Cuckoo search algorithm for short-term hydrothermal scheduling

Applied Energy 132 (2014) 276–287

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Cuckoo search algorithm for short-term hydrothermal scheduling Thang Trung Nguyen a, Dieu Ngoc Vo b,⇑, Anh Viet Truong c a

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

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

 A new cuckoo search method is

Hydro Plants

Thermal Plants Minimize fuel cost

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.

~

~

Cuckoo Search Algorithm

Electrical Load X1

X2 … XNd-1 Net solution

Randomize a number of nests, Nd

XNd

Calculate slack thermal and hydro units

Evaluate fitness funcon Iteraon=1 Generate new eggs via Lévy Flights

Calculate slack thermal and hydro units Evaluate fitness funcon Discover alien egg

Host bird’s nest

Host bird’s egg

Alien bird’s egg

Generate new eggs And randomize Calculate slack thermal and hydro units Evaluate fitness funcon

Iteraon=Max

Iteraon = Iteraon + 1 No Yes

5

4.15

x 10

4.1

STOP Fitness Function ($)

4.05 4 3.95 3.9 3.85 3.8 3.75 1 10

a r t i c l e

i n f o

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

2

10 Number of iterations = 1000

3

10

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 control 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 ⇑ Corresponding author. Tel.: +84 88 657 296x5730; fax: +84 88 645 796. E-mail address: [email protected] (D.N. Vo). http://dx.doi.org/10.1016/j.apenergy.2014.07.017 0306-2619/Ó 2014 Elsevier Ltd. All rights reserved.

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 satisfying both hydraulic and electrical operational constraints such as the quantity of available water, limits on generation, and power balance [1]. Several conventional methods have been implemented for solving the hydrothermal scheduling problem such as effective conventional method (ECM) based on Lagrange multiplier theory

T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287

277

Nomenclature ahj, bhj, chj asi, bsi, csi dsi, esi Bij, B0i, B00 PD,m Phj,m Phj,max Phj,min

water discharge coefficients of hydro plant j fuel cost coefficients of thermal plant i fuel cost coefficients of thermal plant i reflecting valve-point effects B-matrix coefficients for transmission power loss total system load demand at subinterval m power output of hydro plant j in subinterval m maximum power output of hydro plant i minimum power output of hydro plant i

[1], k–c iteration method, dynamic programming (DP) [2], Lagrange relaxation (LR) [3], decomposition and coordination method [4], and mixed integer programming (MIP) [5], Newton’s method [6,7]. In the ECM method, the coordination equations are linearized and solved for the water availability constraint separately from generating units, thus the Lagrangian multiplier associated with water availability constraint is separately from the outputs of generating units. Based on the obtained Lagrangian multiplier of water constraint, the Lagrangian multiplier associated with power balance constraint is determined and the outputs of thermal and hydro units are finally calculated. In the k–c method, the c values of the different hydro plants are initially chosen and 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 hydrothermal scheduling problem. However, computational and dimensional requirements in the DP method increase drastically with large-scale system planning horizon [8]. On the contrary to the DP method, the LR method is more efficient for dealing with large-scale problems. However, the LR method may suffer to duality 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 hydrothermal 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 economic 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 as evolutionary programming (EP) [8], genetic algorithm (GA) [9–12], differential evolution (DE) [13], artificial immune system (AIS) [14], and Hopfield neural network (HNN) [7]. Both the GA and EP algorithms are evolutionary based method for solving optimization 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 offspring may be detrimental to actually reaching an optimal solution. In this regard, the EP is more likely better when overcoming these disadvantages. In the EP method, the mutation is a key

PL,m Psi,m Psi,max Psi,min qj,m tm Wj

total transmission loss at subinterval m power output of thermal plant i in subinterval m maximum power output of thermal plant i minimum power output of thermal plant i rate of water flow from hydro plant j in subinterval m duration for subinterval m volume of water available for generation by hydro unit j during the scheduling period

search operator which generates new solutions from the current ones [15]. However, one disadvantage of the EP method in solving some of the multimodal optimization problems is its slow convergence to a near optimum. The DE method has the ability to search in very large spaces of candidate solutions with few or no assumptions about the considered problem. However, the DE method is difficult to deal with large-scale problems with slow or no convergence to the near optimum solution. The AIS method is one of the efficient methods for solving the nonconvex short-term hydrothermal 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 premature 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 genetic algorithm (OGB-GA) [9] is an improvement of GA for efficiently solving the HTS problem. In the OGB-GA method, the c values of the hydro plants are considered as the GA variables and the k iterations over the scheduling period can be called to find the thermal 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 on the number of intervals in the scheduling period [9]. The HNN method is an efficient neural network for dealing with optimization problems. However, it encounters a difficulty of predetermining 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 constraints of the problem must be linearized when applying in HNN [16]. In general, most of the artificial intelligence techniques usually suffer slow convergence to the near optimum solution for the HTS problems. The cuckoo search algorithm (CSA) developed by Yang and Deb [17] is a new meta-heuristic algorithm for solving optimization 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 optimization (PSO) and GA for ten standard optimization benchmark functions [17]. As observed from the obtained results, the CSA 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 generator and system characteristics including valve point loading effects, multiple fuel options, prohibited operating zones, spinning reserve and power loss [18]. In addition, CSA has been also used for solving

