Optimal load management of autonomous power systems in conditions of water shortage

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 14, No. 1, February 2024, pp. 99~109
ISSN: 2088-8708, DOI: 10.11591/ijece.v14i1.pp99-109  99
Journal homepage: http://ijece.iaescore.com
Optimal load management of autonomous power systems in
conditions of water shortage
Firdavs Rahimov1
, Alifbek Kirgizov1
, Murodbek Safaraliev2
, Inga Zicmane3
, Nikita Sergeev4
,
Pavel Matrenin4,5
1
Department of Electric stations, Tajik Technical University named after academic M. S. Osimi, Dushanbe, Tajikistan
2
Department of Automated Electrical Systems, Ural Federal University, Yekaterinburg, Russia
3
Faculty of Electrical and Environmental Engineering, Riga Technical University, Riga, Latvia
4
Department of Industrial Power Supply Systems, Novosibirsk State Technical University, Novosibirsk, Russia
5
Department of Electrical Engineering, Ural Federal University, Yekaterinburg, Russia
Article Info ABSTRACT
Article history:
Received Jul 3, 2023
Revised Aug 31, 2023
Accepted Sep 6, 2023
The issues of optimizing the operation of micro hydropower plants in
conditions of water scarcity, performed by additional connection to the grid
of an energy storage system and wind power turbine, as well as optimal load
management, are considered. It is assumed that the load of the system is a
concentrated autonomous power facility that consumes only active power.
The paper presents a rigorous mathematical formulation of the problem, the
solution of which corresponds to the minimum cost of an energy storage
system and a wind turbine, which allows for uninterrupted supply of
electricity to power facilities in conditions of water shortage necessary for
the operation of micro hydropower plants (under unfavorable hydrological
conditions). The problem is formulated as a nonlinear multi-objective
optimization problem to apply metaheuristic stochastic algorithms. At the
same time, a significant part of the problem is taken out and framed as a
subproblem of linear programming which will make it possible to solve it by
a deterministic simplex method that guarantees to find the exact global
optimum. This approach will significantly increase the efficiency of solving
the entire problem by combining metaheuristic algorithms and taking into
account expert knowledge about the problem being solved.
Keywords:
Domain-specific optimization
problem
Electrical load
Energy storage system
Micro hydropower plants
Optimal control
Wind power turbine
This is an open access article under the CC BY-SA license.
Corresponding Author:
Pavel Matrenin
Department of Electrical Engineering, Ural Federal University, Yekaterinburg, Russia
19 Mira Street, Ekaterinburg 620002, Russia
Email: p.v.matrenin@urfu.ru
1. INTRODUCTION
Recently, close attention has been paid to small-scale energy using renewable energy sources. The
interest in the use of renewable energy sources (solar, wind, river water.) is explained by the lack of fuel
purchases and the possibility of power energy supply to hard-to-reach areas. The latter is especially important
for countries with mountainous terrain and sparsely populated areas, where laying a power grid is
economically impractical. At the same time, these countries and regions are characterized by a large
hydropower potential, which has led to the accelerated development of small and micro hydropower plants
(HPPs) in them. The active construction and operation of these stations, which, as a rule, do not have
reservoirs, is also associated with the desire to avoid [1], [2].
Micro HPPs (power up to 100 kW) can be installed almost anywhere. The hydraulic unit includes a
water intake device, a power unit and an automatic control device. Micro HPPs are used as sources of power

