.
- 61. The untouched plate of gruel is like the priceless gift of liberty presented to the Russian people by the Revolution. Was it, after all, to be expected that after centuries of despotism, and amid circumstances of world cataclysm, the Russian nation would all at once be inspired with knowledge of how to use the new-found treasure, and of the duties and responsibilities that accompany it? But I am convinced that during these dark years of affliction the Russian peasant is, so to speak, fashioning for himself a spoon, and when again the dream occurs, he will possess the wherewithal to eat his gruel. Much faith is needed to look ahead through the black night of the present and see the dawn, but eleven years of life amongst all classes from peasant to courtier have perhaps infected me with a spark of that patriotic love which, despite an affectation of pessimism and self-deprecation, does almost invariably glow deep down in the heart of every Russian. I make no excuse for concluding this book with the oft-quoted lines of “the people’s poet,” Tiutchev, who said more about his country in four simple lines than all other poets, writers, and philosophers together. In their simplicity and beauty the lines are quite untranslatable, and my free adaptation to the English, which must needs be inadequate, I append with apologies to all Russians: Umom Rossii nie poniatj; Arshinom obshchym nie izmieritj; U niei osobiennaya statj— V Rossiu mozhno tolko vieritj. Seek not by Reason to discern The soul of Russia: or to learn Her thoughts by measurements designed For other lands. Her heart, her mind, Her ways in suffering, woe, and need, Her aspirations and her creed, Are all her own— Depths undefined,
- 62. To be discovered, fathomed, known By Faith alone. THE END
- 63. INDEX Agents-provocateurs, 72, 217 Agitator, professional, 245 f. Alexandrinsky Theatre, 258 Allies: economic intervention, 299; military intervention, 46, 65, 81 f., 92, 226; political control by, 225 f. Anglo-Russian Commission, 3 Apfelbaum. See Zinoviev. Archangel, 4, 11, 83 Arms, right to carry, 268, 277 Army, Red, 34, 42, 69, 116, 207, 210, 215 ff.; classification of regiments, 234; Communist “Cells,” 234; discipline, 237; essential features, 248; mobilisation, 231; mutiny, 229; oath, 232; political organisation, 234; promotion, 227; revolutionary tribunals, 230, 233; terror in, 229; uniform, 215. See also Desertion, officers. Arrest, avoidance of, 199, 209, 286; in Finland, 167; of workers, 257
- 64. Badaev, 281 Balahovitch, 249 Ballot, secret, 236, 274 Baluev, 248 Bielo’ostrof, 22, 98, 118 Bielorusia, 249 Bolshevik, demonstrations, 136 f., 142; employment of bourgeois, 97, 115, 229, 269; government, 251 ff.; and the people, 43, 251 ff., 263; and Third International, 254; leaders helped by war, 225, 298; love of theory, 264, 304; negotiations with non-Bolsheviks, 64; party, 34, 251, 271; purge, 265 f.; power, basis of, 303; saints, 131 Bolshevism, and Soviet, 273 f., 296; despotism, 201; international aspect of, 252; a social experiment, 251 Bourgeois, employed by Bolsheviks, 97, 115, 229, 269; enterprise, need of, 294; Lenin’s attitude to, 260 Boy scouts, 3 Bribery, of commissars, 98; investigators, 126; prison warders, 59, 96 Bronstein. See Trotzky. Brusilov’s son shot, 224, 248 Cadet schools, 226 Certificates, of identification, 21 ff.; exemption from military service, 122 ff.
- 65. Chaliapin, 158, 258 Children, league for protection of, 297 Clubs, Communist, 268 Commerce. See Trading. Commissars, bribery of, 98; Communist, 222; regimental, 210, 230, 235; Soviet of, 213, 271 f. Committee of the poor, 43, 46, 242; of village poor, 185, 260 Communal booths, 43; eating-houses, 44, 62 Commune, village, 185 Communist, a typical, 243; women, 267; Youth, Union of, 268, 271 Communist “cells,” 234, 275; clubs, 268; franchise, 273 Communist Party, 68, 71, 79, 114, 131, 211, 248, 251 ff.; apart from people, 251, 264; in army, 218, 227 ff.; change of name, 34; discipline, 270; Jews in, 271; membership, 266 f., 270; meeting, 79; and Soviet Government, 254; and Third International, 254, 271; training school, 267 Contre-espionnage, 64 f. Co-operative Societies, 115, 255, 271, 305 f. Counter-revolution, 66, 193, 204. See also Extraordinary Commission. Couriers, service of, 169, 171, 209 Crombie, Captain, 12, 65 f.
