# Dynamic Economic Dispatch Using Hybrid DE-SQP for Generating Units with Valve-Point Effects

Authors: A. M. Elaiw; X. Xia; A. M. Shehata
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper presents hybrid differential evolution (DE) and sequential quadratic programming (SQP) for solving the dynamic economic dispatch (DED) problem for generating units with valve-point effects. DE is used as a global optimizer and SQP is used as a fine tuning to determine the optimal solution at the final. The feasibility of the proposed method is validated with five-and ten-unit test systems. Results obtained by DE-SQP method are compared with other techniques in the literature.


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## Body

## 1. Introduction

The primary objective of the static economic dispatch (SED) problem of electric power generation is to determine the optimal schedule of online generating units' outputs so as to meet the load demand at a certain time at the minimum operating cost under various system and generator operational constraints. Plant operators, to avoid life-shortening of the turbines and boilers, try to keep thermal stress on the equipments within the safe limits. This mechanical constraint is usually transformed into a limit on the rate of change of the electrical output of generators. Such ramp rate constraints link the generator operation in two consecutive time intervals. Optimal dynamic dispatch problem is an extension of SED problem which is used to determine the generation schedule of the committed units so as to meet the predicted load demand over a time horizon at minimum operating cost under ramp rate constraints and other constraints (see [1–31]). Since the ramp rate constraints couple the time intervals, the optimal dynamic dispatch problem is a difficult optimization problem. If the ramp rate constraints are not included in the optimization problem, the optimal dynamic dispatch problem is reduced to a set of uncoupled SED problems that can be easily solved.Optimal dynamic dispatch problem was first formulated by Bechert and Kwatny [1] in 1972 and was followed by [2–5]. In these papers, the problem was formulated as an optimal control problem. The optimal control dynamic dispatch formulation models the power system generation by means of state equations where the state variables are the electrical power outputs of the generators and the control inputs are the ramp rates of the generators. In this approach, the optimization is done with respect to the ramp rates and the solution produces an optimal output generator trajectory for a given initial generation. Since the 1980s, the optimal dynamic dispatch problem has been formulated as a minimization problem of the total cost over the dispatch period under some constraints and has been known as the dynamic economic dispatch (DED) problem (see [6–31]). Since the DED problem was introduced, several optimization techniques and procedures have been used for solving the DED problem with complex objective functions or constraints (see the review paper [6]). There were a number of classical methods that have been applied to solve this problem such as the lambda iterative method [16], gradient projection method [25], Lagrange relaxation [26], linear programming [24], and interior point method [11, 13]. Most of these methods are not applicable for nonsmooth or nonconvex cost functions. To overcome this problem, many stochastic optimization methods have been employed to solve the DED problem, such as simulated annealing (SA) [27], genetic algorithms (GA) [28], differential evolution (DE) [18, 19], particle swarm optimization (PSO) [10, 31], and artificial immune system (AIS) [15]. Many of these techniques have proven their effectiveness in solving the DED problem without any or fewer restrictions on the shape of the cost function curves. Hybrid methods which combine two or more optimization methods have been successfully applied to DED problems with valve-point effects such as EP-SQP [9] and PSO-SQP [29, 30].DE which was proposed by Storn and Price [32] is a population-based stochastic parallel search technique. DE uses a rather greedy and less stochastic approach to problem solving compared to other evolutionary algorithms. DE has the ability to handle optimization problems with nonsmooth/nonconvex objective functions [32]. Moreover, it has a simple structure and a good convergence property, and it requires a few robust control parameters [32]. DE has been successfully applied to the DED problem with nonsmooth and nonconvex cost functions (see [18–21]).DE is one of the good methods which have been used for solving the DED problem with nonsmooth and nonconvex cost functions; however, the obtained solutions are just near global optimum with long computation time. Therefore, hybrid methods such as DE-SQP can be effective in solving the DED problem with valve-point effects. The aim of this paper is to propose hybrid DE-SQP method to solve the DED problem with valve-point effects. DE is used as a base level search for global exploration and SQP is used as a local search to fine-tune the solution obtained from DE. In the DE-SQP techniques, DE will thoroughly search the solution space and stops when the specified maximum iteration count is reached. Thereafter, the SQP technique will be used to fine-tune the final solution obtained by the DE method.The remainder of this paper is organized as follows: in Section2, we introduce the DED problem formulation. An overview of the differential evolution and sequential quadratic programming algorithms is presented in Sections 3 and 4. In Section 5, numerical examples and simulation results are presented. Finally, conclusions are drawn in Section 6.