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 14, No. 1, February 2024: 99-109
100
energy for farms, country villages, farms and small industries in hard-to-reach areas, where it is unprofitable
to lay networks [3], [4]. The main advantage of small hydropower is safety from an environmental point of
view. During the construction of facilities in this industry and their further operation, there are no harmful
effects on the quality and properties of water. Modern stations have simple designs and are fully automated,
i.e., they do not require the presence of a person during operation. The electric power of micro hydroelectric
power plants meets the requirements of regulatory documents on voltage and frequency, and the stations are
able to operate autonomously. The total service life of the station is more than 40 years (at least 5 years
before the overhaul) [5], [6].
As is known, there is no common universal solution for the construction of microgrids in isolated
areas. The layout of systems depends on financial and economic conditions, logistical opportunities and
potential for the use of relevant renewable energy source (RES) on the ground, and the availability of
relevant technological competences of engineering companies building microgrids. At the same time, there is
an obvious tendency of transition from simpler solutions to more complex and technically advanced ones.
For instance, the initial stage of evolution of "old" local systems is the addition of additional generating
capacities based on RES to the diesel generator. The most common and popular types of such systems are
wind-diesel and solar-diesel complexes. Since the generation of wind and solar power plants is stochastic, it
is quite logical to move to the next stage, the efficiency of the system is increased by energy storage. For
example, the lack of generation at night and the need for load balancing during the day do not allow
photovoltaic generation to replace more than 20-30% of diesel capacity. The use of energy storage can
extend the period of "clean" electricity use during the day and further reduce diesel consumption, as well as
improve overall system reliability. As the cost of energy storage decreases in the future, it will increasingly
be used in isolated power systems, reducing dependence on imported energy resources [7]–[10].
The problem of optimal load management in modern distribution systems is becoming particularly
relevant in the era of Smart grid and Microgrid. In particular, this is due to the fact that the operation of
autonomous power systems with a high degree of renewable energy sources can create significant system
balancing problems. As a consequence, from the point of view of cost minimization, decentralized
integration of renewable energy sources on the basis of Smart grid and Microgrid smart energy systems is
currently seen as the most promising way to improve the stability and reliability of such systems [7]–[10].
Existing research in this area can be divided into several categories. A significant part of the work is
aimed at development, optimization and integration of individual elements or devices of the power system
such as multi terminal interlinking converters (ILCs) [11], [12], maximum power point tracker controllers
[13]–[15], different other types of battery controllers [14], controller of six pulse three phase rectifier [15],
maximum power point tracking [16], generators [17]. Another area of research is the optimization of
placement from the modes of work of various elements, for example, reactive power compensation units
[18]. The resulting optimization problems are often solved using population algorithms. In 2023, Myintzu
et al. [19] presents a technique to allocate a shunt capacitor using the particle swarm optimization algorithm.
A method of the optimal allocation of an energy storage system using the wild geese algorithm is proposed in
[20]. Metaheuristic optimization algorithms are also used to solve optimization problems at a higher level, for
example an economic dispatch problem in microgrid system [21] or a designing power system stabilizer [22].
Synchronization, ensuring optimally balanced operation of such hybrid generation facilities
integrated into microgrids, is a complex engineering task requiring the use of appropriate automation and
software. Improvement of these automation tools and software, increasing the efficiency of interaction of all
elements of microgrids, is an important task of their further development. A number of works are devoted to
the problems of optimizing the operation of generating stations of power supply systems using renewable
energy sources, as well as to the technical and economic assessment of power supply to autonomous
consumers. They mainly either provide economic justification of the efficiency of connection to the
centralized power supply or consider the possibility of using local sources of small-scale energy [23]–[30].
The authors of many works on similar topics, using mathematical modeling methods, propose to
create a technical and economic model to analyze the feasibility of large energy complementarity between
different stations: to solve the problem of uninterrupted power supply at photovoltaic power plants at night,
the authors [29], [30] propose to use the theory of complementarity of hydro and solar energy, which allows
to solve the problem of unstable solar energy generation in the dark and at night, when solar insolation is less
than the average daily. To reduce risks, increase the reliability and stability of such power systems, they are
additionally equipped with energy storage devices [31]–[34]. As for the complementarity between wind and
HPPs, everything depends on the average wind speed, so due to the instability of wind resources, the authors
[35], [36] combined pumped storage power plants with wind power plants, striving for an optimal mode of
complementary operation and maximizing profits [35].
In this paper, the authors consider the optimization of the operation of a hybrid autonomous system:
micro hydropower plant, energy storage system and wind power plant in conditions of severe water shortage.

Int J Elec & Comp Eng ISSN: 2088-8708 
Optimal load management of autonomous power systems in conditions of water shortage (Firdavs Rahimov)
101
Optimization is performed in order to maintain a stable level of power generation and is also carried out by
optimal load management of the following power objects: a group of private houses, an apartment building, a
farm, a small industrial enterprise with predominant consumption of active power. The load power is
conditionally divided into two components: strictly defined and variable, i.e., variable during the
optimization process. At the same time, power consumption schedules have a daily maximum and vary
depending on the day of the week.
2. METHOD
2.1. Accepted assumptions and initial data
The autonomous power system considered in the work is presented in a simplified form in Figure 1.
Power energy generation is produced by a wind turbine and micro HPPs, there is also an energy storage
system (ESS) device in the power system containing the energy storage itself and a converter connecting it to
the grid. For the sake of certainty, will be assumed that a battery of accumulators or supercapacitors acts as
an energy storage device [36], which, however, is not mandatory in this study.
Figure 1. The model of the considered autonomous power system
When creating a mathematical model for optimal load management of an autonomous power
system, the following assumptions are taken as a basis:
− The consumption schedule 𝑃𝑙𝑜𝑎𝑑(𝑡) is heterogeneous in time both during the day and for different days of
the optimization interval, which is assumed to be equal to 168 hours–one week. A typical view of the
dependence of 𝑃𝑙𝑜𝑎𝑑(𝑡) during the day is shown in Figure 2, where the average experimental curve of the
energy consumption of the village in winter is presented;
− The power of the load can be conditionally represented as the sum of strictly specified 𝑃𝑐𝑜𝑛𝑠𝑡(𝑡) and
variable 𝑃𝑣𝑎𝑟(𝑡) parts, of which only the latter can change during optimization. Obviously: 𝑃𝑙𝑜𝑎𝑑(𝑡) =
𝑃𝑐𝑜𝑛𝑠𝑡(𝑡) + 𝑃
𝑣𝑎𝑟(𝑡);
− Since the total amount of power energy 𝑊𝑙𝑜𝑎𝑑 = ∫ 𝑃𝑙𝑜𝑎𝑑(𝑡)𝑑𝑡
𝑇
0
consumed during the optimization
interval T is known, respectively, the average power consumption 𝑃
𝑎𝑣𝑒 =
𝑊𝑙𝑜𝑎𝑑
𝑇
and, in addition, the value
of energy consumed by the variable part of the load 𝑊𝑙𝑜𝑎𝑑,𝑣𝑎𝑟 = ∫ 𝑃
𝑣𝑎𝑟(𝑡)𝑑𝑡
𝑇
0
can be set;
− The initial 𝑊𝑏𝑎𝑡(0) and the final value of energy 𝑊𝑏𝑎𝑡(𝑇) = 𝑊𝑏𝑎𝑡(0) + ∫ 𝑃𝑏𝑎𝑡(𝑡)𝑑𝑡
𝑇
0
of the ESS are the
same, from which it follows that ∫ 𝑃𝑏𝑎𝑡(𝑡)𝑑𝑡
𝑇
0
= 0;
− The best mode of power energy generation for micro HPPs is the mode in which the minimum amount of
water is consumed in the absence of a power shortage in the load nodes;
− The power generated by the wind turbine into the network 𝑃𝑤𝑖𝑛𝑑(𝑡) is set based on the average statistical
values for the winter period of the corresponding area. The average value of this power can be defined as
𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑 =
1
𝑇
∫ 𝑃𝑤𝑖𝑛𝑑(𝑡)𝑑𝑡
𝑇
0
;
− Losses in the elements of the autonomous power system under consideration are not taken into account,
this simplifying assumption can be easily removed and accepted here so as not to overload the
presentation with details.