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T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287

the ED problems in practical power system and micro grid power dispatch problem [19]. For ED problems [18,19], CSA has been 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. Moreover, for micro grid power dispatch problem [19], CSA also obtains higher solution quality than DE and PSO. On the other hand, for Photovoltaic system, CSA has been used to track Maximum Power Point [20]. A comprehensive assessment is carried out against two 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 large and medium-sized PV systems. It is stated in [20] that CSA outperforms both P&O and PSO with respect to tracking capability, transient behavior and convergence. Consequently, CSA is an efficient 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 proposed 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 OGB-GA in [9], Newton’s method and HNN in [7], and PSO, DE, EP and AIS in [14]. 2. Problem formulation The objective of the HTS problem is to minimize the total fuel cost of thermal generators while satisfying hydraulic, power balance, and generator operating limits constraints. The short-term fixed-head hydrothermal scheduling problem having N1 thermal units and N2 hydro units scheduled in M time sub-intervals is formulated as follows. The objective is to minimize the total cost of thermal generators [14]:

MinC T ¼

N1 M X X

tm ½asi þ bsi Psi;m þ cs P2si;m þ jdsi  sinðesi  ðP min si  P si;m ÞÞj

m¼1 i¼1

ð1Þ subject to: – Power balance constraint: The total power generation from thermal and hydro plants must satisfy the total load demand and power loss in each subinterval: N1 N2 X X Psi;m þ Phj;m  PL;m  PD;m ¼ 0; i¼1

m ¼ 1; . . . ; M

ð2Þ

j¼1

where the power losses in transmission lines are calculated using Kron’s formula:

PL;m ¼

NX 1 þN 2 NX 1 þN 2 i¼1

P i;m Bij P j;m þ

j¼1

NX 1 þN 2

tm qj;m ¼ W j ;

B0i Pi;m þ B00

ð3Þ

i¼1

j ¼ 1; . . . ; N2

ð4Þ

m¼1

i ¼ 1; . . . ; N1 ;

Phj;min 6 Phj;m 6 Phj;max ;

j ¼ 1; . . . ; N2 ;

m ¼ 1; . . . ; M m ¼ 1; . . . ; M

ð6Þ 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 Deb [17]. In comparison with other meta-heuristic search algorithms, 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 during the search process as follows [21]. 1. Each cuckoo lays one egg (a design solution) at a time and dumps its egg in a randomly chosen nest among the fixed number 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 discover an alien egg with a probability pa e [0, 1]. In this case, it 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 approximated by a fraction pa of the n host nests are replaced by new 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 calculated 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) subintervals of N2 hydro units are obtained, the water discharge for hydro unit j at subinterval M is calculated using the available water constraint (4) as follows:

qj;M ¼

Wj 

M1 X

!

t m qj;m =t M ;

j ¼ 1; . . . ; N2

ð8Þ

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

Phj;m ¼

ð5Þ

bhj 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 bhj  4chj ðahj  qj;m Þ 2chj

j ¼ 1; 2; . . . ; N2 where

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

qj;m ¼ ahj þ bhj Phj;m þ cj P2hj;m

Psi;min 6 Psi;m 6 P si;max ;

m¼1

– Water availability constraint: The total available water discharge of each hydro plant for the whole scheduled time horizon is limited by: M X

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

2 ðbhj

;

m ¼ 1; . . . ; M; ð9Þ

 4chj ðahj  qj;m ÞÞ P 0.

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 remaining N1  1 thermal units and N2 hydro units in the system.

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T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287

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:

Ps1;m ¼ PD;m þ PL;m 

N1 N2 X X Psi;m  Phj;m i¼2

ð10Þ

j¼1

Eq. (3) is rewritten in terms of the slack thermal unit 1 as follows:

PL;m ¼

BTT;11 P2s1;m þ 2

Psi;m BTT;ij Psj;m þ

N2 X N2 X

Phi;m BHH;ij P hj;m

i¼1 j¼1

i¼2 j¼1

i¼2

j¼1

þ B00

ð11Þ

where

HT;ij

 BTH;ij  ;  B

   BT;0i   B0i ¼  B 

HH;ij

H;0i

BTT,ij, BT,0i Power loss coefficients due to thermal units; BHH,ij, BH,0i Power loss coefficients due to hydro units; BTH,ij, BHT,ij Power loss coefficients due to thermal and hydro units, BTH,ij = BHT,ijT

A  P 2s1;m þ B  Ps1;m þ C ¼ 0 where

ð13Þ

N1 N2 X X BTT;1i Psi;m þ 2 BTH;1j P hj;m þ BT;01  1 i¼2

ð14Þ

j¼1

N1 X N1 N2 X N2 X X C¼ ‘ P si;m BTT;ij P sj;m þ P hi;m BHH;ij Phj;m i¼2 j¼2

þ2

FT d ¼

þ B00 þ PD;m 

i¼2 N2 X

i¼2

j¼1

Psi;m 

ð15Þ

The solution of the second order Eq. (12) is obtained by:

Ps1;m ¼

B 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2  4AC 2A

M X 2 ðPs1;m;d  P lim s1 Þ m¼1

ð19Þ

where Ks and Kq are penalty factors for the slack thermal unit 1 and available water at subinterval M, respectively; Ps1,m,d is the power output of the slack thermal unit calculated from Section 3.2 corresponding to nest d in the population; qj,M is the water discharge of all hydro plants at the subinterval M calculated from Eq. (8) corresponding 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:

j¼1

Phj;m

F i ðPsi;m;d Þ þ K s

N2 X 2 þ K q ðqj;M;d  qlim j Þ

N1 X N2 N1 N2 X X X Psi;m BTH;ij Phj;m þ BT;0i Psi;m þ BH;0j Phj;m N1 X

N1 M X X m¼1 i¼1

Plim s1

i¼1 j¼1

i¼2 j¼1

ð18Þ

j¼1

ð12Þ

A ¼ BTT;11

m

where rand1 and rand2 are uniformly distributed random numbers in [0, 1]. Consider vector X d ¼ ½P s2;m;d ; Ps3;m;d ; . . . ; P sN1 ;m;d ; q1;m;d ; q2;m;d ; . . . ; qN2 ;m:d  of nest d including the thermal units from 2 to N1 for M subintervals and water discharges for hydro units from 1 to N2 for the first (M  1) subintervals. At the subinterval M, the nest d only contains thermal units from 2 to N1. The power output of the thermal units and water discharges in the Np nests are randomly chosen satisfying Psi,min 6 Psi,m,d 6 Psi,max and qj,min 6 qj,m,d 6 qj,max. Based on the initialized population of the nests, the fitness function to be minimized corresponding to each nest for the considered problem is calculated:

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

B¼2

j ¼ 1; . . . ; N2 ;

¼ 1; . . . ; M  1

N1 X N2 N1 N2 X X X Psi;m BTH;ij Phj;m þ BT;0i Psi;m þ BH;0j Phj;m

  BTT;ij Bij ¼  B

m ð17Þ

qj;m;d ¼ qj;min þ rand2  ðqj;max  qj;min Þ;

j¼1

i¼2 j¼2

þ2

i ¼ 2; . . . ; N1 ;

!

i¼2

þ

Psi;m;d ¼ P si;min þ rand1  ðP si;max  Psi;min Þ; ¼ 1; . . . ; M

N1 N2 X X BTT;1i Psi;m þ 2 BTH;1j Phj;m þ BT;01 Ps1;m

N1 X N1 X

power out of thermal unit i at subinterval m corresponding to nest d and qj,m,d is the water discharge for hydro unit j at subinterval m corresponding to nest d. In the CSA, each egg can be regarded as a solution which is randomly generated in the initialization. Therefore, each element in nest d of the population is randomly initialized as follows:

ð16Þ

where B2  4AC P 0 3.3. Implementation of cuckoo search algorithm Based on the three rules in Section 3.1, the CSA method is implemented for solving the short-term fixed-head HTS problem as follows. 3.3.1. Initialization A population of Np host nests is represented by X = [X1, X2, . . ., XNp]T, in which each Xd (d = 1, . . ., Np) represents a solution vector of variables given by Xd = [Psi,m,d, qj,m,d], where Psi,m,d is the

qlim j

8 Ps1;max > > >

P s1;m;d > > : 8 qj;max > > > < qj;min ¼ > qj;M;d > > :

if Ps1;m;d > Ps1;max if Ps1;m < Ps1;min otherwise

ð20Þ

if qj;M;d > qj;max if qj;M;d < qj;min otherwise

ð21Þ

where Ps1,max and Ps1,min are the maximum and minimum power outputs of slack thermal unit 1, respectively; qj,max and qj,min are the maximum and minimum water discharges of hydro plant j. The initialized population of the host nests is set to the best value of each nest Xbestd (d = 1, . . ., Nd) and the nest corresponding to the best fitness function in (19) is set to the best nest Gbest 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 the Lévy flights is calculated by Mantegna’s algorithm [22]. The new solution by each nest is calculated as follows:

X new ¼ Xbestd þ a  rand3  DX new d d

ð22Þ

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T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287

where a > 0 is the updated step size; rand3 is a normally distributed random number in [0, 1] and the increased value DXnew is deterd mined by:

DX new ¼v d

rx ðbÞ  ðXbest d  GbestÞ ry ðbÞ

ð23Þ

where

m¼

randx

ð24Þ

jrandy j1=b

where randx and randy are two normally distributed stochastic variables with standard deviation rx(b) and ry(b) given by:

 31=b Cð1 þ bÞ  sin p2b 5 4 rx ðbÞ ¼   b1 C 1þb  b  2ð 2 Þ 2

ð25Þ

ry ðbÞ ¼ 1

ð26Þ

The flowchart of the proposed CSA for solving the problem is given in Fig. 1. 4. Numerical results The proposed CSA has been tested on five systems with quadratic fuel cost function of thermal units and two systems with nonconvex fuel cost function of thermal units. The proposed algorithm is coded in Matlab platform and run on a 2 GHz PC with 2 GB of RAM. 4.1. Selection of parameters

2

where b is the distribution factor (0.3 6 b 6 1.99) and C() is the gamma distribution function. For the newly obtained solution, its lower and upper limits should be satisfied according to the unit’s limits:

8 > < qj;max if qj;m;d > qj;max qj;m;d ¼ qj;min if qj;m;d < qj;min ; j ¼ 1;. .. ;N2 ; m ¼ 1;. .. ; M  1 > : qj;m;d otherwise ð27Þ 8 > < Psi;max if Psi;m;d > Psi;max P si;m;d ¼ Psi;min if Psi;m;d < Psi;min ; i ¼ 2;. .. ;N1 ; m ¼ 1;. .. ;M > : Psi;m;d otherwise

Initialize population of host nests Psi ,m ,d = Psi ,min + rand1 * ( Psi ,max − Psi ,min )

ð28Þ The power output of N2 hydro units and the slack thermal unit 1 are then obtained as in Section 3.2. The fitness value is calculated using (19) and the nest corresponding to the best fitness function is set to 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 with the probability of pa also creates a new solution for the problem similar to the Lévy flights. The new solution due to this action can be found out in the following way:

X dis d

¼ Xbestd þ K 

DX dis d

ð29Þ

where K is the updated coefficient determined based on the probability of a host bird to discover an alien egg in its nest:

 K¼

1 if rand4 < pa 0

otherwise

ð30Þ

¼ rand5  ½randp1 ðXbestd Þ  randp2 ðXbestd Þ

q j ,m ,d = q j ,min + rand 2 * (q j ,max − q j ,min )

Calculate all thermal and hydro generations based on the initialization. - Set Xd to Xbestd for each nest - Set the best of all Xbestd to Gbest - Set iteration counter iter = 1.

Generate new solution via Lévy flights

X dnew = Xbest d + α × rand 3 × ΔX dnew

- Check for limit violations and repairing. - Calculate all hydro and thermal generation outputs. - Evaluate fitness function to choose new Xbestd and Gbest

Discover alien egg and randomize

and the increased value DXdis d is determined by:

DX dis d

In the proposed CSA method, three main parameters which have to be predetermined are the number of nests Np, maximum number of iterations Nmax, and the probability of an alien egg to be discovered pa. Among the three parameters, the number of nests has significantly effects on the obtained solution quality. Generally, the higher number of NP is chosen the higher probability for a better 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 100 depending on system size. Similar to NP, the maximum number of iterations Nmax also has an impact on the obtained solution quality and computation time. It is chosen based on the complexity and scale of the considered problems. For the test systems in this paper, the maximum number of Nmax ranges from 300 for small

X ddis = Xbestd + K × ΔX ddis

ð31Þ

where rand4 and rand5 are the distributed random numbers in [0, 1] and randp1(Xbestd) and randp2(Xbestd) are the random perturbation for positions of the nests in Xbestd. For the newly obtained solution, 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.

- Check for limit violations and repairing. - Calculate all hydro and thermal generation outputs. - Evaluate fitness function to choose new Xbestd and Gbest

No

Iter
3.3.4. Stopping criteria The algorithm is stopped when the number of iterations (Iter) reaches the maximum number of iterations (Itermax).

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

Iter = Iter + 1

281

T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287 Table 1 Results by CSA for the first system with quadratic fuel cost of thermal units with different values of Pa.

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

Pa

Min. cost ($)

Avg. cost ($)

Max. cost ($)

Std. dev. ($)

CPU (s)

Pa

Min. cost ($)

Avg. cost ($)

Max. cost ($)

Std. dev. ($)

CPU (s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

96024.6877 96024.6826 96024.6817 96024.6816 96024.6816 96024.6816 96024.6816 96024.6817 96024.6816

96030.083 96024.845 96024.704 96024.684 96024.723 96024.686 96024.729 96024.717 96024.988

96053.9166 96026.0801 96024.7553 96024.6914 96024.954 96024.7213 96025.1145 96024.9907 96026.0008

8.8589314 0.4141929 0.0273447 0.0027169 0.0802032 0.011685 0.1286358 0.091912 0.4308014

19.2 19.8 19.1 19.3 19.6 19.6 19.4 19.2 19.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

53051.4988 53051.4787 53051.4768 53051.4765 53051.4765 53051.4765 53051.4765 53051.4768 53051.4763

53051.52 53051.48 53051.48 53051.48 53051.48 53051.48 53051.51 53051.48 53051.54

53051.54 53051.48 53051.48 53051.48 53051.48 53051.51 53051.65 53051.48 53051.89

0.012 0.001 0.001 0.000 0.001 0.009 0.061 0.002 0.124

87 88 86 87 85 86 85 85.8 86.4

systems to 2500 for the large-scale systems. The value of the probability for an alien egg to be discovered pa can be chosen in the range [0, 1]. However, different values of pa may lead to different optimal solutions for a problem. For the complicated or large-scale problems, the selection of the value for the probability has an obvious effect on the optimal solution. In contrast, the effect of the probability is inconsiderable for the simple problems, that is different values of the probability can also lead the same optimal solution. Therefore, the best value of pa has to be tuned in its range [0, 1]. Besides, the value of distribution factor b needs be determined 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 [22]. As experience in [18], the value of distribution factor in the 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 data for the first four system is from [1] consisting of one thermal 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 for the fourth one. The fifth system from [7] consists of two thermal plants and two hydro plants with four scheduling subintervals. All data for the five systems are given in Appendix A. The number of nests is set to 20 for system 1, 50 for systems 2 and 3, 10 for system 4, and 100 for system 5. The maximum number of iterations for the CSA is set to 1000, 1500, 2500, 300 and 1000 for the five systems, respectively. The value of the probability pa will be analyzed 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 can obtain optimal solution for the values of pa from 0.2 to 0.9.