- 66. Cronstadt, 92, 121, 163, 285 Cross-examination, 58, 73 f. Cultural-Enlightenment Committees, 240 f., 248 Denikin, 215, 224 f., 249 Denunciations, 238 Deriabinskya Prison, 91 Desertion from Red army, 116, 225 f., 229 f. —— Commission for Combating, 231, 288 Disenfranchising peasants, 273 Disguise, change of, 150, 197 f., 285 f. Disturbances, suppression of, 212 “Doctor,” the, 131 ff., 169, 194, 207 Dostoievsky, 304 Duma, 2, 257, 277 Dvinsk, 287 Eating-houses, Communal, 44, 283; private, 44, 81, 120 Ekaterina Canal, 155 Elections, 186, 265 ff. Electric light scarcity, 53 Engels, 131 Evdokimov, 280 Extraordinary Commission for the suppression of the counter-revolution, 19 f., 23, 50, 67, 70 ff., 91, 99, 101, 119, 125, 152, 193, 200, 205, 209, 230, 249, 276, 296 Finland, 12, 34, 92, 98, 110, 113 f., 117, 164 ff., 284 —— Railway station, 28 First International, 252 Fleet, British, 285 f. Food, cards, 43;
- 67. rations, 41, 46, 56, 64, 114, 212, 261, 282; sale of, 34, 40, 43 f., 100, 154, 235, 255 f., 262 f.; smuggling, 188 f., 261 Formulas, Bolshevik love of, 264, 304 Franchise, 273 Freedom of speech, 79, 133, 141, 255, 277 —— of trade, 154, 255, 263 f. —— suppression of, 303 “Freight weeks,” 261 Friedmann, 23 Frontier, Finnish, 13, 20, 98, 108 ff., 114, 166 f., 171 f., 285 —— Latvian, 292 —— Patrols, 18, 20, 114, 165, 167, 178 Fuel, 56, 69, 80, 114 Furniture confiscated, 149, 157 German Soviet of Petrograd, 139 Germans understand Russians, 92 Glinka, 304 Goróhovaya Street, No. 2. See Extraordinary Commission. Grand Duke Paul Alexandrovitch, 111 Green Guards, 291 f. Grodno, 229 Grusino station, 77, 106 Gutov, 248 Helsingfors, 11 f., 113, 168 f. Hiding documents, 75 f., 95, 209, 289 Hold-ups in street, 79 Home, molestation in, 149, 157, 224 Hostages, 58, 93, 94, 216, 223 f., 231, 257 f., 289 House committee. See Committee of the poor.
- 68. Ice-route to Finland, 163 ff. Instructions, 6 f. Intervention. See Allies. Intrigues in Finland, 171 Investigators of the Extraordinary Commission, 70, 95, 126 Ivan Sergeievitch, 16 f., 38 f., 118, 159, 169, 202 ff. Jews, in army, 218; Communist Party, 271 “Journalist,” the, 55, 80, 123, 148, 153, 194 Kamenostrovsky Prospect, 32, 44, 131 Kazan Cathedral, 38, 51, 160, 162 Kazanskaya Street, 38, 160 Klembovsky, 249 Kolchak, 210, 225 f., 249 Kresty Prison, 2 Kuznetchny Pereulok, 100 Labour Party, British, 257 Ladoga, Lake, 108, 172, 178 Latvia, 287 ff. Lawyers, Tsarist, employed, 233 Legislation, Soviet, 279 Lenin, 3, 34, 218, 241, 254, 257 ff.; bodyguard, 259; and bourgeoisie, 260, 294; and peasants, 260; and Trotzky, 272 Licences to trade, 33 Lida, 229 Liebknecht, Karl, 131 f., 258 Lissy Nos, 164
- 69. Liteiny Bridge, 30 —— Prospect, 62 Lithuania, 303 Luban, Lake, 292 Lunacharsky, 79, 260 Luxembourg, Rosa, 131 f., 258 Makhno, 249 Maria, 52, 73, 94, 99 ff., 113, 153, 195 Marinsky Opera, 159, 258, 275 Market of speculators, 100; raid on, 154 Marsh, Mr., 31 f., 48 ff., 66, 73, 199; escape of, 77 ff. Marsh, Mrs., arrest of, 50, 73, 74; danger of, 94; escape of, 97 ff. Marx, Karl, 131, 242, 252, 260 Matches, smuggling, 188 Melnikoff, 12 ff., 18 f., 48 f., 67, 89 f., 117 f., 125, 131 ff., 205 Menshevist Party, 65, 237, 243, 261, 266, 281 Mihailovsky Square, 77 Military service. See Army. Exemption from, 117, 122 Miliukoff, 257 Mines in Gulf of Finland, 121 Mirbach, Count, 4 Mobilisation. See Army. Moscow, 4, 97, 263 Music, love of, 183 f., 244, 258 Mutiny in Red army, 229 f.
