DSpace at VNU: Cuckoo search algorithm for short-term hydrothermal scheduling

Cuckoo search algorithm for short-term hydrothermal schedulingThang Trung Nguyena, Dieu Ngoc Vob,⇑, Anh Viet Truongc a Faculty of Electrical and Electronics Engineering, Ton Duc Thang Un

Cuckoo search algorithm for short-term hydrothermal scheduling

Thang Trung Nguyena, Dieu Ngoc Vob,⇑, Anh Viet Truongc

a

Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, 19 Nguyen Huu Tho Str., 7th Dist., Ho Chi Minh City, Viet Nam

b

Department of Power Systems, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Str., 10th Dist., Ho Chi Minh City, Viet Nam

c

Faculty of Electrical and Electronics Engineering, University of Technical Education Ho Chi Minh City, 1 Vo Van Ngan Str., Thu Duc Dist., Ho Chi Minh City, Viet Nam

h i g h l i g h t s

A new cuckoo search method is

proposed for solving hydrothermal

scheduling problem

There are few control parameters for

the proposed method

The proposed method can properly

deal with nonconvex short-term

hydrothermal scheduling problem

The robustness and effectiveness of

the proposed method have been

validated for different test systems

Hydro Plants Minimize fuel cost

Thermal Plants

Electrical Load

Cuckoo Search Algorithm

Calculate slack thermal and hydro units Evaluate fitness funcon Generate new eggs via Lévy Flights Calculate slack thermal and hydro units Evaluate fitness funcon Discover alien egg And randomize Generate new eggs

Randomize a number of nests, N d

Iteraon=1

Iteraon = Iteraon + 1 No

Iteraon=Max

STOP Yes

10 1

10 2

10 3 3.75

3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 5

Number of iterations = 1000

Calculate slack thermal and hydro units Evaluate fitness funcon

Host bird’s nest Alien bird’s egg

Host bird’s egg

Net solution

Article history:

Received 19 February 2014

Received in revised form 4 July 2014

Accepted 7 July 2014

Keywords:

Cuckoo search algorithm

Short-term hydrothermal scheduling

Convex fuel cost function

Nonconvex fuel cost function

Lévy flights

a b s t r a c t

This paper proposes a cuckoo search algorithm (CSA) for solving short-term fixed-head hydrothermal scheduling (HTS) problem considering power losses in transmission systems and valve point loading effects in fuel cost function of thermal units The CSA method is a new meta-heuristic algorithm inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds of other species for solving optimization problems The advantages of the CSA method are few con-trol parameters and effective for optimization problems with complicated constraints The effectiveness

of the proposed CSA has been tested on different hydrothermal systems and the obtained test results have been compared to those from other methods in the literature The result comparison has shown that the CSA can obtain higher quality solutions than many other methods Therefore, the proposed CSA can

be an efficient method for solving short-term fixed head hydrothermal scheduling problems

Ó 2014 Elsevier Ltd All rights reserved

1 Introduction

The short term hydro-thermal scheduling (HTS) problem is to

determine the power generation among the available thermal

and hydro power plants so that the total fuel cost of thermal units

is minimized over a schedule time of a single day or a week satis-fying both hydraulic and electrical operational constraints such as the quantity of available water, limits on generation, and power

for solving the hydrothermal scheduling problem such as effective conventional method (ECM) based on Lagrange multiplier theory

http://dx.doi.org/10.1016/j.apenergy.2014.07.017

0306-2619/Ó 2014 Elsevier Ltd All rights reserved.

⇑Corresponding author Tel.: +84 88 657 296x5730; fax: +84 88 645 796.

E-mail address: vndieu@hcmut.edu.vn (D.N Vo).