Table 2 Results by CSA for the second system with quadratic fuel cost of thermal units with different values of Pa. Pa

Min. cost ($)

Avg. cost ($)

Max. cost ($)

Std. dev. ($)

CPU (s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

848.3503 848.3464 848.3463 848.3463 848.3463 848.3463 848.3463 848.3463 848.3463

848.3614 848.3485 848.347 848.36 848.3825 848.3468 848.3473 848.3503 848.35

848.3825 848.3619 848.3491 848.4779 848.7028 848.348 848.3497 848.3733 848.369

0.008 0.004 0.001 0.039 0.107 0.001 0.001 0.008 0.007

45.5 46.4 44.8 45.6 45.5 46.7 44.8 47.1 45.6

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

Min. cost ($)

Avg. cost ($)

Max. cost ($)

Std. dev. ($)

CPU (s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6

169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6

169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6 169637.6

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.2 0.25 0.27 0.31 0.24 0.26 0.28 0.31 0.32

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

Min. cost ($)

Avg. cost ($)

Max. cost ($)

Std. dev. ($)

CPU (s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

376149.2619 376140.9741 376142.1596 376080.7325 376055.7279 376040.9513 376040.9513 375967.2741 375960.5756

376269.321 376293.4898 376235.8054 376315.8979 376290.9361 376194.2023 376194.2023 376259.8584 376190.9635

376406.9 376544.7 376379.6 376511.4 376676.6 376454.2 376454.2 376537.6 376408.8

93.633 131.299 67.918 118.197 205.239 148.870 125.953 171.856 176.633

12.3 12.4 11.8 13.0 12.6 12.6 12.5 12.4 12.1

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)

1

EGA [9] OGB-GA CSA EGA [9] OGB-GA CSA EGA [9] OGB-GA CSA EGA [9] OGB-GA CSA

96028.651 96024.344 96024.6816 848.027 848.326 848.3463 53055.712 53053.708 53051.4763 169637.944 169637.593 169637.6

96050.154 96024.368 96024.684 850.187 848.488 848.3468 53081.517 53053.798 53051.54 169643.468 169637.599 169637.6

96086.695 96024.424 96024.6914 852.114 849.552 848.348 53092.566 53053.894 53051.89 169653.127 169637.602 169637.6

220 52 19.3 210 90 46.7 312 92 86.4 37 19 0.2

2

3

4

[9]

[9]

[9]

[9]

For the fifth system, the best value of pa for CSA to obtain the optimal solution is 0.9. The values of Pa to obtain the best solution chosen are respectively 0.4, 0.6, 0.9, 0.1 and 0.9 for the five systems corresponding to the bold values in Tables 1–5. The minimal cost, average cost, maximum cost and average computational time obtained by the proposed CSA are then compared to those from EGA [9] and

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Table 7 Result comparison for the fifth test system with quadratic fuel cost function of thermal units. Method

Newton’s method [7]

HNN [7]

CSA

Cost ($)

377,374.67

377,554.94

375,960.5756

x 10

4

9.95

Fitness Function ($)

x 10

4

5.4

Fitness Function ($)

10

5.42

5.38

5.36

5.34

9.9 5.32

9.85 9.8

5.3 2 10

10

3

10

4

Number of iterations = 2500

9.75

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

9.7 9.65 9.6 0 10

1

10

2

10

3

10

Number of iterations = 1000 Fig. 2. Convergence characteristic of test system 1 with quadratic fuel cost function.

OGB-GA [9] given in Table 6 for the first four systems and from HNN [7] and Newton’s method [7] given in Table 7 for the fifth system. The result comparison has shown that the proposed CSA can obtain approximate or better total than EGA [9] and OGB-GA [9] for the first four test systems. Obviously, the proposed CSA obtains much better total cost than both Newton’s method and HNN. Furthermore, the proposed CSA is faster than EGA [9] and OGB-GA [9]. Therefore, the proposed CSA is effective for solving different hydrothermal systems with quadratic fuel cost function of thermal units. There is no computer reported in [9]. The optimal solution for each system by the proposed CSA is given in Appendix B. Figs. 2–6 show

8

x 10

the convergence respectively.

characteristic

of

CSA

for

systems

1–5,

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 systems from [14] where the first system comprises two hydro plants 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 in Appendix 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 CSA is set to 1500 for the both systems. For the probability pa, 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,

10

2

x 10

11

1.8

7

Fitness Function ($)

Fitness Function ($)

1.6 6 5 4 3

1.4 1.2 1 0.8 0.6

2

0.4

1 0 1 10

0.2 2

10

3

10

4

10

Number of iterations = 1500 Fig. 3. Convergence characteristic of test system 2 with quadratic fuel cost function.

0 1 10

2

10

3

10

Number of iterations = 300 Fig. 5. Convergence characteristic of test system 4 with quadratic fuel cost function of thermal units.