## 2. Formulation of the DED Problem

The objective of the DED problem is to determine the generation levels for the committed units which minimize the total fuel cost over the dispatch period[0,T],
(2.1)minCT=∑t=1T‍∑i=1N‍Ci(Pit)
subject to the following constraints:(i)
power balance constraint:(2.2)∑i=1N‍Pit=Dt+Losst,t=1,…,T,(ii)
generation limits:(2.3)Pimin≤Pit≤Pimax,i=1,…,N,t=1,…,T,(iii)
generating unit ramp rate limits:(2.4)-DRi≤Pit-Pit-1≤URi,i=1,…,N,t=2,…,T, where N is the number of committed units; T is the number of intervals in the time horizon; Pit is the generation of unit i during the tth time interval [t-1,t); Dt is the demand at time t (i.e., the t-th time interval); URi and DRi are the maximum ramp up/down rates for unit i; Pimin and Pimax are the minimum and maximum capacity of unit i, respectively. The fuel cost of unit i considering valve-point effects can be expressed as
(2.5)Ci(Pit)=ai+biPit+ci(Pit)2+|disin(ei(Pimin-Pit))|,
where ai,bi, and ci are positive constants, and di and ei are the coefficients of unit i reflecting valve-point effects.TheB-coefficient method is one of the most commonly used by power utility industry to calculate the network losses. In this method, the network losses are expressed as a quadratic function of the unit's power outputs that can be approximated by
(2.6)Losst=∑i=1N‍∑j=1N‍PitBijPjt,t=1,…,T,
where Bij is the ijth element of the loss coefficient square matrix of size N.


## 3.   Overview of Differential Evolution Algorithm

DE is a simple yet powerful heuristic method for solving nonlinear, nondifferentiable, and nonsmooth optimization problems. DE algorithm is a population-based-algorithm using three operators mutation, crossover, and selection to evolve from randomly generated initial population to final individual solution. The key idea behind DE is that it starts with an initial population of feasible target vectors (parents) and new solutions (offsprings) are generated (by mutation, crossover, and selection operations) until the optimal solution is reached. In the mutation operation, three different vectors are selected randomly from the population and a mutant vector is created by perturbing one vector with the difference of the two other vectors. In the crossover operation, a new trial vector (offspring) is created by replacing certain parameters of the target vector by the corresponding parameters of the mutant vector on the bases of a probability distribution. In DE, the competition between the parents and offspring is one to one. The individual with best fitness will remain till the next generation. The iterative process continues until a user-specific stopping criterion is met. DE algorithm has three control parameters, which are differentiation (or mutation) factorF, crossover constant CR, and size of population NP. According to Storn and Price [32], the basic strategy of DE for m-dimensional optimization problem can be described as follows.(1) Initialization: generate a population ofNP initial feasible target vectors (parents) Xi={x1i,x2i,…,xmi}, i=1,2,…,NP randomly as
(3.1)xji=xjmin+s1·(xjmax-xjmin),j=1,2,…,m,i=1,2,…,NP,
where s1 is uniform random number in [0,1]; xjmin and xjmax are the lower and upper bounds of the jth component of the target vector.(2) Mutation: letXiG={x1iG,x2iG,…,xmiG} be the individual i at the current generation G. A mutant vector ViG+1=(v1iG+1,v2iG+1,…,vmiG+1) is generated according to the following:
(3.2)ViG+1=Xr1G+F·(Xr2G-Xr3G),r1≠r2≠r3≠i,i=1,2,…,NP
with randomly chosen integer indexes r1,r2,r3∈{1,2,…,NP}.(3) Crossover: according to the target vectorXiG and the mutant vector ViG+1, a new trial vector (offspring) UiG+1={u1iG+1,u2iG+1,…,umiG+1} is created with
(3.3)ujiG+1={vjiG+1if(rand(j)≤CR)orj=rnb(i),xjiGotherwise,
where j=1,2,…,m, i=1,2,…,NP and rand(j) is the jth evaluation of a uniform random number generator between [0,1]. CR is the crossover constant between [0,1] which has to be determined by the user. rnb(i) is a randomly chosen index from 1,2,…,m which ensures that UiG+1 gets at least one parameter from ViG+1 [32].(4) Selection: This process determines which of the vectors will be chosen for the next generation by implementing one-to-one competition between the new generated trial vectors and their corresponding parents. The selection operation can be expressed as follows:(3.4)XiG+1={UiG+1iff(UiG+1)≤f(XiG),XiGotherwise,
where i=1,2,…,NP and f is the objective function to be minimized. The value of f of each trial vector UiG+1 is compared with that of its parent target vector XiG. If the value of f, of the target vector XiG, is lower than that of the trial vector, the target vector is allowed to advance to the next generation. Otherwise, the target vector is replaced by a trial vector in the next generation. Thus, all the individuals of the next generation are as good as or better than their counterparts in the current generation. The above steps of reproduction and selection are repeated generation after generation until some stopping criteria are satisfied.In this paper, we define the evaluation function for evaluating the fitness of each individual in the population in DE algorithm as follows:(3.5)f=CT+λ∑t=1T‍(∑i=1N‍Pit-(Dt+Losst))2,
where λ is a penalty value. Then the objective is to find fmin, the minimum evaluation value of all the individuals in all iterations. The penalty term reflects the violation of the equality constraint. Once the minimum of f is reached, the equality constraint is satisfied. Also, the generation power output of each unit at time t should be adjusted to satisfy the following constraints which combine constraints (2.3) and (2.4) as
(3.6)Pit={Pit,minifPit<Pit,min,PitifPit,min≤Pit≤Pit,max,Pit,maxifPit>Pit,max,
where
(3.7)Pit,min={Piminift=1,max(Pimin,Pit-1-DRi)others,Pit,max={Pimaxift=1,min(Pimax,Pit-1+URi)others.