Applied Energy

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a p e n e r g y

[1], k–c iteration method, dynamic programming (DP) [2],

linearized and solved for the water availability constraint

sepa-rately from generating units, thus the Lagrangian multiplier

associ-ated with water availability constraint is separately from the

outputs of generating units Based on the obtained Lagrangian

mul-tiplier of water constraint, the Lagrangian mulmul-tiplier associated

with power balance constraint is determined and the outputs of

thereafter the k iterations are invoked for the given power demand

at each interval of the scheduling period The DP method is a

popular optimization method implemented for solving the

hydro-thermal scheduling problem However, computational and

dimen-sional requirements in the DP method increase drastically with

DP method, the LR method is more efficient for dealing with

large-scale problems However, the LR method may suffer to

dual-ity gap oscillation resulting from the dual problem formulation,

leading to divergence for some problems with operation limits

and non-convexity of incremental heat rate curves of generators

In the decomposition and coordination method, the problem is

decomposed into thermal and hydro sub-problems and they are

solved by network flow programming and priority list based

dynamic programming methods In order to solve the

hydrother-mal scheduling problem, MIP requires linearization of equations

whereas the decomposition and coordination method may

encounter difficulties when dealing with the operation limits and

non-linearity of objective function and/or constraints The

Newton’s method is computationally stable, effective, and fast for

solving a set of nonlinear equations Therefore, it has a high

potential for implementation on optimization problems such

eco-nomic load dispatch in hydrothermal power systems However,

the Newton’s method mainly depends on the formulation and

inversion of Jacobian matrix, leading to restriction of applicability

on large-scale problems In general, these conventional methods

can be applicable for only the HTS problems with differentiable

fuel cost function and constraints

Recently, several artificial intelligence techniques have been

proposed for solving the hydrothermal scheduling problems such

[9–12], differential evolution (DE)[13], artificial immune system

and EP algorithms are evolutionary based method for solving

opti-mization problems However, the essential encoding and decoding

schemes in the both methods are different In the GA method, the

crossover and mutation operations required to diversify the

off-spring may be detrimental to actually reaching an optimal

solu-tion In this regard, the EP is more likely better when overcoming

these disadvantages In the EP method, the mutation is a key

search operator which generates new solutions from the current

some of the multimodal optimization problems is its slow conver-gence to a near optimum The DE method has the ability to search

in very large spaces of candidate solutions with few or no assump-tions about the considered problem However, the DE method is difficult to deal with large-scale problems with slow or no conver-gence to the near optimum solution The AIS method is one of the efficient methods for solving the nonconvex short-term hydrother-mal scheduling The most important step of the AIS method is the application of the aging operator to eliminate the old antibodies, to maintain the diversity of the population, and to avoid the prema-ture convergence The advantages of the AIS method are few parameters and small maximum number of iterations However, the AIS method is also difficult to deal with large-scale problems like other meta-heuristic search methods Optimal gamma based

val-ues of the hydro plants are considered as the GA variables and the k iterations over the scheduling period can be called to find the ther-mal and hydro generations for each chromosome in the population

to calculate the value of the fitness function Therefore, the number

of the GA variables is drastically reduced and does not even depend

method is an efficient neural network for dealing with optimiza-tion problems However, it encounters a difficulty of predetermin-ing the synaptic interconnections among neurons which may lead

to constraint mismatch if the weighting coefficients in its energy function are not carefully selected Moreover, the HNN method also suffers slow convergence to optimal solution and the con-straints of the problem must be linearized when applying in

usually suffer slow convergence to the near optimum solution for the HTS problems

The cuckoo search algorithm (CSA) developed by Yang and Deb

problems inspired from the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds

of other species To verify the effectiveness of the CS algorithm, Yang and Deb compared its performance with particle swarm opti-mization (PSO) and GA for ten standard optiopti-mization benchmark

method has been outperformed both PSO and GA methods for all test functions in terms of success rate in finding optimal solution and the number of required objective function evaluations The highlighted advantages of the CSA method are fine balance of randomization and intensification and less number of control parameters Recently, CSA has been successfully applied for solving non-convex economic dispatch (ED) problems considering genera-tor and system characteristics including valve point loading effects, multiple fuel options, prohibited operating zones, spinning reserve

Nomenclature

valve-point effects

unit j during the scheduling period

the ED problems in practical power system and micro grid power

tested on many systems and obtained better solution quality than

several methods in the literature such as HNN, GA, EP, Taguchi

method, biogeography-based optimization, and PSO, etc

higher solution quality than DE and PSO On the other hand, for

Photovoltaic system, CSA has been used to track Maximum Power

other methods, namely Perturbed and Observed (P&O) and PSO

The evaluations include (1) gradual irradiance and temperature

changes, (2) step change in irradiance and (3) rapid change in both

irradiance and temperature These tests are carried out for both

outperforms both P&O and PSO with respect to tracking capability,

transient behavior and convergence Consequently, CSA is an

effi-cient method for solving optimal problems

In this paper, a cuckoo search algorithm (CSA) is proposed for

solving short-term fixed head HTS problem considering power

losses in transmission systems and valve point loading effects in

fuel cost function of thermal units The effectiveness of the

pro-posed CSA has been tested on different hydrothermal systems

and the obtained results have been compared to those from other

methods available in the literature such as existing GA (EGA), and

2 Problem formulation

The objective of the HTS problem is to minimize the total fuel

cost of thermal generators while satisfying hydraulic, power

bal-ance, and generator operating limits constraints The short-term

for-mulated as follows

The objective is to minimize the total cost of thermal generators

[14]:

m¼1

i¼1

ð1Þ subject to:

– Power balance constraint: The total power generation from

ther-mal and hydro plants must satisfy the total load demand and

power loss in each subinterval:

i¼1

j¼1

where the power losses in transmission lines are calculated using

Kron’s formula:

i¼1

X

N 1 þN 2

j¼1

i¼1

– Water availability constraint: The total available water discharge

of each hydro plant for the whole scheduled time horizon is

lim-ited by:

m¼1

where the rate of water flow from hydro plant j in interval m is

determined by:

– Generator operating limits: Each thermal and hydro units have their upper and lower generation limits:

Psi;min6Psi;m6Psi;max; i ¼ 1; ; N1; m ¼ 1; ; M ð6Þ

Phj;min6Phj;m6Phj;max; j ¼ 1; ; N2; m ¼ 1; ; M ð7Þ

3 Cuckoo search algorithm for short-term fixed-head HTS problem

3.1 Cuckoo search algorithm The cuckoo search algorithm (CSA) was developed by Yang and

algo-rithms, the CSA is a new and efficient population-based heuristic evolutionary algorithm for solving optimization problems with the advantages of simple implement and few control parameters This algorithm is based on the obligate brood parasitic behavior

of some cuckoo species combined with the Lévy flight behavior

of some birds and fruit flies There are mainly three principal rules

1 Each cuckoo lays one egg (a design solution) at a time and dumps its egg in a randomly chosen nest among the fixed num-ber of available host nests

2 The best nests with high quality of egg (better solution) will be carried over to the next generation

3 The number of available host nests is fixed, and a host bird can

can either throw the egg away or abandon the nest so as to build a completely new nest in a new location

As a further approximation, the last assumption can be

nests (with new random solutions) For maximization problems, the quality or fitness of a solution can simply be proportional to the value of the objective function Other forms of fitness can be defined in a similar way to the fitness function in genetic algorithms

3.2 Calculation of power output for slack thermal and hydro units

In this research, the output power for slack hydro units is calcu-lated based on the availability water constraint while the power output of thermal units is determined using the power balance constraint

Suppose that the water discharges of the first (M  1)

unit j at subinterval M is calculated using the available water con-straint (4) as follows:

m¼1

!

Therefore, the power output of hydro unit j at subinterval m is determined using (5):

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

To guarantee that the power balance constraint (2) is always satisfied, a slack thermal unit is arbitrarily selected and thus its power output will be dependent on the power output of the

Suppose that the power outputs of (N1 1) thermal unit and N2

hydro units at subinterval m are known, the power output of the

slack thermal unit 1 is calculated by:

i¼2

j¼1

PL;m¼ BTT;11P2s1;m

i¼2

BTT;1iPsi;mþ 2XN 2

j¼1

BTH;1jPhj;mþ BT;01

!

i¼2

j¼2

Psi;mBTT;ijPsj;mþXN 2

i¼1

j¼1

Phi;mBHH;ijPhj;m

i¼2

j¼1

Psi;mBTH;ijPhj;mþXN 1

i¼2

j¼1

where

Bij¼ BTT;ij BTH;ij

BHT;ij BHH;ij





units,

Substituting (11) into (10), a quadratic equation is obtained:

where

i¼2

BTT;1iPsi;mþ 2XN2

j¼1

i¼2

j¼2

Psi;mBTT;ijPsj;mþXN 2

i¼1

j¼1

Phi;mBHH;ijPhj;m

i¼2

j¼1

Psi;mBTH;ijPhj;mþXN 1

i¼2

j¼1

i¼2

j¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 4AC p

3.3 Implementation of cuckoo search algorithm

implemented for solving the short-term fixed-head HTS problem

as follows

3.3.1 Initialization

power out of thermal unit i at subinterval m corresponding to nest

corresponding to nest d

In the CSA, each egg can be regarded as a solution which is ran-domly generated in the initialization Therefore, each element in nest d of the population is randomly initialized as follows:

Psi;m;d¼ Psi;minþ rand1 ðPsi;max Psi;minÞ; i ¼ 2; ; N1; m

qj;m;d¼ qj;minþ rand2 ðqj;max qj;minÞ; j ¼ 1; ; N2; m

in [0, 1]

first (M  1) subintervals At the subinterval M, the nest d only

Based on the initialized population of the nests, the fitness func-tion to be minimized corresponding to each nest for the considered problem is calculated:

m¼1

i¼1

FiðPsi;m;dÞ þ Ks

m¼1

ðPs1;m;d Plims1 Þ2

j¼1

cor-responding to nest d in the population

The limits for the slack thermal unit 1 and water discharge at the subinterval M in (19) are determined as follows:

Ps1;max if Ps1;m;d>Ps1;max

Ps1;min if Ps1;m<Ps1;min

Ps1;m;d otherwise

8

>

<

>

:

ð20Þ

qj;max if qj;M;d>qj;max

qj;min if qj;M;d<qj;min

qj;M;d otherwise

8

>

<

>

:

ð21Þ

the maximum and minimum water discharges of hydro plant j The initialized population of the host nests is set to the best

among all nests in the population

3.3.2 Generation of New Solution via Lévy Flights The new solution is calculated based on the previous best nests via Lévy flights In the proposed CSA method, the optimal path for

new solution by each nest is calculated as follows:

ð22Þ

wherea> 0 is the updated step size; rand3is a normally distributed

deter-mined by:

d ¼vrxðbÞ

where

m¼ randx

pb 2

 

2

2

4

3 5

1=b

ð25Þ

gamma distribution function For the newly obtained solution, its

lower and upper limits should be satisfied according to the unit’s

limits:

qj;m;d¼

qj;max if qj;m;d>qj;max

qj;min if qj;m;d<qj;min

qj;m;d otherwise

8

>

ð27Þ

Psi;m;d¼

Psi;max if Psi;m;d>Psi;max

Psi;min if Psi;m;d<Psi;min

Psi;m;d otherwise

8

>

ð28Þ

the best nest Gbest

3.3.3 Alien egg discovery and randomization

The action of discovery of an alien egg in a nest of a host bird

prob-lem similar to the Lévy flights The new solution due to this action

can be found out in the following way:

where K is the updated coefficient determined based on the

proba-bility of a host bird to discover an alien egg in its nest:



ð30Þ

its lower and upper limits should be also satisfied constraints (27)

and (28) The value of the fitness function is calculated using (19)

and the nest corresponding to the best fitness function is set to

the best nest Gbest

3.3.4 Stopping criteria

The algorithm is stopped when the number of iterations (Iter)

The flowchart of the proposed CSA for solving the problem is

4 Numerical results The proposed CSA has been tested on five systems with qua-dratic fuel cost function of thermal units and two systems with nonconvex fuel cost function of thermal units The proposed algo-rithm is coded in Matlab platform and run on a 2 GHz PC with 2 GB

of RAM

4.1 Selection of parameters

In the proposed CSA method, three main parameters which

Among the three parameters, the number of nests has signifi-cantly effects on the obtained solution quality Generally, the

optimal solution is obtained However, the computational time for obtaining the solution for case with the large numbers is long

By experiments, the number of nests in this paper is set from 10 to

quality and computation time It is chosen based on the complexity and scale of the considered problems For the test systems in this

Initialize population of host nests

) (

* , max , min 1

min ,

m

) (

* ,max ,min

2 min ,

m

Calculate all thermal and hydro generations based on the initialization

- Set X d to Xbest dfor each nest

- Set the best of all Xbest d to Gbest

- Set iteration counter iter = 1.

Generate new solution via Lévy flights

new d d

new

d Xbest rand X

X = +α× 3×Δ

- Check for limit violations and repairing

- Calculate all hydro and thermal generation outputs

- Evaluate fitness function to choose new Xbest d and Gbest

Discover alien egg and randomize

X =Xbest + × ΔK X

- Check for limit violations and repairing

- Calculate all hydro and thermal generation outputs

- Evaluate fitness function to choose new Xbest d and Gbest

Iter<Iter max?