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T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287

4.15

x 10

5

6.74

4

6.72

4.05

Fitness Function ($)

Fitness Function ($)

4.1

x 10

4 3.95 3.9

6.7 6.68 6.66 6.64

3.85

6.62

3.8 3.75 1 10

2

6.6 1 10

3

10

10

2

3

10

4

10

10

Number of iterations = 1500

Number of iterations = 1000 Fig. 6. Convergence characteristic of test system 5 with quadratic fuel cost function of thermal units.

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

Table 8 Results by CSA for the first system with nonconvex fuel cost of thermal units with different values of pa. Min. cost ($)

Avg. cost ($)

Max. cost ($)

Std. dev. ($)

CPU (s)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

66115.5431 66115.8717 66115.531 66115.4743 66115.485 66115.4554 66115.4633 66115.4451 66115.4471

66125.9988 66128.0202 66126.6982 66127.9227 66127.8139 66128.7083 66135.7946 66133.856 66119.034

66154.1203 66146.3192 66153.3879 66157.8827 66161.9327 66158.1144 66154.445 66158.1011 66150.4011

15.640 12.998 14.910 16.834 16.593 18.242 16.769 18.544 10.456

6.8 6.4 7.1 7.1 6.9 7.1 7.1 6.7 7.15

x 10

4

10

Fitness Function ($)

pa

10.1

9.9 9.8 9.7 9.6 9.5 9.4

Table 9 Results by CSA for the second system with nonconvex fuel cost of thermal units with different values of pa. Pa

Min. cost ($)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

92742.9266 92743.3945 92788.722 92726.014 92771.6158 92783.3591 92738.8334 92725.1343 92760.9481

Avg. cost ($) 92876.24 92839.59 92879.16 92814.26 92855.4 92912.8 92803.97 92779.32 92801.61

Max. cost ($) 92973.556 93072.39 93185.371 93009.627 93048.514 93271.887 92995.727 92934.773 92893.961

Std. dev. ($) 66.584 96.628 108.426 80.574 74.846 169.999 67.918 56.324 47.418

9.3 9.2 2 10

CPU (s)

3

4

10

10

Number of iterations = 1500

19.2 18.8 19.4 19.6 20.3 19.6 19.8 18.6 19.3

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

maximal total cost, standard deviation, and average computational time are respectively given in Tables 8 and 9. Based on the analysis, the value of pa for CSA to obtain the best solution is 0.8 for the both systems. The obtained results are compared to those from other methods available in the literature including AIS, EP, PSO, and DE in [14]

given in Table 10. The result comparison in the table has indicated 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 in [14] have been implemented on a Pentium-IV 3.0 GHz PC. The optimal solutions for the systems are given in Appendix B. Figs. 7 and 8 show the convergence characteristic of CSA for both systems, respectively.

Table 10 Result comparison for two test systems with non-convex fuel cost of thermal units. Method

AIS [14]

EP [14]

PSO [14]

DE [14]

CSA

System 1

Cost ($) CPU time (s)

66,117 53.43

66,198 75.48

66,166 71.62

66,121 60.76

66,115.44 6.7

System 2

Cost ($) CPU time (s)

93,950 59.14

94,250 67.82

94,126 80.37

94,094 83.54

92,725.1343 18.6

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T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287

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 problems. The highlighted advantages of the CSA method are few control parameters and effective for optimization problems with complicated constraints. The proposed CSA has been tested on several 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 quadratic fuel cost function (see Tables A1–A9).

 B¼

0:00005 0:00001 0:00001 0:00015

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

2

0:0

System

Thermal plant

asi ($/h)

bsi ($/MW h)

csi ($/MW2 h)

1 2 3

1 1 1 2

373.7 15 25 30

9.606 3 3.2 3.4

0.001991 0.01 0.0025 0.0008

0:000

0:0

6 B ¼ 4 0:0 0:001 0:0

Table A1 Thermal generator data of test systems 1, 2 and 3 with quadratic fuel cost function of thermal units.



0:0

3 7 5

0:000 0:0005

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

2

0:00014

0:000010 0:000015 0:000015

3

6 0:000010 0:00006 0:000010 0:000013 7 7 6 B¼6 7 4 0:000015 0:000010 0:000068 0:000065 5 0:000015 0:000013 0:000065

0:00007

Table A2 Hydro system data of test systems 1, 2 and 3 with quadratic fuel cost function of thermal units. System

Hydro plant

ahj (MCF/h)

bhj (MCF/MW h)

chj (MCF/(MW)2h)

Wj (MCF)

1 2

1 1 2 1 2

61.53 0.2 0.4 1.98 0.936

0.009079 0.03 0.06 0.306 0.612

0.0007749 0.00005 0.0001 0.000216 0.00036

2559.6 25 35 2500 2100

3

Table A3 Thermal generator and hydro system data of the test system 4 with quadratic fuel cost function of thermal units. Thermal plant

Hydro plant

asi ($/h)

bsi ($/MW h)

csi ($/MW2 h)

ahj (acre-ft/h)

bhj (acre-ft/MW h)

chj (acre-ft/MW2 h)

Wj (acre-ft)