## 4. Sequential Quadratic Programming

SQP method can be considered as one of the best nonlinear programming methods for constrained optimization problems. It outperforms every other nonlinear programming method in terms of efficiency, accuracy, and percentage of successful solutions over a large number of test problems. The method closely resembles Newton's method for constrained optimization, just as is done for unconstrained optimization. At each iteration, an approximation of the Hessian of the Lagrangian function is made using Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton updating method. The result of the approximation is then used to generate a quadratic programming (QP) subproblem whose solution is used to form a search direction for a line search procedure. Since the objective function to be minimized is nonconvex, SQP ensures a local minimum for an initial solution. SQP has been combined with stochastic optimization techniques to constitute hybrid methods for solving the DED problem with nonsmooth cost functions (see [9, 22]). In this paper, DE is used as a global search and finally the best solutions obtained from DE is given as initial condition for SQP method as a local search to fine-tune the solution. SQP simulations are computed by the fmincon code of the MATLAB Optimization Toolbox.


## 5. Simulation Results

In this paper, to assess the efficiency of the proposed DE-SQP method, two case studies (5 units with losses and 10 units without losses, resp.) of DED problems have been considered in which the objective functions are nonsmooth. In each case study, the simulation parameters chosen are population sizeNP=60, maximum iteration Gmax=20000, mutation factor F=0.423, and crossover factor CR=0.885 and the results represent the average of 30 runs of the proposed method. All computations are carried out by MATLAB program.


