Module-1 Part-2 Unit Commitment and Hydro-Thermal Coordination.pdf

ECE666: Power Systems Operation
Module-1: Power System Economic Operation
Part-2: Unit Commitment and Hydro-Thermal Coordination
Prof. Kankar Bhattacharya
Department of Electrical & Computer Engineering
University of Waterloo, Waterloo, N2L 3G1, Canada
kankar@uwaterloo.ca
Module-1 Part-2 ECE666: Winter 2017
1

Coverage
• The Unit Commitment (UC) problem
• Need of UC, different shut-down modes, spinning reserves
• UC mathematical model
• UC solution: complete enumeration, priority list method
• The Hydro-Thermal Coordination Problem
• Introduction to hydro generation, characteristics, classes
• Hydro Scheduling Problems
• Hydro-Thermal Scheduling
• Energy Scheduling problem
• Short-Term Hydro-Thermal Scheduling problem
2

The Unit Commitment Problem
Module-1 Part-2
3 ECE666: Winter 2017

G1
GN
G2
P1
P2
PN
PD
What is Unit Commitment (UC)?
4
1 6 12 18 24 Hours
PD
Chronological Load Curve
UC is a decision making problem on generating unit
ON / OFF status over a day. This is required because
of the load variations over a day and UC helps
minimize overall operating costs

What is UC? Contd..
• UC is a scheduling activity usually covering a time range
from 24 hours (1 day) to 168 hours (1 week) ahead,
carried out by the system operator in the pre-dispatch
stage
• UC is essential because system load varies significantly over
a day / weekly period
• Hence it is not economical to keep all generators on-line for
the entire duration
• A proper schedule for start up or shut down of the generators
can save costs significantly
5

What is UC? …contd.
• In the UC problem the operator seeks to minimize its
system costs over the period
• While meeting the forecast demand
• To decide the unit up / down status for every hour
• The UC solution provides decisions on:
• Committing a generator- unit ON
• De-committing a generator- unit OFF for each hour
• For the pre-dispatch period, i.e. next 24 hours / 48 hours /
168 hours, etc.
• UC Input
• Hourly system demand forecast for the pre-dispatch period
6

UC Decision Variables




OFF
Unit
0
ON
Unit
1
,k
i
W
7
• A variable Wi,k is introduced which denotes the unit
ON/OFF status decision of generator i at hour k
A Typical UC Solution Matrix
Hour
Gen
1 2 3 4 … … … … 22 23 24
G1 1 1 1 1 1 1 1 1 0 0 0
G2 0 0 0 1 1 1 1 1 1 1 0
.. 0 0 1 1 0 0 1 1 1 1 1
.. -- -- --
GN-1 -- -- --
GN 0 0 1 1 1 1 1 1 1 1 1

• Corresponding to Wi,k, two other decision variable matrices
are required:
• Ui,k denoting generator start-up at an hour
• Vi,k denoting generator shut-down at an hour
UC Decision Variables… contd.





Unit
of
up
Start
No
0
up
-
Start
Unit
1
,k
i
U
8




Unit
of
down
-
Shut
No
0
down
-
Shut
Unit
1
,k
i
V
Hour k 1 2 3 4 5 6 7 8 9
Wi,k
1 1 0 0 1 1 1 0 1
Ui,k
0 0 0 0 1 0 0 0 1
Vi,k
0 0 1 0 0 0 0 1 0

UC Decision Variables… contd
• Therefore, the set of UC decisions include
• Generator ON/OFF (1 or 0) schedule
• Wi,k  i  N, k  T
• Generator start-up and shut-down decisions
• Ui,k and Vi,k  i  N, k  T
• Power generation schedule
• Pi,k  i  N, k  T
• Although this may be deviated from in real-time dispatch
• Spinning reserve allocation
• Hourly import / export schedules, etc.
9

What is Spinning Reserve?
• A reserve available to system operator from amongst its spinning
generating units
• Available within 10 minutes (as per NERC)
• The operator is responsible for maintaining adequate spinning
reserves in the system
• Not only on a total-MW basis but also location of the reserve, taking
into account transmission capacities available in the system
• Operator experience or certain rules are in place for determining
the amount of reserve to be maintained in the system
• This could typically comprise a base component, a fraction of the load
requirement and a fraction of the high operating limit of the largest
on-line unit
10