 ISSN: 2088-8708
102
Figure 2. Experimental dependence 𝑃𝑙𝑜𝑎𝑑(𝑡) of village consumption in winter
2.2. Problem formulation
The goal of optimizing the local power system can be formulated as follows. It is necessary to
control the variable part of the load 𝑃
𝑣𝑎𝑟(𝑡) and the energy storage 𝑃𝑏𝑎𝑡(𝑡) in such a way that two conditions
are met simultaneously:
− The limit mode of power generation of micro HPPs takes place when their generating capacity does not
change over time: 𝑃
𝑔(𝑡) = 𝑃𝑔,0 = 𝑐𝑜𝑛𝑠𝑡 . This condition corresponds to the situation of the winter period,
when the power energy produced by micro HPPs is in short supply, and consumption is always the
maximum possible, that is, uneven load means underutilization of available water;
− With a given power energy consumption 𝑊𝑙𝑜𝑎𝑑, the installed capacities of the wind turbine 𝑃𝑤𝑖𝑛𝑑,𝑚𝑎𝑥,
ESS 𝑃𝑏𝑎𝑡,𝑚𝑎𝑥 and ESS capacity 𝑊𝑏𝑎𝑡,𝑚𝑎𝑥 should be minimal, this condition ensures the minimum cost of
the entire system.
The formulated optimization goal can be written as (1).
( )
,max
,max
,max
,
,max ,max ,max
min,
min,
min,
,
0, 0, 0.
wind
bat
bat
g ave
wind bat bat
P
P
W
P
P P W

 →

 →

→


 =


  

(1)
Here the variation δ is defined as (2).
𝛿 = 𝑚𝑎𝑥
𝑡∈[0,𝑇]
|𝑃𝑔,𝑎𝑣𝑒−𝑃𝑔(𝑡)|
𝑃𝑔,𝑎𝑣𝑒
, 𝑃
𝑔,𝑎𝑣𝑒 =
1
𝑇
∫ 𝑃
𝑔(𝑡)𝑑𝑡,
𝑇
0
(2)
and Δ, the value of this variation specified in the equality type constraint.
The solution of this nonlinear multi-objective optimization problem (1) should be performed under a
constraint 𝛥 = 𝛿(𝑃
𝑔,𝑎𝑣𝑒), the calculation of which significantly exceeds the complexity of the calculation of
the minimized functions. Therefore, let's consider this restriction itself and the way it is calculated in more
detail. The equality of the variation Δ to zero corresponds to the fact that the micro HPP operates in an
optimal mode. However, in this case, the values𝑃𝑤𝑖𝑛𝑑,𝑚𝑎𝑥, 𝑃𝑏𝑎𝑡,𝑚𝑎𝑥, 𝑊𝑏𝑎𝑡,𝑚𝑎𝑥 can be large and the whole
system as a whole will be very expensive. Reducing the installed capacity of the wind turbine 𝑃𝑤𝑖𝑛𝑑,𝑚𝑎𝑥, ESS
𝑃𝑏𝑎𝑡,𝑚𝑎𝑥, as well as the ESS 𝑊𝑏𝑎𝑡,𝑚𝑎𝑥 of batteries reduces the cost of the system, but increasingly loads
micro hydropower plants and, starting with some values, leads to the inability to maintain power energy
generation 𝑃
𝑔 at a constant level and, consequently, the appearance of non-zero values of variation𝛿(𝑃
𝑔,𝑎𝑣𝑒).
Of considerable interest is the solution of the multi-purpose task at various specified levels of
variation Δ. The higher this level, the lower the cost of the entire system, but at the same time, the less
efficient the micro hydropower plant works. The compromise between these mutually contradictory criteria
depends on the specific situation (technical capabilities available for the implementation of the project, long-
term plans for the development of this local network, the planned amount of funding) of the customer. Let's
move on to the method of calculating the constraint 𝛥 = 𝛿(𝑃
𝑔,𝑎𝑣𝑒). In fact, it is necessary to specify an
algorithm for calculating the optimal 𝑃
𝑔(𝑡). Since with a known 𝑃
𝑔(𝑡), calculating the constraint is not a
0
20
40
60
80
100
0 12 24 36 48 60 72 84 96 108 120 132 144
Active
power,
kW
Time, hours