### 5.1.  Five-Unit System

This example presents an application of the DE-SQP method to the DED problem consisting of five units with valve point effects and transmission line losses. The technical data of the units are taken from [17]. The optimal solution of the DED problem among 30 runs is over, for example, 24h(T=24), and is given in Table 1.Table 1
Hourly generation (MW) schedule of 5-unit system with losses (MW) using DE-SQP.
Hour
P
1
P
2
P
3
P
4
P
5
Loss
1
21.6845
99.3693
113.2191
40.0047
139.3571
3.6348
2
10.0001
98.4091
112.8296
78.0092
139.7746
4.0225
3
10.0000
93.0102
112.6376
124.5983
139.5085
4.7547
4
10.0208
98.9394
112.7739
174.5821
139.6980
6.0141
5
10.0000
94.6503
111.1824
209.3376
139.5944
6.7647
6
39.9323
98.7250
112.7829
210.1512
154.2991
7.8906
7
10.0000
98.0631
112.7434
209.7011
203.9654
8.4729
8
12.5522
98.9663
112.9829
209.5933
229.1619
9.2567
9
42.5522
101.9286
114.7389
210.6817
230.2829
10.1843
10
64.8347
98.4322
112.3767
209.6753
229.2401
10.5591
11
75.0000
100.3215
114.8471
211.1004
229.7591
11.0283
12
75.0000
98.9111
112.9582
234.6857
230.1685
11.7235
13
64.2203
97.9704
112.5101
210.1351
229.7242
10.5602
14
50.0381
98.4511
112.5499
209.7742
229.3542
10.1675
15
35.4162
98.8278
112.6282
186.4410
229.8143
9.1277
16
10.0000
98.4782
112.6986
136.4410
229.6153
7.2332
17
10.0000
88.2951
112.6200
124.6074
229.1609
6.6833
18
35.3553
98.7119
112.7691
139.3772
229.6497
7.8633
19
33.6668
98.5493
111.9794
189.3772
229.5668
9.1396
20
63.6668
98.5990
112.5890
210.0303
229.6762
10.5612
21
39.2846
98.5966
112.7120
209.8748
229.4336
9.9017
22
10.0437
98.6538
112.9970
161.5680
229.6066
7.8692
23
10.0000
98.8754
112.6148
124.7293
186.6872
5.9067
24
10.0000
81.0109
112.1181
124.8490
139.5118
4.4899


### 5.2.   Ten-Unit System

This example presents an application of the DE-SQP method to the DED problem consisting of ten units without losses. The data of the ten-unit system are taken from [9]. The optimal solution of the DED problem is over, for example, 24h(T=24), and is given in Table 2.Table 2
Hourly generation (MW) schedule of 10-unit system without losses using DE-SQP.
Hour
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
1
226.2680
135.0000
73.3721
64.0280
123.9052
123.9200
129.7852
84.7215
20.0000
55
2
223.7127
137.2833
75.0068
114.0280
173.5295
126.5124
129.8165
55.0235
20.0871
55
3
301.2838
217.1962
148.4157
64.4870
123.8118
120.6932
129.5217
77.5907
20.0000
55
4
302.6657
225.8898
197.9257
114.4870
122.4412
160.0000
130.0000
47.5907
50.0000
55
5
302.9885
305.7038
217.5037
119.5778
122.8379
159.9003
129.4879
47.0000
20.0000
55
6
379.2909
308.5845
297.3863
125.2933
172.8379
122.3651
99.4879
47.7542
20.0000
55
7
376.1425
313.2181
299.8249
175.2933
124.3167
160.0000
100.2252
77.7542
20.2252
55
8
455.9173
392.9929
246.5742
130.5711
170.4569
119.9589
130.0000
54.5287
20.0000
55
9
456.3707
401.4499
326.5742
180.5711
173.0766
124.5330
101.8959
84.5287
20.0000
55
10
383.4956
400.8323
328.1355
230.5711
221.4583
160.0000
127.9784
114.5287
50.0000
55
11
455.4912
460.0000
302.9254
245.0551
173.4924
160.0000
128.7601
85.2758
80.0000
55
12
457.9600
460.0000
339.5183
237.7736
223.4387
160.0000
129.3094
77.0000
80.0000
55
13
454.8727
460.0000
300.1122
187.7918
229.2453
160.0000
127.9416
47.0182
50.0182
55
14
378.7648
389.8503
298.0170
226.9459
218.6326
160.0000
129.6473
47.0711
20.0711
55
15
380.4469
309.8503
284.0959
176.9459
221.4171
122.4799
128.8134
76.9505
20.0000
55
16
300.6291
309.9676
204.7668
187.0835
173.1144
126.1948
130.0000
47.1326
20.1112
55
17
305.6309
309.8471
185.3632
186.0791
123.1144
118.3037
129.2531
47.4085
20.0000
55
18
305.1398
311.8823
216.0384
236.0227
124.3315
122.3716
129.9350
77.3435
49.9350
55
19
383.3568
313.4480
293.2465
233.9297
173.3850
121.5403
128.3334
53.7371
20.0233
55
20
385.2071
393.3921
339.9674
281.5235
223.3292
159.9674
129.9320
83.6812
20.0000
55
21
383.3007
395.5746
289.7491
231.5539
222.3756
119.5615
129.8692
76.9978
20.0176
55
22
303.9705
385.3051
209.7887
181.5935
172.4152
122.9631
99.9088
47.0374
50.0176
55
23
224.3924
305.3051
175.5064
131.5960
122.4177
124.8285
95.9138
77.0000
20.0399
55
24
150.0000
225.3051
186.0551
180.6403
73.3908
121.6506
124.9581
47.0000
20.0000
55Comparisons between our proposed method (DE-SQP) and other methods for both examples (five units with losses and ten units without losses) are given in Table3. It is observed that the proposed method reduces the total generation cost better than the other methods reported in the literature. These methods can be classified into (1) heuristic methods such as pattern search [23], particle swarm optimization [17], differential evolution [18], evolutionary programming [9], and modified differential evolution [20], (2) mathematical programming-based methods such as sequential quadratic programming [9], and (3) hybrid methods such as hybrid evolutionary programming and sequential quadratic programming [9], and hybrid particle swarm optimization and sequential quadratic programming [30]. Moreover, it is observed that the transmission line losses calculated by our method are smaller than those of other methods. For more details about these methods and their way of working we refer the reader to the review paper [6].Table 3
Comparison of the results with other methods.
Optimization technique
5-unit system with losses
10-unit system without losses
Cost ($)
Total losses (MW)
Cost ($)
Pattern search [23]
46530
192.21
—
Particle swarm optimization [17]
47852
—
—
Differential evolution [18]
45800
194.35
—
Sequential quadratic programming [9]
—
—
1051163
Evolutionary programming [9]
—
—
1048638
Hybrid evolutionary programming and
Sequential quadratic programming [9]
—
—
1031746
Modified differential evolution [20]
—
—
1031612
Hybrid particle swarm optimization and
Sequential quadratic programming [30]
—
—
1030773
Proposed hybrid differential evolution and
Sequential quadratic programming
43231
193.81
1030500