UC Mathematical Model
• Objective Function: Minimize Total System Cost over the
scheduling horizon (24-hours / 48-hours, etc.), comprising:
• Operating or Fuel Cost
• This is similar to the cost discussed in the ELD problem
• Start-up cost (CUP), Shut-down Cost (CDN)
• These are considered for each start-up/shut-down decision in the scheduling period
• Depends on type of start-up, from its shut-down state
• Shut-down cost is typically a constant cost
•  is a fixed cost associated with unit start-up,  is the cost of a cold start-up, TOFF
is the time for which the unit has been off and  is time-constant representing
cooling speed of the unit
11










  i
OFF
i
T
i
i
i
UP e
C 

 /
1

Linear Generator Cost Representation
• The operation cost Ci(Pi) is represented as a linear function
of Pi
• Ai is the no-load cost, Bi is the incremental cost of unit i
• This is a simplified representation, in practical applications, 2- or 3-
section stepped incremental characteristics may be used
12
PMin PMax P, MW
No-Load
cost
C(P)
Ci(Pi)=No-load cost + Incremental cost x P
= Ai + BiPi

UC Model- Objective Function
• Composite Total System Cost would be
• Non-linear/linear, depending on whether quadratic/ linear fuel
cost function is considered
• For simplicity, linear functions are usually used
• When Wi,k=1, the operational cost is accounted for
• When Ui,k=1, the start-up cost is accounted for
• When Vi,k=1, the shut-down cost is accounted for
• N is the total number of generators
• K is the total scheduling horizon (24-hours/48-hours, etc.)
• A, B are explained next
13
 
  



 
K
k
N
i
k
i
DN
k
i
UP
k
i
i
k
i
i V
C
U
C
P
B
W
A
J i
i
1 1
,
,
,
,

UC Model Constraints
• Demand Supply Balance: Ensures enough generation
capacity is scheduled at an hour to meet the forecasted
demand
• May include pre-decided import (Im,k) / export (Em,k) contracts
with other utilities (M)
• Generation Limits: Power generation from a unit is governed by
the upper and lower limits, PMax and PMin respectively
14
 
  


 
N
i
M
m
k
k
m
k
m
k
i PD
E
I
P
1 1
,
,
,
Max
i
k
i
k
i
Min
i
k
i P
W
P
P
W 


 ,
,
,

UC Model Constraint: Ramp Rate Limits
• RUP is ramp-up and RDN the ramp-down rates of a generator.
They denote the MW increase / decrease allowable across a
given hour as per the unit’s technical characteristic
• For large thermal units these are in the order of ±30% to
±40% of PMax while, hydro / gas turbine units have larger
ramping capabilities
• Let us denote:
• Power generation by
• Ramp up capacity by
• Ramp down capacity by
15

• The ramping range (up and down limits), when the generator is
producing Pk-2 MW at k-2, is shown by the dotted lines
• Pk-1 MW is the optimal schedule determined when these
constraints are applied
16
K-2 k-1 k k+1 k+2
Hours
Pk-2
Pk-1

• The ramping range is moved forward based on Pk-1 obtained
earlier, and applied to Pk
• Pk MW is the optimal schedule determined when these
17
Pk-2
K-2 k-1 k k+1 k+2
Hours
Pk-1
Pk

• The ramping range is moved forward based on Pk obtained
earlier, and applied to Pk+1
• Pk+1 MW is the optimal schedule determined when these
18
K-2 k-1 k k+1 k+2
Hours
Pk-2
Pk-1
Pk Pk+1

• The ramping range is further moved forward based on Pk+1
obtained earlier, and applied to Pk+2
• It is shown here (as an example) that Pk+2 violates the ramp-up
constraints. Thus Pk+2 needs to be re-calculated
19
K-2 k-1 k k+1 k+2
Hours
Violated at hour
K+2
Pk-2
Pk-1
Pk Pk+1
Pk+2

Ramp Rate Limits… contd.
• Ramping constraints
• The constraints link the generation variables of the previous
hour to that of the present hour
• Introduces a dynamic characteristic in the UC models
20
0
0
,
,
1
, 1
,...,
2
,
1
,
0
,
i
k
k
i
i
UP
k
i
k
i
P
P
K
k
i
R
P
P








1
...,
2,
1,
0,
,
,
1
,
, 



  K
k
i
R
P
P i
DN
k
i
k
i

• Minimum Up-Time: Minimum time for which unit has to
remain in ON state before de-committing
• Xk-1
ON : time duration for which unit i has been ON up to hour
k-1
• TON : minimum up time of the unit
• Minimum Down-Time: Minimum time for which unit has to
remain in OFF state before commitment
• Xk-1
OFF : time duration for which unit i has been OFF up to
hour k-1
• TOFF : minimum down time of the unit
UC Model Constraints… contd.
   0
,
1
,
1
, 


 
 k
i
k
i
ON
i
ON
k
i W
W
T
X
21
   0
1
,
,
1
, 


 
 k
i
k
i
OFF
i
OFF
k
i W
W
T
X

Minimum Up-Time Constraint, TON = 4
• Not Satisfied
22
• Satisfied
    0
,
1
,
1
, 


 
 k
i
k
i
ON
i
ON
k
i W
W
T
X
k 1 2 3 4 5 6
Wk 0 0 1 1 1 0
Xk-1
ON 0 0 1 2 3
Xk-1
ON – TON -4 -4 -3 -2 -1
Wk-1 – Wk 0 -1 0 0 1
(Xk-1
ON – TON) x
(Wk-1 – Wk)
0 4 0 0 -1
OK OK OK OK Not
OK
k 1 2 3 4 5 6 7
Wk 0 0 1 1 1 1 0
Xk-1
ON 0 0 1 2 3 4
Xk-1
ON – TON -4 -4 -3 -2 -1 0
Wk-1 – Wk 0 -1 0 0 0 1
(Xk-1
ON – TON) x
(Wk-1 – Wk)
0 4 0 0 0 0
OK OK OK OK OK OK