575

9.2

0.00184

330

4.97

0

100,000

Table A4 Thermal generator data of the test system 5 with quadratic fuel cost function of thermal units. Thermal plant

asi ($/h)

bsi ($/MW h)

csi ($/MW2 h)

Psi,min (MW)

Psi,max (MW)

1 2

380 600

6.75 5.28

0.00225 0.0055

47.5 100

450 1000

Table A5 Hydro system data of the test system 5 with quadratic fuel cost function of thermal units. Hydro plant

ahj (acre-ft/h)

bhj (acre-ft/MW h)

chj (acre-ft/MW2 h)

Wj (acre-ft)

Phj,min (MW)

Phj,max (MW)

1 2

260 250

8.5 9.8

0.00986 0.0114

125,000 286,000

0 0

250 500

Table A6 Thermal generator data of the test system 1 with non-convex fuel cost of thermal units. Thermal plant

asi ($/h)

bsi ($/MW h)

csi ($/MW2 h)

dsi ($/h)

esi (1/MW)

Psi,min (MW)

Psi,max (MW)

1 2

25 30

3.2 3.4

0.0025 0.0008

12 14

0.0550 0.0450

50 50

300 700

285

T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287 Table A7 Hydro system data of the test system 1 with non-convex fuel cost of thermal units. Hydro plant

ahj (MCF/h)

bhj (MCF/MW h)

chj (MCF/MW2 h)

Wj (MCF)

Phj,min (MW)

Phj,max (MW)

1 2

1.980 0.936

0.306 0.612

0.000216 0.000360

2500 2100

0 0

400 300

Table A8 Thermal generator data of the test system 2 with non-convex fuel cost of thermal units. Thermal plant

asi ($/h)

bsi ($/MW h)

csi ($/MW2 h)

dsi $/h)

esi (rad/MW)

Psi,min (MW)

Psi,max (MW)

1 2 3 4

10 10 20 20

3.25 2.00 1.75 1.00

0.0083 0.0037 0.0175 0.0625

12 18 16 14

0.0450 0.0370 0.0380 0.0400

20 30 40 50

125 175 250 300

Table A9 Hydro system data of the test system 2 with non-convex fuel cost of thermal units. Hydro plant

ahj (acre-ft/h)

bhj (acre-ft/MW h)

chj (acre-ft/MW2 h)

Wj (acre-ft)

Phj,min (MW)

Phj,max (MW)

1 2

260 250

8.5 9.8

0.00986 0.01140

125,000 286,000

0 0

250 500

Table B1 Optimal solution obtained by CSA for test system 1 with quadratic fuel cost function of thermal units. Hour

1

2

3

4

5

6

7

8

Load (MW) Ps1 (MW) Ph1 (MW) Hour Load (MW) Ps1 (MW) Ph1 (MW) Hour Load (MW) Ps1 (MW) Ph1 (MW)

455 231.9745 235.0974 9 665 431.3039 254.946 17 721 485.1017 260.3584

425 203.8466 232.2705 10 675 440.8983 255.9005 18 740 500.0000 265.7511

415 194.4400 231.3806 11 695 460.0931 257.8356 19 700 464.9050 258.3124

407 186.9274 230.6628 12 705 469.7189 258.7899 20 678 443.7818 256.1836

400 180.3598 230.0338 13 580 350.2702 246.7235 21 630 397.8721 251.5351

420 199.1280 231.8404 14 605 374.0194 249.1503 22 585 354.9624 247.2638

487 262.0946 238.0912 15 616 384.4858 250.2215 23 540 312.2406 243.0097

604 372.9889 249.1359 16 653 419.8281 253.7759 24 503 277.2186 239.5605

Table B2 Optimal solution obtained by CSA for test system 2 with quadratic fuel cost function of thermal units. Hour

1

2

3

4

5

6

7

8

Load (MW) Ps1 (MW) Ph1 (MW) Ph2 (MW) Hour Load (MW) Ps1 (MW) Ph1 (MW) Ph2 (MW) Hour Load (MW) Ps1 (MW) Ph1 (MW) Ph2 (MW)

30 1.2893 20.2058 8.9533 9 61 9.6808 30.6706 21.8275 17 71 12.4622 34.0119 26.0212

33 2.092 21.236 10.1747 10 58 8.8634 29.6343 20.5925 18 62 9.9608 31.0082 22.2398

35 2.6217 21.8951 11.0234 11 56 8.3076 28.9621 19.7645 19 55 8.0392 28.6399 19.328

38 3.4237 22.9174 12.2592 12 57 8.5855 29.3094 20.1675 20 50 6.672 26.969 17.2348

40 3.9658 23.5859 13.0903 13 60 9.4133 30.3294 21.4062 21 43 4.777 24.5981 14.3326

45 5.313 25.2664 15.1741 14 61 9.69 30.6531 21.8349 22 33 2.0909 21.2269 10.1847

50 6.6708 26.9517 17.2527 15 65 10.7903 31.9814 23.5278 23 31 1.5517 20.5412 9.3729

59 9.1385 29.9905 20.9908 16 68 11.6331 32.991 24.771 24 30 1.2801 20.2179 8.9508

The transmission loss formula coefficients of test system 4 with quadratic fuel cost function