103
problem and in this case the optimality is understood in the sense that using all the capabilities of the local
power system, namely: control of the variable part of the load 𝑃
𝑣𝑎𝑟(𝑡), control of the ESS 𝑃𝑏𝑎𝑡(𝑡) and using
the energy 𝑃𝑤𝑖𝑛𝑑(𝑡) received from the wind turbine 𝑃𝑤𝑖𝑛𝑑(𝑡) such values 𝑃
𝑣𝑎𝑟(𝑡), 𝑃𝑏𝑎𝑡(𝑡), 𝑃
𝑔(𝑡) must be
found at which the smallest deviation 𝑃
𝑔(𝑡) from its average value 𝑃
𝑔,𝑎𝑣𝑒 is possible for a given time
interval [0, 𝑇].
The complexity of task (1) can be somewhat reduced if the ratio of the cost of a kilowatt of installed
power of a 𝑆𝑤𝑖𝑛𝑑 wind turbine and an 𝑆𝑏𝑎𝑡 ESS is known. Let
𝑆𝑤𝑖𝑛𝑑
𝑆𝑏𝑎𝑡=𝑚
, then, due to the fact that 𝑃𝑤𝑖𝑛𝑑,𝑚𝑎𝑥 and
𝑃𝑏𝑎𝑡,𝑚𝑎𝑥 are the installed capacities of the wind turbine and ESS, respectively, and the meaning of the task
(1), to find the cheapest optimal configuration of the local power system, we can proceed to (3).
( )  
( )
,max ,max
,max
, , ,
0,
,max ,max ,max
min,
min,
max ( ) ,
0, 0, 0.
wind bat
bat
g ave g ave g g ave
t T
wind bat bat
P m P
W
P P P t P
P P W


+  →


→



= − = 


  


(3)
It is a nonlinear multi-objective problem, therefore the solution of (3) is the Pareto set (set of all Pareto
efficient situations).
3. RESULTS AND DISCUSSION
Thus, “inside” (high-level) the nonlinear multi-objective problem formulated above, it is a multiple
solution to a simpler problem needed: determining the optimal one 𝑃
𝑔(𝑡) and calculating using it: 𝛿(𝑃
𝑔,𝑎𝑣𝑒).
Let us now consider the formulation of the “internal” (low-level) problem. The initial data for the task are:
− Dependencies 𝑃𝑐𝑜𝑛𝑠𝑡(𝑡) and 𝑃𝑤𝑖𝑛𝑑(𝑡), their characteristic form is shown in Figure 2;
− The total amount of power energy 𝑊𝑙𝑜𝑎𝑑 consumed during T and the total amount of energy 𝑊𝑙𝑜𝑎𝑑,𝑣𝑎𝑟
consumed by the variable part of the load.
The statement of the problem also includes a restriction on the variables of the task and the
relationships between them. The power balance [23] in the considered network is expressed by the ratio:
∀𝑡∈[0,𝑇]: 𝑃𝑐𝑜𝑛𝑠𝑡(𝑡) + 𝑃
𝑣𝑎𝑟(𝑡)
⏟
𝑃𝑙𝑜𝑎𝑑(𝑡)
= 𝑃
𝑔(𝑡) + 𝑃𝑤𝑖𝑛𝑑(𝑡) + 𝑃𝑏𝑎𝑡(𝑡). (4)
The load must consume energy 𝑊𝑙𝑜𝑎𝑑 during the time T. In this case, the part of the energy of the load that is
consumed by the variable part of the load 𝑊𝑙𝑜𝑎𝑑,𝑣𝑎𝑟 in accordance with the assumptions adopted above is also
given in (5):
𝑊𝑙𝑜𝑎𝑑 = ∫ 𝑃𝑙𝑜𝑎𝑑(𝑡)𝑑𝑡
𝑇
0
, 𝑊𝑙𝑜𝑎𝑑,𝑣𝑎𝑟 = ∫ 𝑃
𝑣𝑎𝑟(𝑡)𝑑𝑡
𝑇
0
. (5)
given as the initial data, and the value of is actually determined. For the ESS:
∀𝑡∈[0,𝑇]: −𝑃𝑏𝑎𝑡𝑏𝑎𝑡,𝑚𝑎𝑥𝑏𝑎𝑡,𝑚𝑎𝑥
(6)
∀𝑡∈[0,𝑇]: 0 ≤ 𝑊𝑏𝑎𝑡(𝑡) ≤ 𝑊𝑏𝑎𝑡,𝑚𝑎𝑥 (7)
The latter inequality will continue to be used in another equivalent form.
∀𝑡∈[0,𝑇]: 0 ≤ ∫ 𝑃𝑏𝑎𝑡(𝑡)𝑑𝑡
𝑡
0
≤ 𝑊𝑏𝑎𝑡,𝑚𝑎𝑥 (8)
due to the fact that 𝑊𝑏𝑎𝑡(𝑇) = 𝑊𝑏𝑎𝑡(0):
∫ 𝑃𝑏𝑎𝑡(𝑡)𝑑𝑡
𝑡
0
= 0. (9)
The variable component 𝑃
𝑣𝑎𝑟(𝑡) of the power of the load must be non-negative, as well as the power 𝑃
𝑔(𝑡)
produced by the micro HPP is positive and cannot exceed the total installed capacity of generators.