Minimum Down-Time Constraint, TOFF=3
• Not Satisfied
23
• Satisfied
k 1 2 3 4 5 6
Wk 1 1 0 0 1 1
Xk-1
OFF 0 0 1 2 0
Xk-1
OFF – TOFF -3 -3 -2 -1 -3
Wk – Wk-1 0 -1 0 1 0
(Xk-1
OFF – TOFF)
x (Wk – Wk-1)
0 3 0 -1 0
OK OK OK Not
OK
OK
k 1 2 3 4 5 6
Wk 1 1 0 0 0 1
Xk-1
OFF 0 0 1 2 3
Xk-1
OFF – TOFF -3 -3 -2 -1 0
Wk – Wk-1 0 -1 0 0 1
(Xk-1
OFF – TOFF)
x (Wk – Wk-1)
0 3 0 0 0
OK OK OK OK OK
    0
1
,
,
1
, 


 
 k
i
k
i
OFF
i
OFF
k
i W
W
T
X

UC Constraints… contd.
• Coordination Constraints: Ensures proper transition of UC
states from 0 to 1 and vice versa with unit start-up, shut-
down decisions
• If there is a start-up, Wi,k – Wi,k-1 = 1
• This ensures Ui,k = 1, Vi,k = 0
• If there is a shut-down, Wk – Wk-1 = -1
• This ensures Ui,k = 0, Vi,k = 1
24
i
W
W
K
k
i
W
W
V
U
i
k
k
i
k
i
k
i
k
i
k
i









0
0
,
1
,
,
,
, ...,
2,
1,
,

UC Constraints… contd.
• System Security Constraint: Ensures enough system capacity
committed to meet peak demand while also ensuring spinning
reserve (SR) availability
• RESV denotes the spinning reserve in the system
• Must-run units: Some units have a “must-run” status because of
certain system requirements (voltage support, etc.)
• Crew Constraints: These pertain to the number of units that can
be started at the same time in a particular plant
25
 




N
i
k
k
k
i
Max
i k
RESV
PD
W
P
1
,
MR
i
W k
i 

 ;
1
,

Consolidated UC Model
• Minimize Total System Cost
• Subject to the constraints:
• Demand supply balance
• Maximum and Minimum Generation Limits
• Ramp up and ramp-down limits
• Minimum up-time and down-time constraints
• Coordination constraints
• System security constraints
• Must run units, crew constraints, etc.
• Other constraints which can be considered are:
• Transmission limits
• Environmental emissions, and so on
26

UC Solution Methods
• UC problems are much more complex to solve compared to
the ELD problem
• Because of the presence of binary (0-1) decision variables on
unit status (ON/OFF)
• Depending on system needs and computational tools available,
a utility chooses to use such UC models and solution methods
that suits its requirements
• Complete Enumeration- gives optimal solution but extremely cumbersome to handle
• Priority List Methods- the simplest, but may not be optimal
• Dynamic Programming Based Methods- search technique, can be problematic for
large systems
• Integer Programming
• Lagrange Relaxation
• Genetic Algorithm, Expert Systems, etc.
27

Example-1: Unit Commitment
• Consider a 3-generator system. UC solution is required for a single
hour, when the system load is 550 MW. The unit cost
characteristics are given below.
28
Unit Limits Cost Ch.
1 150<P1<600 561+7.92P1+0.001562P2
1
2 100<P2<400 310+7.85P2+0.00194P2
2
3 50<P3<200 93.6+9.564P3+0.005784P2
3

Solution Example-1a: Complete Enumeration
• Dispatch of generators are obtained using an ELD in each commitment
state
• Case (1-0-0) yields least-cost solution
• This method guarantees least cost solution, but for large systems may
be impossible to calculate
• (2N-1) states to be calculated for N units: More than 33 million states for N=25
29
W1 W2 W3 Total Capacity, MW P1 P2 P3 Total Cost, $
0 0 1 200 Infeasible
0 1 0 400 Infeasible
0 1 1 600 0 400 150 5418
1 0 0 600 550 0 0 5389
1 0 1 800 500 0 50 5497
1 1 0 1000 295 255 0 5471
1 1 1 1200 267 233 50 5617