 B¼

0:00 0:00000



0:00 0:00008

The transmission loss formula coefficients of test system 5 with quadratic fuel cost function

2

4:0

1:0 1:5 1:5

3

6 1:0 3:5 1:0 1:2 7 6 7 B ¼ 105 6 7 4 1:5 1:0 3:9 2:0 5 1:5 1:2 2:0 4:9 The transmission loss formula coefficients of test system 1 with non-convex fuel cost

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T.T. Nguyen et al. / Applied Energy 132 (2014) 276–287

Table B3 Optimal solution obtained by CSA for test system 3 with quadratic fuel cost function of thermal units. Hour

1

2

3

4

5

6

7

8

Load (MW) Ps1 (MW) Ps2 (MW) Ph1 (MW) Ph2 (MW) Hour Load (MW) Ps1 (MW) Ps2 (MW) Ph1 (MW) Ph2 (MW) Hour Load (MW) Ps1 (MW) Ps2 (MW) Ph1 (MW) Ph2 (MW)

400 77.6154 136.7090 168.3957 22.9107 9 1230 213.1207 553.1738 334.0795 184.3387 17 1350 233.4975 615.4858 359.0210 208.5034

300 61.8847 88.1188 149.1679 4.0900 10 1250 216.5019 563.5142 338.2224 188.3521 18 1470 254.0927 678.3826 384.2226 232.8936

250 53.1630 61.2059 138.0007 0.0000 11 1350 233.4984 615.4847 359.0176 208.5069 19 1330 230.0845 605.0643 354.8432 204.4613

250 53.1609 61.2082 138.0004 0.0000 12 1400 242.0525 641.6276 369.4877 218.6326 20 1250 216.5023 563.5176 338.2238 188.3469

250 53.1612 61.2088 137.9994 0.0000 13 1200 208.0583 537.6800 327.8873 178.3354 21 1170 203.0098 522.2288 321.7122 172.3361

300 61.8839 88.1257 149.1592 4.0926 14 1250 216.5019 563.5147 338.2220 188.3520 22 1050 182.9464 460.7356 297.1675 148.5079

450 85.5249 161.1261 178.0585 32.3758 15 1250 216.5004 563.5197 338.2168 188.3537 23 900 158.1442 384.6413 266.8355 119.0086

900 158.1462 384.6399 266.8318 119.0118 16 1270 219.8885 573.8834 342.3635 192.3680 24 600 109.4438 234.8604 207.2947 60.9530

2

0:000049 6 0:000014 6 6 6 0:000015 B¼6 6 0:000015 6 6 4 0:000020 0:000017

Table B4 Optimal solution obtained by CSA for test system 4 with quadratic fuel cost function of thermal units. Subinterval

Duration (h)

PD (MW)

Ps1 (MW)

Ph1 (MW)

1 2

12 12

1200 1500

567.413 685.7244

668.319 875.6112

0:000014 0:000045 0:000016 0:000020 0:000018 0:000015

0:000015 0:000016 0:000039 0:000010 0:000012 0:000012

0:000015 0:000020 0:000010 0:000040 0:000014 0:000010

0:000020 0:000018 0:000012 0:000014 0:000035 0:000011

3 0:000017 0:000015 7 7 7 0:000012 7 7 0:000010 7 7 7 0:000011 5 0:000036

Appendix B. Optimal solution of test systems

2

14

1:0 1:5 1:5

3

See Tables B1–B7.

6 6:0 1:0 1:3 7 7 B ¼ 10 6 7 4 1:5 1:0 6:8 6:5 5 5 6 1:0

1:5 1:3 6:5 7:0 The transmission loss formula coefficients of test system 2 with non-convex fuel cost

Table B5 Optimal solution obtained by CSA for test system 5 with quadratic fuel cost function of thermal units. Subinterval

Duration (h)

PD (MW)

Ps1 (MW)

Ps2 (MW)

Ph1 (MW)

Ph2 (MW)

1 2 3 4

12 12 12 12

1200 1500 1400 1700

444.1752 449.9864 449.9804 449.9865

324.2934 448.3269 396.8426 567.1928

164.8779 239.3499 221.5277 249.9789

298.0429 411.3243 374.2936 496.4541

Table B6 Optimal solution obtained by CSA for test system 1 with non-convex fuel cost of thermal units. Subinterval

Duration (h)

PD (MW)

Ps1 (MW)

Ps2 (MW)

Ph1 (MW)

Ph2 (MW)

1 2 3

8 8 8

900 1200 1100

220.0541 221.3595 221.3595

399.0658 538.6921 538.6921

238.1189 323.9389 274.5214

83.0142 183.8417 125.3249

Table B7 Optimal solution obtained by CSA for test system 2 with non-convex fuel cost of thermal units. Sub-interval

Duration (h)

PD (MW)

Ps1 (MW)

Ps2 (MW)

Ps3 (MW)

Ps4 (MW)

Ph1 (MW)

Ph2 (MW)

1 2 3 4

12 12 12 12

900 1100 1000 1300

89.8132 125 124.997 125

175 175 175 175

106.0374 122.6728 118.9902 219.9307

50 50.0001 50 70.1117

175.2938 249.4151 201.1758 250

322.44 406.31 352.687 500

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