Stop

Iter = Iter + 1

Yes

No

Fig 1 The flowchart of CSA for solving ST-HTS.

systems to 2500 for the large-scale systems The value of the

optimal solutions for a problem For the complicated or large-scale

problems, the selection of the value for the probability has an

obvi-ous effect on the optimal solution In contrast, the effect of the

probability is inconsiderable for the simple problems, that is

differ-ent values of the probability can also lead the same optimal

[0, 1] Besides, the value of distribution factor b needs be

deter-mined has a significant impact on solution quality of CSA and it

is suggested in the range [0.3,1.99] as in the Mantegna’s algorithm

suggested range does not have much effect on the final solution

of economic dispatch problem and it has been fixed at 1.5 for all

test systems Therefore, it also fixed at 1.5 for all test systems in

this paper

4.2 Systems with quadratic fuel cost function of thermal units

In this test case, the proposed CSA is tested on five systems The

and one hydro plant for the first system, one thermal and two

hydro plants for the second system, two thermal and two hydro

plants for the third system, and one thermal and one hydro plants

ther-mal plants and two hydro plants with four scheduling subintervals

of nests is set to 20 for system 1, 50 for systems 2 and 3, 10 for

sys-tem 4, and 100 for syssys-tem 5 The maximum number of iterations

for the CSA is set to 1000, 1500, 2500, 300 and 1000 for the five

in the range from 0.1 to 0.9 with a step of 0.1 For each case, the

proposed CSA is run 10 independent trials and the obtained results

including minimal total cost, average total cost, maximal total cost,

standard deviation, and average computational time are given in

Tables 1–5 For the first four systems, the proposed CSA method

opti-mal solution is 0.9

respec-tively 0.4, 0.6, 0.9, 0.1 and 0.9 for the five systems corresponding

maximum cost and average computational time obtained by the

Table 1

Results by CSA for the first system with quadratic fuel cost of thermal units with

different values of P a

P a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)

0.1 96024.6877 96030.083 96053.9166 8.8589314 19.2

0.2 96024.6826 96024.845 96026.0801 0.4141929 19.8

0.3 96024.6817 96024.704 96024.7553 0.0273447 19.1

0.4 96024.6816 96024.684 96024.6914 0.0027169 19.3

0.7 96024.6816 96024.729 96025.1145 0.1286358 19.4

0.9 96024.6816 96024.988 96026.0008 0.4308014 19.3

Table 2

Results by CSA for the second system with quadratic fuel cost of thermal units with

different values of P a

P a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)

Table 3 Results by CSA for the third system with quadratic fuel cost of thermal units with different values of P a

P a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)

Table 4 Results by CSA for the fourth system with quadratic fuel cost of thermal units with different values of P a

P a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)

Table 5 Results by CSA for the fifth system with quadratic fuel cost of thermal units with different values of P a

P a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)

Table 6 Result comparison for the first four test systems with quadratic fuel cost function of thermal units.

System Method Min cost ($) Avg cost ($) Max cost ($) CPU (s)

OGB-GA [9] 96024.344 96024.368 96024.424 52

OGB-GA [9] 53053.708 53053.798 53053.894 92

OGB-GA [9] 169637.593 169637.599 169637.602 19

OGB-GA[9]given inTable 6for the first four systems and from

sys-tem The result comparison has shown that the proposed CSA can

the first four test systems Obviously, the proposed CSA obtains

much better total cost than both Newton’s method and HNN

Therefore, the proposed CSA is effective for solving different

hydro-thermal systems with quadratic fuel cost function of hydro-thermal units

respectively

4.3 Systems with valve point effects on fuel cost function of thermal units

For this case, cost curve with valve point loading effects for thermal units is considered The proposed CSA is tested on two

and two thermal plants and the second one consists of two hydro plants and four thermal plants The data of the test systems is given

inAppendix A The number of nests is set to 50 and 100 for the first and second system, respectively The maximum number of iterations for the

the proposed CSA is run for its different values from 0.1 to 0.9 with

a step size of 0.1 to find the best one For each case of each system, the proposed CSA is performed 10 independent runs and the obtained results including minimal total cost, average total cost,

Table 7

Result comparison for the fifth test system with quadratic fuel cost function of

thermal units.

9.6

9.65

9.7

9.75

9.8

9.85

9.9

9.95

10x 10

4

Number of iterations = 1000

Fig 2 Convergence characteristic of test system 1 with quadratic fuel cost

function.

0

1

2

3

4

5

6

7

8x 10

10

Number of iterations = 1500

Fig 3 Convergence characteristic of test system 2 with quadratic fuel cost

5.3 5.32 5.34 5.36 5.38 5.4

5.42x 10 4

Number of iterations = 2500

Fig 4 Convergence characteristic of test system 3 with quadratic fuel cost function

of thermal units.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2x 10 11

Number of iterations = 300

Fig 5 Convergence characteristic of test system 4 with quadratic fuel cost function

maximal total cost, standard deviation, and average computational

systems

The obtained results are compared to those from other methods

that the proposed CSA can obtain better solution quality than the others in terms of total cost and computational time Therefore, the proposed is very effective for solving the HTS problem with nonconvex fuel cost function of thermal units Note all methods

and 8show the convergence characteristic of CSA for both systems, respectively

3.75

3.8

3.85

3.9

3.95

4

4.05

4.1

4.15x 10

5

Number of iterations = 1000

Fig 6 Convergence characteristic of test system 5 with quadratic fuel cost function

of thermal units.