• Compute Full Load Average Cost (FLACi) of units i Є N
• FLACi is obtained by calculating generating cost when the unit
is operating at full load, PMax; and dividing by the full load
• The “Priority List” is created by ordering the units in
increasing order of their FLACi
• The unit with the lowest FLACi being of highest priority
UC Solution by Priority List Method
 
Max
i
P
P
i
i
i
P
P
C
FLAC
Max
i
i 

30

• Calculate the Full Load Average Cost of the generators
• The priority order so obtained is thus:
• 1st Priority: Unit-2
• 2nd Priority: Unit-1
• 3rd Priority: Unit-3
Solution Example-1b: Priority List Method
 
MWh
FLAC
P
/
$
401
.
9
400
4
.
3760
$
400
P
0.00194
P
85
.
7
310
400
2
2
2
2
2







31
 
MWh
FLAC
P
/
$
1888
.
11
200
76
.
2237
$
200
P
0.005784
P
546
.
9
93.6
200
2
3
3
3
3







 
MWh
FLAC
P
/
$
7922
.
9
600
32
.
5875
$
600
P
0.001562
7.92P
561
600
2
1
1
1
1








Solution Example-1b… contd.
• Using the Priority List we now consider a set of N-states only,
instead of 2N-1 states
• Note: Generators are loaded in priority order, not by ELD
• Gen-2 is loaded up to its maximum capacity of 400 MW
• Then gen-1 is loaded to meet the remaining load of 150 MW
32
Priority W1 W2 W3 Total Capacity, MW P1 P2 P3 Total Cost, $
1st 0 1 0 400 Infeasible -
2nd 1 1 0 1000 150 400 0 5544.55
3rd 1 1 1 1200 Don’t need to calculate this because
the previous state is of higher priority
and is able to meet the demand

Example-1b (Contd.): UC By Priority List
• Observe that the UC schedule for 550 MW load by Priority
List is
• 1-1-0 (150 MW; 400 MW; 0 MW)
• Total System Cost: $5544.55
• Complete Enumeration Method yields a minimum cost of $5389
• The Complete Enumeration method provides the overall
optimal solution
• But it may be extremely large to handle
• Priority List method is simple but may be expensive
33

Example-1b (contd.): UC by Priority List
Priority Order:
• 1st Priority: Unit-2 Load Range: 100 MW – 400 MW
• 2nd Priority: Unit-1 Load Range: 400 MW – 1000 MW
• 3rd Priority: Unit-3 Load Range: 1000 MW – 1200 MW
34
0
200
400
600
800
1000
1200
1400
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Unit-1
Unit-3 Unit-3
Unit-2

Example-2: UC Problem
Unit PMax
MW
PMin
MW
No-Load
Cost
$
Full-load avg.
Cost
$/MWh
Min Up
Time, h
Min Down
Time, h
Start-up
Cost
$
1 80 25 213.0 23.54 4 2 350
2 250 60 585.62 20.34 5 3 400
3 300 75 684.74 19.74 5 4 1100
4 60 20 252.00 28.00 1 1 0.02
35
• Given, U-2 & U-3 are
committed at time k = 0
Hour 1 2 3 4 5 6 7 8
Load, MW 450 530 600 540 400 280 290 500
0
100
200
300
400
500
600
700
1 2 3 4 5 6 7 8
Hour
Demand,
MW

Example-2 … contd.
 
 
 
  4
4
4
3
3
3
2
2
2
1
1
1
8
.
23
252
4575
.
17
74
.
684
9975
.
17
62
.
585
8775
.
20
213
P
P
C
P
P
C
P
P
C
P
P
C








36
• Linear cost characteristic of the unit is assumed
Ci(Pi)=(No-load cost)i
+ (Incremental cost)i x Pi
No Load Cost
PMin PMax
P
F(P)

Example-2: Case (a)- Priority List Method
Hr U1 U2 U3 U4 PD
MW
P1
MW
P2
MW
P3
MW
P4
MW
Cost
$/h
Start up
Cost, $
Total
Cumulative
Cost, $
1 0 1 1 0 450 - 150 300 - 9208 - 9208
2 0 1 1 0 530 - 230 300 - 10647 - 19855
3 1 1 1 0 600 50 250 300 - 12264 350 32469
4 0 1 1 0 540 - 240 300 - 10827 - 43296
5 0 1 1 0 400 - 100 300 - 8307 - 51603
6 0 0 1 0 280 - - 280 - 5573 - 57176
7 0 0 1 0 290 - - 290 - 5747 - 62923
8 0 1 1 0 500 - 200 300 - 10107 400 73430
37
• The priority order is decided by FLAC of the units.
In this case, the order is:
• Unit-3  Unit-2  Unit-1  Unit-4
• We ignore minimum up- and down times