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104
∀𝑡∈[0,𝑇]: 0 ≤ 𝑃𝑣𝑎𝑟(𝑡) ≤ 𝑃𝑔𝑔,𝑚𝑎𝑥𝑣𝑎𝑟,𝑚𝑎𝑥.
(10)
3.1. Transition to discrete time
Let’s introduce into consideration the vectors 𝑃𝑣𝑎𝑟, 𝑃𝑏𝑎𝑡, 𝑃
𝑔, 𝑃𝑤𝑖𝑛𝑑, 𝑃𝑙𝑜𝑎𝑑, 𝑃𝑐𝑜𝑛𝑠𝑡, the elements of
which are the values, respectively, of the values of 𝑃
𝑣𝑎𝑟(𝑡), 𝑃𝑏𝑎𝑡(𝑡), 𝑃𝑛𝑒𝑡(𝑡), 𝑃𝑤𝑖𝑛𝑑(𝑡), 𝑃𝑙𝑜𝑎𝑑(𝑡), 𝑃𝑐𝑜𝑛𝑠𝑡(𝑡) at
discrete moments of time 𝑡𝑘: {𝑡1 = 0; 𝑡𝑘 = 𝑡𝑘−1 + ℎ; 𝑡𝑁 = 𝑇}, where h = 10 minutes is the observation step
in the calculations. Then 𝑃
𝑣𝑎𝑟, 𝑃𝑏𝑎𝑡 and similarly for all introduced vectors:
𝑃
𝑣𝑎𝑟 = [𝑃𝑣𝑎𝑟,1, 𝑃𝑣𝑎𝑟,2, … , 𝑃𝑣𝑎𝑟,𝑁]
𝑇
, 𝑃𝑏𝑎𝑡 = [𝑃𝑏𝑎𝑡,1, 𝑃𝑏𝑎𝑡,2, … , 𝑃𝑏𝑎𝑡,𝑁]
𝑇
. (11)
In (11), T is the transposition symbol and, for the sake of brevity, the notation 𝑃𝑣𝑎𝑟,𝑘 = 𝑃
𝑣𝑎𝑟(𝑡𝑘), 𝑃𝑏𝑎𝑡,𝑘 =
𝑃𝑏𝑎𝑡(𝑡𝑘) is adopted, which will then be used for all vectors. the unknown vector quantities of the task are the
vectors 𝑃𝑣𝑎𝑟, 𝑃𝑏𝑎𝑡, 𝑃
𝑔. Let's combine them into one vector of unknown Y:
𝑌 = [
𝑃
𝑣𝑎𝑟
𝑃𝑏𝑎𝑡
𝑃
𝑔
] (12)
Let's write now the ratios (3)–(10), using the introduced constraint vectors, starting with equality type
constraints (4), (5), (9). The ratio (4) will be rewritten in such a way that only known quantities are in the
right part:
∀𝑡∈[0,𝑇]: 𝑃
𝑣𝑎𝑟(𝑡) + 𝑃𝑏𝑎𝑡(𝑡) + 𝑃
𝑔(𝑡) = 𝑃𝑐𝑜𝑛𝑠𝑡(𝑡) − 𝑃𝑤𝑖𝑛𝑑(𝑡).
from where for vectors:
[𝐸, 𝐸, 𝐸] ⋅ [
𝑃𝑣𝑎𝑟
𝑃𝑏𝑎𝑡
𝑃
𝑔
] = [𝑃𝑐𝑜𝑛𝑠𝑡 − 𝑃𝑤𝑖𝑛𝑑], (13)
where Е is the unit matrix. Let's rewrite (5) in such a way that only known quantities are in the right part:
∫ (𝑃
𝑔(𝑡) + 𝑃𝑏𝑎𝑡(𝑡))𝑑𝑡
𝑇
0
= 𝑊𝑙𝑜𝑎𝑑 − ℎ ∑ 𝑃𝑤𝑖𝑛𝑑,𝑘
𝑘=1008
𝑘=1 .
Replacing integrals with sums, the following equation for the elements of vectors will be obtained:
ℎ ∑ (𝑃𝑔,𝑘 + 𝑃𝑏𝑎𝑡,𝑘)
𝑘=1008
𝑘=1
= 𝑊𝑙𝑜𝑎𝑑 − ℎ ∑ 𝑃𝑤𝑖𝑛𝑑,𝑘.
𝑘=1008
𝑘=1
The second equality in (5) can also be written for elements of vectors in the form:
𝑊𝑙𝑜𝑎𝑑,𝑣𝑎𝑟 = ℎ ∑ 𝑃𝑣𝑎𝑟,𝑘
𝑘=1008
𝑘=1
.
for further application, it is convenient to present the latter relations in matrix form:
[
1, 0, 0
0, 1, 1
] ⋅ [
𝑃𝑣𝑎𝑟
𝑃𝑏𝑎𝑡
𝑃
𝑔
] = [
ℎ ∑ 𝑃𝑣𝑎𝑟,𝑘
𝑘=1008
𝑘=1
ℎ
− ∑ 𝑃𝑤𝑖𝑛𝑑,𝑘
𝑘=1008
𝑘=1
], (14)
In the ratio (13), h = 10 minutes is the time step of the grid (respectively, one week = 168 hours or 1008 ten-
minute intervals), 0 is the row of the matrix containing 1008 zeros, 1 is the row of the matrix containing 1008
units. Similarly, to (14), (9) for the components of vectors will have the form:

105
ℎ ∑ 𝑃𝑏𝑎𝑡,𝑘
𝑘=1008
𝑘=1
= 0.
or in matrix form:
[0, 1, 0] ⋅ [
𝑃
𝑣𝑎𝑟
𝑃𝑏𝑎𝑡
𝑃
𝑔
] = 0. (15)
3.2. Reducing the procedure for calculating constraints to a linear programming problem
Let's move on to the constraints of the type of inequalities (5), (7) and (9). Constraints (5) and (9) set
the boundaries of the change of the unknowns of the task. The corresponding inequalities for the elements of
vectors have the form:
−𝑃𝑏𝑎𝑡, 𝑘1,1008𝑏𝑎𝑡,𝑚𝑎𝑥𝑏𝑎𝑡,𝑚𝑎𝑥
(16)
0 ≤ 𝑃𝑣𝑎𝑟,𝑘 ≤ 𝑃𝑔, 𝑘1,1008𝑔,𝑚𝑎𝑥𝑣𝑎𝑟,𝑚𝑎𝑥
(17)
Constraint (8), when passing to vectors, generates 1008 double inequalities:
0 ≤ ℎ ∑ 𝑃𝑏𝑎𝑡,𝑘
𝑘=𝑆
𝑘=1 ≤ 𝑊1,1008𝑏𝑎𝑡,𝑚𝑎𝑥
the matrix form of these inequalities has the form:
[
𝑁 𝐷 𝑁
𝑁 − 𝐷 𝑁
] ⋅ [
𝑃𝑣𝑎𝑟
𝑃𝑏𝑎𝑡
𝑃
𝑔
] ≤
𝑊𝑏𝑎𝑡,𝑚𝑎𝑥
ℎ[1𝑇
0𝑇
][
1 0 ⋯ 0
1 1 ⋯ 0
⋮ ⋮ ⋱ ⋮
1 1 ⋯ 1
]
⏟
1008
(18)
in the ratio (18), N is a square matrix of size 1008×1008, all elements of which are zeros.
Thus, the formation of constraints of the type of equalities and inequalities is completed and you can
proceed to writing the objective function. As the objective function, as it was defined in the formulation of
the problem, some functional 𝐹(𝑃
𝑣𝑎𝑟, 𝑃𝑏𝑎𝑡, 𝑃
𝑔) should act that characterizes the heterogeneity of power
energy generation at micro HPPs. The next task is connected with looking for a minimum of this functional.
let's introduce into consideration two vectors A and B, whose components 𝛼𝑘 and 𝛽𝑘 are non-negative:
𝐴 = [𝛼1, 𝛼2, … , 𝛼1008]𝑇
, 𝐵 = [𝛽1, 𝛽2, … , 𝛽1008]𝑇
, 𝛼𝑘 ≥ 0, 𝛽𝑘 ≥ 0, 𝑘 = 1,1008. (19)
using 𝑃
𝑎𝑣𝑒 =
𝑇
, the average power consumed by the load over the time interval T, 𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑 =
1
𝑇
∫ 𝑃𝑤𝑖𝑛𝑑(𝑡)𝑑𝑡
𝑇
0
, the average power supplied to the network by the wind turbine and considering that
∫ 𝑃𝑏𝑎𝑡(𝑡)𝑑𝑡
𝑇
0
= 0 the inequalities for the power consumed from the network can be written as follows:
(𝑃
𝑎𝑣𝑒 − 𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑) − 𝛼𝑘 ≤ 𝑃𝑔,𝑘 ≤ (𝑃𝑎𝑣𝑒 − 𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑) + 𝛽𝑘, 𝑘 = 1,1008,
which, with a tendency to zero 𝛼𝑘 and 𝛽𝑘, guarantee the constancy of or, what is the same thing, the equality
of all among themselves. The matrix form of the notation of the last inequalities has the form:
[
𝑁 𝑁 −𝐸 −𝐸 𝑁
𝑁 𝑁 𝐸 𝑁 −𝐸
] ⋅
[
𝑃𝑣𝑎𝑟
𝑃𝑏𝑎𝑡
𝑃
𝑔
𝐴
𝐵 ]
≤ (𝑃𝑎𝑣𝑒 − 𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑) ⋅ [1𝑇
0𝑇
], (20)
Thus, a linear functional can be taken as a minimized functional:

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106
𝐹(𝑃𝑣𝑎𝑟, 𝑃𝑏𝑎𝑡, 𝑃
𝑔) = ∑ (𝛼𝑘 + 𝛽𝑘)
𝑘=1008
𝑘=1
𝑃𝑣𝑎𝑟, 𝑃𝑏𝑎𝑡, 𝑃𝑔
→ 𝑚𝑖𝑛. (21)
The problem (21) of minimizing a linear functional with linear constraints of the type of the above
equations and inequalities is classified as a linear programming problem. It should be noted additionally that
the result of the above classification is extremely important. It is explained by the fact that the methods of
solving this problem are well studied, moreover, the existence of its only solution providing the global
minimum of the functional F is proved. Let's bring the problem to the standard form used, for example, in the
MATLAB package:
𝐹(𝑋) = ⟨𝐶, 𝑋⟩
𝑋
→ 𝑚𝑖𝑛,
{
𝐴𝑒𝑞𝑋 = 𝑏𝑒𝑞,
𝐴𝑖𝑛𝑒𝑞𝑋 ≤ 𝑏𝑖𝑛𝑒𝑞,
𝑋𝑚𝑎𝑥𝑚𝑖𝑛{
In the expression of the minimized functional, brackets ⟨⋅,⋅⟩ denote the scalar product of vectors. The vector
X of unknowns and the vector of coefficients С from the task have the form:
𝑋 = [𝑃𝑣𝑎𝑟, 𝑃𝑏𝑎𝑡, 𝑃
𝑔, 𝐴, 𝐵]𝑇
,
𝐶 = [0, 0, 1, 1, 1]𝑇
.
In result of combination of the relations (13)–(15) into one matrix equality, a matrix 𝐴𝑒𝑞 and a vector 𝑏𝑒𝑞 are
formed:
[
𝐸 𝐸 𝐸 𝑁 𝑁
1 0 0 0 0
0 1 1 0 0
0 1 0 0 0
]
⏟
𝐴𝑒𝑞
⋅
[
𝑃
𝑣𝑎𝑟
𝑃𝑏𝑎𝑡
𝑃
𝑔
𝐴
𝐵 ]
≤
[
𝑃𝑐𝑜𝑛𝑠𝑡 − 𝑃𝑤𝑖𝑛𝑑
ℎ ∑ 𝑃𝑣𝑎𝑟,𝑘
𝑘=1008
𝑘=1
ℎ
− ∑ 𝑃𝑤𝑖𝑛𝑑,𝑘
𝑘=1008
𝑘=1
0 ]
⏟
𝑏𝑒𝑞
. (22)
by combining the relations (18) and (20) into one matrix inequality, the matrix 𝐴𝑖𝑛𝑒𝑞 and vector 𝑏𝑖𝑛𝑒𝑞 will be
formed:
[
𝑁 𝑁 −𝐸 −𝐸 𝑁
𝑁 𝑁 𝐸 𝑁 −𝐸
𝑁 𝐷 𝑁 𝑁 𝑁
𝑁 −𝐷 𝑁 𝑁 𝑁
]
⏟
𝐴𝑖𝑛𝑒𝑞
⋅
[
𝑃
𝑣𝑎𝑟
𝑃𝑏𝑎𝑡
𝑃
𝑔
𝐴
𝐵 ]
≤
[
−(𝑃𝑎𝑣𝑒 − 𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑) ⋅ 1𝑇
(𝑃
𝑎𝑣𝑒 − 𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑) ⋅ 1𝑇
𝑊𝑏𝑎𝑡,𝑚𝑎𝑥
ℎ𝑇
0𝑇
[]
]
⏟
𝑏𝑖𝑛𝑒𝑞
(23)
Equalities (16), (17), and (19) make it possible to form vectors 𝑋𝑚𝑖𝑛 and 𝑋𝑚𝑎𝑥:
𝑋𝑚𝑖𝑛 = [0, −𝑃𝑏𝑎𝑡,𝑚𝑎𝑥 1, 0, 0, 0],
𝑋𝑚𝑎𝑥 = [𝑃
𝑣𝑎𝑟,𝑚𝑎𝑥 1, 𝑃𝑏𝑎𝑡,𝑚𝑎𝑥 1,𝑃
𝑔,𝑚𝑎𝑥 1, (𝑃
𝑎𝑣𝑒 − 𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑)1,(𝑃
𝑔,𝑚𝑎𝑥 − (𝑃
𝑎𝑣𝑒 − 𝑃𝑎𝑣𝑒,𝑤𝑖𝑛𝑑))1]
Thus, the internal issue of determining the optimal 𝑃
𝑔, and accordingly 𝑃
𝑔(𝑡), is reduced to the standard form
for linear programming tasks and the task (1) is fully posed.
4. CONCLUSION
The mathematical formulation of the problem is formulated of optimal load management for an
autonomous power system with a micro HPP, a wind power turbine, and an energy storage system in
conditions of water deficit. The difference between the proposed formulation of the problem is the use of the
stability of the generation of micro-HPPs as one of the criteria for optimality and minimum costs for a wind
power turbine, and an energy storage system as the second criterion. The resulting nonlinear multi-criteria
problem with a complex system of constraints is supposed to be solved using stochastic metaheuristic
optimization methods such as the Genetic algorithm or Swarm Intelligence algorithms, which are able to