- 70. Names used by author, 21, 40, 53, 55, 57, 63, 105, 117, 144, 196 Narodny Dom, 159, 217, 258, 262 National Centre, Moscow, 213 Nevsky Prospect, 34, 62, 79, 190, 207 Nicholas station, 190, 232, 253, 255 Obuhov works, 276 Odessa, 220 Officers, Tsarist, in Red army, 91, 211, 216 ff., 220, 223 f., 226 Okhta Bridge, 190 —— station, 77, 105, 189 Opera, 158, 258, 275 Parliament, British, 257 Parsky, 249 Passports, 16, 21, 76, 116, 135, 144 Peasant “capitalist,” 181 f. Peasantry, against both “Red” and “White,” 225, 227 f., 250; feud with Bolsheviks, 263, 301 f.; Finnish, 174; leaders of, 249; Lenin understands, 260 f.; power of, 193; propaganda among, 47, 228; treatment of, 68, 185 ff. Pertz, Otto, 139 Petrograd, return to, 30, 111, 113, 194 f. —— Soviet, 236, 265 ff. Petropavlovka, 31 Pilsudski, 249 Poland, 228, 249
- 71. Police, Tsarist, re-engaged, 50, 71, 117, 191 “Policeman,” the, 54, 57 f., 73, 85 f., 99, 105, 119, 123, 151, 169 Political guides, 238 Political organisation in army, 232 f. Press, monopoly of, 34, 277 Professional classes. See Bourgeoisie. Propaganda, in army, 143, 239 f.; author to undertake, 285; Communist, 271, 276; by gramophone, 241; among peasants, 47 Protopopoff, 190 Provisional Government, 242 Provocation by police, 72 Public Works, Department of, 54 Punctuality, lack of, 278 Putilov works, 255 f., 276 “Radishes,” 265 Raid, on eating-house, 62; on market, 154; on train, 188, 288 f. Railway travelling, 77 f., 106, 187 f., 261 f., 288 f. Rajajoki, 17 Rasputin, 257 Rations. See Food. Rattel, 248 Rautta, 172 Red army. See Army. Regiments, classification of, 234 Relatives as hostages, 216, 223 f. Religion, failure to suppress, 305 ff. Revolt against Bolsheviks, 71, 248, 257 Revolution of March, 1917, 1, 132, 190, 255, 275, 277
- 72. Revolutionary Tribunals, 230, 233 Rezhitsa, 289 “Sackmen,” 188 ff., 261 Sadovaya Street, 45 St. Izaac’s Cathedral, 103 St. Peter and St. Paul Fortress, 31, 111 Samara, 3, 278 Scheidemann, 140 ff. Schools, Communist training, 267 Scriabine, 304 Seals, importance of, 22 Ship committee, 222 Shpalernaya Prison, 91 Siennaya market, 52, 154 Smuggling, 162, 187, 188, 200 Socialist-Revolutionary Party, 65 f., 237, 243, 261, 266, 281 Soldiers’ committees, 221 Soviet, connection with Bolshevism, 273, 296; election, 186, 265, 272 ff; Executive Committee, 279; legislation, 279; meeting of a, 278 ff.; of people’s commissars, 213, 271; Republic’s president, 262; what is a, 272 ff. Speculation, 70, 106, 154 f., 201, 235 Speech, freedom of, 79, 133, 141, 255, 277 Speeches, Communist, 133 Spies in Red army, 217 Stage, privileges of the, 64 Stáraya Derévnya, 162 Stepanovna, 39 f., 137 ff., 146, 152 f. Stravinsky, 304