Page-38
State Units
1-2-3-4
Capacity,
MW
1
450 MW
15 1111 690 $9861
$10211
14 1110 630 $9492
$9842
13 0111 610 $9582
$9582
12
State at
Hr-0
0110 550 $9208
$9208
11 1011 440
10 1101 390
9 1010 380
8 0011 360
7 1100 330
6 0101 310
5 0010 300
The initial state is state-12, i.e.
0-1-1-0
From state-12, there are four
feasible states for hour-1, i.e.
states-12, 13, 14 and 15
Transition from State-12 to State-
14, and State-12 to State-15
involves start-up cost of unit-1
Hour
Demand, MW
Stage Cost, $
Cumulative Cost, $
Case-b: Complete Enumeration
Denotes Lowest Cost

Page-39
State Units
1-2-3-4
Capacity,
MW
1
450 MW
2
530
15 1111 690 $9861
$10211
$11300
$20858
14 1110 630 $9492
$9842
$10932
$20490
13 0111 610 $9582
$9582
$11015
$20223
12
State at
Hr-0
0110 550 $9208
$9208
$10647
$19855
11 1011 440
10 1101 390
9 1010 380
8 0011 360
7 1100 330
6 0101 310
5 0010 300
Case-b: Complete Enumeration
Transition from hour-1 to
hour-2 also has four feasible
states.
The cheapest option is
shown by the circled state
The optimal path is shown
by the bold arrows

Page-40
State Units
1-2-3-4
Capacity,
MW
1
450 MW
2
530 MW
3
600 MW
15 1111 690 $9861
$10211
$11300
$20858
$12574
$32779
14 1110 630 $9492
$9842
$10932
$20490
$12264
$32469
13 0111 610 $9582
$9582
$11015
$20223
$12449
$32304
12
State at
Hr-0
0110 550 $9208
$9208
$10647
$19855
11 1011 440
10 1101 390
9 1010 380
8 0011 360
7 1100 330
6 0101 310
5 0010 300
Transition from hour-
2 to hour-3 has three
feasible states.
The cheapest option
is shown by the
circled state
The optimal path is
shown by the bold
arrows

Page-41
State Units
1-2-3-4
Capacity,
MW
1
450 MW
2
530 MW
3
600 MW
4
540 MW
15 1111 690 $9861
$10211
$11300
$20858
$12574
$32779
$11480
$44170
14 1110 630 $9492
$9842
$10932
$20490
$12264
$32469
$11112
$43776
13 0111 610 $9582
$9582
$11015
$20223
$12449
$32304
$11195
$43499
12
State at Hr-0
0110 550 $9208
$9208
$10647
$19855
$10827
$43131
11 1011 440
10 1101 390
9 1010 380
8 0011 360
7 1100 330
6 0101 310
5 0010 300

Page-42
State Units
1-2-3-4
Capacity,
MW
1
450 MW
2
530 MW
3
600 MW
4
540 MW
5
400 MW
15 1111 690 $9861
$10211
$11300
$20858
$12574
$32779
$11480
$44170
9313
52444
14 1110 630 $9492
$9842
$10932
$20490
$12264
$32469
$11112
$43776
8942
52441
13 0111 610 $9582
$9582
$11015
$20223
$12449
$32304
$11195
$43499
8675
51806
12
State at
Hr-0
0110 550 $9208
$9208
$10647
$19855
$10827
$43131
$8307
51438
11 1011 440 $8533
52014
10 1101 390
9 1010 380
8 0011 360
7 1100 330
6 0101 310
5 0010 300

Page-43
State Units
1-2-3-4
Capacity,
MW
1
450 MW
2
530 MW
3
600 MW
4
540 MW
5
400 MW
6
280 MW
15 1111 690 $9861
$10211
$11300
$20858
$12574
$32779
$11480
$44170
9313
52444
7218
58656
14 1110 630 $9492
$9842
$10932
$20490
$12264
$32469
$11112
$43776
8942
52441
6839
58277
13 0111 610 $9582
$9582
$11015
$20223
$12449
$32304
$11195
$43499
8675
51806
12
State at
Hr-0
0110 550 $9208
$9208
$10647
$19855
$10827
$43131
$8307
51438
11 1011 440 $8533
52014
10 1101 390
9 1010 380
8 0011 360
7 1100 330 6274/58288
6 0101 310 5993/58006
5 0010 300 5573/57587

Multi-area UC
• UC has been used in multi-area systems where individual utilities
are interconnected by tie lines and dispatch is carried out jointly
• The additional constraint is the inter-area transmission constraint,
which can impose severe restrictions on the optimal solution
• the inter-area transmission lines can be modeled using a linear flow
network model or dc power flow representation
• joint scheduling of multi-area systems can bring about significant
reduction in system costs
• Such systems are seen to be vulnerable to transmission capacity
availability
• a critical parameter in determining the level of savings achievable from joint dispatch
44