Table 8

Results by CSA for the first system with nonconvex fuel cost of thermal units with

different values of p a

p a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)

Table 9

Results by CSA for the second system with nonconvex fuel cost of thermal units with

different values of p a

P a Min cost ($) Avg cost ($) Max cost ($) Std dev ($) CPU (s)

Table 10

Result comparison for two test systems with non-convex fuel cost of thermal units.

6.6 6.62 6.64 6.66 6.68 6.7 6.72

6.74x 10 4

Number of iterations = 1500

Fig 7 Convergence characteristic of test system 1 with non-convex fuel cost of thermal units.

9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10 10.1x 10 4

Number of iterations = 1500

Fig 8 Convergence characteristic of test system 2 with non-convex fuel cost of thermal units.

5 Conclusions

In this paper, the CSA method has been successfully applied for

solving short-term hydrothermal scheduling problem with smooth

and nonsmooth fuel cost curves of thermal units The CSA method

is a new meta-heuristic algorithm inspired from the obligate brood

parasitism of some cuckoo species by laying their eggs in the nests

of other host birds of other species for solving optimization

prob-lems The highlighted advantages of the CSA method are few

con-trol parameters and effective for optimization problems with

complicated constraints The proposed CSA has been tested on

sev-eral hydrothermal systems with different fuel cost functions of

thermal units The result comparisons with other methods in the

literature have indicated that the proposed CSA is more efficient

than many other methods Therefore, the proposed CSA can be a

very favorable method for solving the short-term hydrothermal scheduling problem, especially for nonsmooth fuel cost function

of thermal units

Appendix A Data of test systems The transmission loss formula coefficients of test system 1 with

0:00001 0:00015

The transmission loss formula coefficients of test system 2 with quadratic fuel cost function

B ¼

2 6

3 7

The transmission loss formula coefficients of test system 3 with quadratic fuel cost function

B ¼

2 6 6

3 7 7

Table A2

Hydro system data of test systems 1, 2 and 3 with quadratic fuel cost function of thermal units.

Table A3

Thermal generator and hydro system data of the test system 4 with quadratic fuel cost function of thermal units.

a si ($/h) b si ($/MW h) c si ($/MW 2 h) a hj (acre-ft/h) b hj (acre-ft/MW h) c hj (acre-ft/MW 2 h) W j (acre-ft)

Table A4

Thermal generator data of the test system 5 with quadratic fuel cost function of thermal units.

Table A5

Hydro system data of the test system 5 with quadratic fuel cost function of thermal units.

Hydro plant a hj (acre-ft/h) b hj (acre-ft/MW h) c hj (acre-ft/MW 2 h) W j (acre-ft) P hj,min (MW) P hj,max (MW)

Table A6

Thermal generator data of the test system 1 with non-convex fuel cost of thermal units.

Table A1

Thermal generator data of test systems 1, 2 and 3 with quadratic fuel cost function of

thermal units.

System Thermal plant a si ($/h) b si ($/MW h) c si ($/MW 2 h)

The transmission loss formula coefficients of test system 4 with

quadratic fuel cost function

0:00 0:00008

The transmission loss formula coefficients of test system 5 with

quadratic fuel cost function

1:0 3:5 1:0 1:2 1:5 1:0 3:9 2:0 1:5 1:2 2:0 4:9

2 6 6

3 7 7

The transmission loss formula coefficients of test system 1 with non-convex fuel cost

Table A7

Hydro system data of the test system 1 with non-convex fuel cost of thermal units.

Table A8

Thermal generator data of the test system 2 with non-convex fuel cost of thermal units.

Table A9

Hydro system data of the test system 2 with non-convex fuel cost of thermal units.

Hydro plant a hj (acre-ft/h) b hj (acre-ft/MW h) c hj (acre-ft/MW 2

Table B1

Optimal solution obtained by CSA for test system 1 with quadratic fuel cost function of thermal units.

Table B2

Optimal solution obtained by CSA for test system 2 with quadratic fuel cost function of thermal units.