- 73. Street-sweepers, ladies as, 30, 33 Strikes, 84, 193, 256, 259, 274, 276 Suvorov Prospect, 190 Tauride Palace, 2, 277 Tchaikovsky, 83 Tche-Ka. Tchrezvitchaika. See Extraordinary Commission. Telephone, 90, 206 Terijoki, 168, 171 Theatre, Alexandrinsky, 258. See also Opera stage. Third International Workers’ Association, 138, 252 ff., 262, 295 ff.; headquarters, 253; relation to Soviet Government, 254, 271 Thought, freedom of, 133 Tihon, the patriarch, 307 Tihonov, 23 Tiutchev, 308 Tolstoy, 304 Trade unions, 115 Trading, private, 70; licences, 33 Travelling. See Railway. Troitzkaya Street, 63 Trotzky, 44, 69, 79, 90 f., 211, 219 ff., 227, 229, 241, 248, 259 ff., 272 Tsarskoeselsky station, 208 Turgeniev, 304 “Two-pooding,” 262 Ukraine, 228, 231, 249 Uniform of Red army, 215;
- 74. use of British, 215 Uritzky, 20, 266 —— Palace, 277 Vasili Island, 66 Viborg, 12, 134, 174, 202 Vladimirovsky Prospect, 100 Volodarsky, 266 White army, 211, 223, 225, 227, 303 Winter Palace, 137, 202 Women Communists, 267; electors, 277 Workers invited to join Communists, 266 f. Wrangel, 225, 228, 249 Yaroslavl, 13, 205 Y.M.C.A., 3, 243 Yudenitch, 211 Zabalkansky Prospect, 156 Zagorodny Prospect, 208 Zinoviev, 20, 34, 79, 137 ff., 216 f., 241, 253, 257 ff., 278 ff. Ziv, Dr., 219 Znamenskaya Square, 190 Zorinsky, 37, 62, 117 ff., 135 ff., 168 f., 202 f. Printed in Great Britain by Richard Clay Sons, Limited, bungay, suffolk.
- 75. FOOTNOTES: 1 In March, 1918, the Bolsheviks changed their official title from “Bolshevist Party” to that of “Communist Party of Bolsheviks.” Throughout this book, therefore, the words Bolshevik and Communist are employed, as in Russia, as interchangeable terms. 2 A prominent pre-revolutionary journal. 3 The Bolsheviks assert that I lent the National Centre financial assistance. This is unfortunately untrue, for the British Government had furnished me with no funds for such a purpose. I drew the Government’s attention to the existence of the National Centre, but the latter was suppressed by the Reds too early for any action to be taken. 4 Trotzky, by Dr. G. A. Ziv, New York, Narodopravsto, 1921, p. 93. 5 Ibid., p. 26. 6 In such company I was regarded as an invalid, suffering in body and mind from the ill-treatment received at the hands of a capitalistic Government. The story ran that I was born in one of the Russian border provinces, but that my father, a musician, had been expelled from Russia for political reasons when I was still young. My family had led a nomadic existence in England, Australia, and America. The outbreak of the war found me in England, where I was imprisoned and suffered cruel treatment for refusal to
- 76. fight. Bad food, brutality, and hunger-striking had reduced me physically and mentally, and after the Revolution I was deported as an undesirable alien to my native land. The story was a plausible one and went down very well. It accounted for mannerisms and any deficiency in speech. It also relieved me of the necessity of participation in discussions, but I took care that it should be known that there burned within me an undying hatred of the malicious Government at whose hands I had suffered wrong. 7 Published in the New York Times, August 24, 1921. Transcriber’s Note: Obvious printer errors corrected silently. Inconsistent spelling and hyphenation are as in the original.
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