Hydro-Thermal Coordination
Module-1 Part-2
45 ECE666: Winter 2017

Introduction
• Systematic co-ordination of a hydro-thermal system is
more complex than scheduling of an all-thermal generation
system
• Hydro-electric plants may be coupled both electrically (all
serve the same load) and hydraulically (outflow from one unit-
is inflow to the next unit)
• Coordination of their operation involves scheduling of water
release
• No two hydraulic systems in the world are alike
• Natural differences in watersheds, differences in storage and release elements
46

Main Features of Hydro-Electric Generation
• High capital costs
• Especially from civil engineering works
• Operational and maintenance costs are very low
• Useful life  50 yrs compared to 25-30 yrs for thermal plants
• Energy cost is almost independent of the load factor
• Can be start-up and synchronized in a few minutes
• Conserve fuel
• Independent of fuel transportation bottlenecks, no air pollution
• Long gestation periods- about 7 years
• From project commissioning to commercial operation
• Generally these power stations are part of multi-purpose
projects
• irrigation, flood control, etc.
• Provides system black-start capability services
47

Classification of Hydro-electric Plants
Classification according to
Water flow regulation a) Run-off the river
b) Reservoir plants
Load a) Base load plants
b) Peak load plants
c) Pumped-storage plants
Head a) High head plants (head > 100 m)
b) Medium head plants (head 30- 100 m)
c) Low head plants (head < 30 m)
48

Simple Configuration of a Hydro Plant
49
Gross Head
Forebay
Penstock
Afterbay
Reservoir
Generator
Turbine
Draft Tube

Hydroelectric Generators
• Input: Volume of water
per unit of time
• Typically represented in
acre-ft/h
• 1 acre-ft  1233.5 m3
• Output: Electric power,
MW
• For a constant head hydro,
the required water
discharge rate increases
linearly with power output,
up to the rated capacity
50
• Beyond the rated capacity,
the water discharge
requirement increases,
since efficiency falls
Output, P MW
Input,
q
acre-ft/h
Net head = constant
 
H
P
f
q 

Hydroelectric Generators… contd.
• I/O characteristics for
variable head hydro
generation is shown
• Variation in forebay or
afterbay elevations is
significant
• Scheduling tasks are
more complex
51
Output, P MW
Input,
Q
acre-ft/h
Net head = c1
Net head = c2
Net head = c3
c1< c2 < c3
Maximum output

Power Output from a Hydro Unit
   
m
h
s
m
P 









 2
3 81
.
9
m
kg
density,
s
cu.m
discharge,

52
• Power = rate of doing work. For hydro energy it is rate of
change of potential energy
• P =  x m x g x h/s












 m
s
m
h
Q 2
s
kg
81
.
9
1000

kW
,
81
.
9
Watts
,
9810
Qh
Qh





 Notations:
 Efficiency of the unit
m mass of water, kg
g acceleration due to gravity, m/s2
h water head, m
s time in sec
Q water discharge in cu.m/s

Types of Scheduling Problems
• Scheduling problems depend on hydro-thermal mix
• 100% hydro, no thermal (eg. Norway)
• Involves scheduling of water release to satisfy all hydraulic constraints
and meet the power demand
• Pre-dominantly hydro (Brazil, Quebec)
• Scheduling carried out to dispatch thermal production minimizing the cost
• It is basically an energy scheduling problem
• Closer balance between hydro and thermal or where hydro is a
small fraction of total capacity (e.g. Sweden, Ontario, etc.)
• Schedules to minimize thermal generation costs, recognizing all diverse
hydraulic constraints, mathematical models are more elaborate
53

• For pre-dominantly hydro systems with some
thermal generation. Objective is to minimize
total cost of thermal generation
• J = Cost of Steam Energy
• The hydro unit can supply the load by itself, PH, for a
part of the time. For any period, j
• But, over the total time horizon, Tmax
• Where nj: total number of hours in period j
Hydro Energy Scheduling Problem
,...,T
,
j
P
P j
j Load
Max
H 2
1


54
  


 
T
j
T
j
j
j
Load
j
j
H n
P
n
P
1 1
Steam
Hydro
PS
PH
PD
 

T
j
Max
j T
n
1
Hydro
Steam
PLoad
TMax, h
TS
*
PS
*
P, MW

• If all hydro energy available is used up, and remaining load
energy required is met by steam unit, the steam energy
ESteam can be written as:
• The first term on the right hand side is the load energy, and
the second term is the hydro energy. Also, that,
• Now, if the steam unit runs for Ns time intervals,
Energy Scheduling Problem… contd.
 


s
j
N
j
j
S
Steam n
P
E
1
55
 

s
N
j
Max
j T
n
1
 
 




T
j
T
j
j
H
j
Load
Steam n
P
n
P
E j
j
1 1

• Objective function:
• Subject to,
• The Lagrangian is,
• The conditions for optimum:
• This implies that the thermal unit should operate at constant
incremental cost of  for each of the Ns intervals.
• Which means
Energy Scheduling Problem… contd.
 