107
effectively solve optimization problems of technical systems with nonlinear constraints. The subtask of
determining the optimal micro HPP generation schedule for the given parameters of the power system is
considered separately. By taking into account the peculiarities of technological processes in the system and
the transition to a discrete time step, it was possible to convert this subtask into a linear programming
problem without significant losses in accuracy. In turn, this makes it possible to apply a deterministic
solution algorithm, such as the simplex method, and is guaranteed to find the global optimum of the problem.
The next stage of the study will be the approbation of the developed model on the data of a real power system
(Gorno-Badakhshan Autonomous Oblast of the Republic of Tajikistan) with the selection and adaptation of
the high-level stochastic optimization algorithm to solve a multi-criteria problem, as well as the calculation of
the economic effect of the proposed approach for the power system. To do this, a software implementation
will be created, and computational experiments will be conducted on real data.
The proposed method of rational distribution of disconnected power among consumers during the
elimination of emergency situations in the power system allows for an objective assessment of the
capabilities of each consumer and can serve as a basis for introducing market relations, for example, when
developing tariffs differentiated by reliability. The proposed method is based on original mathematical
models of production systems of industrial electricity consumers.
ACKNOWLEDGEMENTS
The research funding from the Ministry of Science and Higher Education of the Russian Federation
(Ural Federal University Program of Development within the Priority-2030 Program) is gratefully
acknowledged.
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BIOGRAPHIES OF AUTHORS
Firdavs Rahimov received the specialist and PhD degrees in electrical
engineering from Tajik Technical University named after academic M. S. Osimi, Tajikistan, in
2003 and 2023 respectively. Currently, he is an associate professor at the Department of
Electric Stations, Tajik Technical University named after academic M. S. Osimi. His research
interests include renewable energy, power generation, battery chargers, circuit breakers,
hydropower plants, and load flow control. He can be contacted at email: rm-firdavs@mail.ru.

109
Alifbek Kirgizov received the specialist degree in electrical engineering from
Tajik Technical University named after academic M. S. Osimi, Tajikistan, in 2003 and PhD
degree in electrical engineering from Tomsk Polytechnic University, Russia in 2017.
Currently, he is an associate professor at the Department of Electric Electric Stations, Tajik
Technical University named after academic M. S. Osimi. Main directions of investigations:
methods of artificial intelligence for planning and optimization of operating regimes of power
systems. He can be contacted at email: alifbek@mail.ru.
Murodbek Safaraliev received the B.S. and M.S. degrees electrical engineering
from Tajik Technical University named after academic M. S. Osimi, Tajikistan, in 2014 and
2016, respectively, and PhD degree in electrical power engineering from Ural Federal
University, Russia in 2022. Currently, he is a researcher at the Department of Automated
Electrical Systems Department, Ural Energy Institute, Ural Federal University, Yekaterinburg,
Russian. His fields of interests include, optimization of the development, and modes of power
system, planning of hybrid renewable energy systems. He can be contacted at email:
murodbek_03@mail.ru.
Inga Zicmane was graduated from Riga Technical University (RTU) in 2000 and
received Dr. sc. ing. in 2005. Since 2000 she works as lector and assistant professor, from
2008 as associate professor of RTU, since 2014 as a professor at the Institute of Power
Engineering of RTU Faculty of Electrical and Environmental Engineering. Her research is
concerned with the quality of education, evaluation of the sensibility of electric power
systems, transient processes, stability of power systems, local power networks, renewable
energy sources. She can be contacted at email: inga.zicmane@rtu.lv.
Nikita Sergeev received a B.S. degree electrical engineering from Novosibirsk
State Technical University, Russia in 2023. He is a laboratory assistant Novosibirsk State
Technical, Novosibirsk, Russia. His current research areas are forecasting and optimization
problems in the power industry. He can be contacted at email: veegresatikin3102@gmail.com.
Pavel Matrenin received the M.S. and Ph.D. degrees information technologies
from Novosibirsk State Technical University, Russia in 2014 and 2018, respectively. He is a
leading researcher at Ural Federal University, Ekaterinburg, Russia and an associate professor
at Novosibirsk State Technical, Novosibirsk, Russia. His current research areas are stochastic
optimization algorithms and machine learning in electric power systems. He can be contacted
at email: p.v.matrenin@urfu.ru.

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