s
N
j
j
j
S
Steam n
P
E
1
0
56
 
 







 




 
s s
j
j
N
j
N
j
j
S
Steam
j
S n
P
E
n
P
C
F
1 1

  j
N
j
S n
P
C
J
s
j



1
Minimize,
 
 
s
j
S
j
S
s
j
S
j
S
j
S
, ..., N
,
j
λ
dP
P
dC
, ..., N
,
j
dP
P
dC
P
F
2
1
2
1
0












s
S
j
S N
j
P
P ,...,
2
,
1
*
*




• Applying the optimality condition, the total cost function can now
be expressed as follows:
• Considering an usual quadratic function for steam generation cost,
given by,
• The total cost J is:
• Also, we know that,
Scheduling of Energy… contd.
  2
*
*
*
S
P
C S
S cP
bP
a 


57
      S
*
S
s
N
j
s
N
j
j
S
j
S T
P
C
n
P
C
n
P
C
J 

 




 
1 1
*
*
  S
S
S T
cP
bP
a
J 2
*
*



*
*
1 1
*
S
Steam
S
Steam
S
S
s
N
j
s
N
j
j
S
j
j
S
P
E
T
E
T
P
n
P
n
P




 

 

• Hence we have,
• By minimizing J, the optimum generation schedule for PS is
obtained
• Now if fc is the fuel cost and H(.) is the unit heat input-output
characteristic, then we can write
c
a
P
cE
P
aE
dP
dJ
S
Steam
S
Steam
S






*
2
*
*
0
58
   
c
S
S
S
S
C
S
f
cP
bP
a
P
H
P
H
f
P
C
2
)
(





  











*
2
*
*
S
Steam
S
S
P
E
cP
bP
a
J

• The unit heat-rate characteristic, signifying the amount of
heat required for every unit of generation, is given as:
• The heat-rate is minimum when,
• Thus the optimum (least cost) generation schedule for the
steam unit is when it operates at the least heat-rate
condition, which is the point of highest efficiency
59
    
















 S
S
C
S
S
S
C
S
S
P
c
b
P
a
f
P
c
P
b
a
P
f
P
P
H 1
1 2
 
*
2
Thus,
0
S
S
S
S
S
S
P
c
a
P
c
P
a
P
P
H
dP
d













Hydro-Thermal Scheduling with Storage Limitations



max
1
j
j
j
jC
n
J
60
• The more general short-term hydro-thermal scheduling problem
involves minimizing the thermal operating cost subject to given
hydraulic constraints including reservoir limits
• The basic objective is minimization of thermal generation costs
over nj scheduling intervals
Vj
qj
rj sj
r = inflow
V = volume of reservoir
q = discharge
s = spillage
j = index for time interval

• Subject to the following constraints
• Where,
Hydro-Thermal Short-term Scheduling … contd.
 
specified
hour,
an
at
discharge
Fixed
q
limits
Discharge
specified
volume
reservoir
Ending
V
specified
volume
reservoir
Starting
V
T
j
balance
Demand
0
discharge
water
Total
j
max
min
j
0
j
max
1















 




j
j
E
T
j
S
j
j
S
j
H
j
j
j
TOT
j
H
j
j
Q
q
q
q
V
V
P
P
PD
Q
P
q
n
61
 

max
1
j
j
Max
j T
n

• The Lagrangian can be formulated as follows-
• For j = k,
• If network losses are considered-
• The Lagrangian is now-
Hydro-Thermal Short-term Scheduling … contd.
      







 

  




 
T
j
TOT
j
H
j
j
j
j
T
j
j
S
j
H
j
D
j
j
S
j Q
P
q
n
P
P
P
P
C
n
F
1
max
1 1


62
  0
0
,
,











k
H
k
H
k
k
k
k
H
k
k
S
k
S
k
k
S
dP
P
dq
n
P
F
dP
dC
n
P
F



0
, 


 j
j S
H
j
Loss
j P
P
P
PD
      







 

  





 
T
j
TOT
j
H
j
j
j
j
T
j
j
S
j
H
j
Loss
j
D
j
j
S
j Q
P
q
n
P
P
P
P
P
C
n
F
1
max
1 1



Short-term Scheduling … contd.
 
 
k
k
H
k
Loss
k
k
H
k
H
k
k
k
S
k
Loss
k
k
S
k
S
k
P
P
dP
P
dq
n
dP
P
dP
P
dC
n












,
,
63
• The coordination equations can be written as follows:

- Iterative Solution Method
64
  ?
, 



 j
Loss
j
H
S P
PD
P
P j
j
Choose starting value of k, k, Ps,k
Set J =1
Solve coordination equations
Find qj(PHj)
?
1
1






T
j
TOT
j
j q
q
n
Project
new 
NO
YES
STOP
YES
NO
Project
new 
J = Jmax?
J = J+1
NO
YES

Pumped Storage Hydro Units
• Designed to save fuel costs
• By serving the peak load with hydro
energy
• The water is pumped back to the
reservoir
• At light load periods
• May involve separate pumps &
turbines
• Recent development of reversible
pump turbines
65
F
Steam
Pumped
Hydro
PSj
PHk
q
PHi

Pumped Storage Hydro
66
Added cost rate
Savings rate
Cost
PSteam
Load
Pump Energy
Generation Energy
Time

Model for Pumped Storage Hydro
Notations
• rj = rate of water inflow in pumped storage reservoir, acre-ft/hour
• Vj = volume of reservoir at the end of time interval j (acre-ft)
• VS = volume at the start of the scheduling horizon (acre-ft)
• VE = volume at the end of the scheduling horizon (acre-ft)
• qj = water discharge rate, when operating as generator (acre-ft/hour)
• wj = water pumping rate, when operating as pump (acre-ft/hour)
• j = index for hour
• k = intervals of generation
• i = intervals of pumping
• PS = generation from steam generating unit, MW
• PH = generation from hydro generating unit, MW
67

• Objective of the pumped storage hydro scheduling
problem is to minimize the sum of the hourly costs (J) for
steam generation over the day
• The function J includes the total costs of generation during
the generation interval (Ck) as well as the total cost incurred
during the pumping interval (Ci) because of purchased
electricity from grid, etc.
Model for P.S. Hydro… contd.
  i
k
j
S
j C
C
P
C
J j





24
1
68

Generation Interval {k}
• The electrical and hydraulic constraints are:
• Lagrangian is formulated as follows:
0
1 


  k
k
k
k q
r
V
V
69
0



 k
k
k H
S
Loss
k P
P
P
PD Electrical Demand balance
Reservoir Hydraulic Balance
 
 
k
k
k
k
k
H
S
k
Loss
k
k
k
q
r
V
V
P
P
P
PD
C
F k
k









1
,
1



Pumping Interval {i}
• The electrical and hydraulic constraints are:
• Lagrangian is formulated as follows:
   
i
i
i
i
i
S
i
Loss
i
H
i
i w
r
V
V
P
P
PD
P
C
F i
i








 1
,
2 

70
0



 i
i
i H
S
Loss
i P
P
P
PD Electrical Demand balance
Reservoir Balance
0
1 


  i
i
i
i w
r
V
V

• Reservoir Volume constraints
• The volume of the reservoir at the start of the scheduling
period is specified, VS
• The volume of the reservoir at the end of the scheduling
period is constrained, VE
S
j
j V
V 
0
71
E
j
j V
V 
24

 
 
 
  































i
i
i
i
i
S
H
i
Loss
i
i
i
k
k
k
k
k
S
H
k
Loss
k
k
k
w
r
V
V
P
P
P
PD
C
q
r
V
V
P
P
P
PD
C
F
i
i
k
k
1
,
1
,




72
• The composite Lagrangian function can be written as
follows-

Some Important References
• H. Ma and S. M. Shahidehpour, Unit commitment with transmission security and voltage
constraints, IEEE Trans. Power Systems, May ’99
• F. N. Lee and Q. Feng, Multi-area unit commitment, IEEE Trans. Power Systems, May’92
• S. Vemouri and L. Lemonidis, Fuel constrained unit commitment, IEEE Trans. Power Systems, Feb.
’92
• S. Y. Lai and R. Baldick, Unit commitment with ramp multipliers, IEEE Trans. on Power Systems,
Feb. ’99
• J. Batut and A. Renaud, Daily generation scheduling optimization with transmission constraints: A
new class of algorithms”, IEEE Trans. Power Systems, Aug.’92
• R. Baldick, The generalized unit commitment problem, IEEE Trans. Power Systems, Feb. ’95
• G. B. Sheble and G. N. Fahd, Unit commitment literature synopsis, IEEE Trans. Power Systems,
Feb. ’94
• C. Wang and S.M. Shahidehpour, Power generation scheduling for multi-area hydro-thermal
systems with tie-line constraints, cascaded reservoirs and uncertain data, IEEE Trans. on Power
Systems, Aug.’93
• N. J. Redondo and A. J. Conejo, Short-term hydro-thermal coordination by Lagrangian relaxation:
solution of the dual problem, IEEE Trans. Power Systems, Feb. 1999
• G. Skugge, J. A. Bubenko and D. Sjelvgren, Optimal seasonal scheduling of natural gas units in a
hydro-thermal power system, IEEE Trans. Power Systems, May 1994
• G. X. Luo, H. Habibollahzadeh and A. Semlyen, Short-term hydro-thermal dispatch detailed model
and solutions, IEEE Trans. Power Systems, Nov. 1989
73

Module-1 Part-2 Unit Commitment and Hydro-Thermal Coordination.pdf

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