## 5.1.  Five-Unit System

This example presents an application of the DE-SQP method to the DED problem consisting of five units with valve point effects and transmission line losses. The technical data of the units are taken from [17]. The optimal solution of the DED problem among 30 runs is over, for example, 24h(T=24), and is given in Table 1.Table 1
Hourly generation (MW) schedule of 5-unit system with losses (MW) using DE-SQP.
Hour
P
1
P
2
P
3
P
4
P
5
Loss
1
21.6845
99.3693
113.2191
40.0047
139.3571
3.6348
2
10.0001
98.4091
112.8296
78.0092
139.7746
4.0225
3
10.0000
93.0102
112.6376
124.5983
139.5085
4.7547
4
10.0208
98.9394
112.7739
174.5821
139.6980
6.0141
5
10.0000
94.6503
111.1824
209.3376
139.5944
6.7647
6
39.9323
98.7250
112.7829
210.1512
154.2991
7.8906
7
10.0000
98.0631
112.7434
209.7011
203.9654
8.4729
8
12.5522
98.9663
112.9829
209.5933
229.1619
9.2567
9
42.5522
101.9286
114.7389
210.6817
230.2829
10.1843
10
64.8347
98.4322
112.3767
209.6753
229.2401
10.5591
11
75.0000
100.3215
114.8471
211.1004
229.7591
11.0283
12
75.0000
98.9111
112.9582
234.6857
230.1685
11.7235
13
64.2203
97.9704
112.5101
210.1351
229.7242
10.5602
14
50.0381
98.4511
112.5499
209.7742
229.3542
10.1675
15
35.4162
98.8278
112.6282
186.4410
229.8143
9.1277
16
10.0000
98.4782
112.6986
136.4410
229.6153
7.2332
17
10.0000
88.2951
112.6200
124.6074
229.1609
6.6833
18
35.3553
98.7119
112.7691
139.3772
229.6497
7.8633
19
33.6668
98.5493
111.9794
189.3772
229.5668
9.1396
20
63.6668
98.5990
112.5890
210.0303
229.6762
10.5612
21
39.2846
98.5966
112.7120
209.8748
229.4336
9.9017
22
10.0437
98.6538
112.9970
161.5680
229.6066
7.8692
23
10.0000
98.8754
112.6148
124.7293
186.6872
5.9067
24
10.0000
81.0109
112.1181
124.8490
139.5118
4.4899


## 5.2.   Ten-Unit System

This example presents an application of the DE-SQP method to the DED problem consisting of ten units without losses. The data of the ten-unit system are taken from [9]. The optimal solution of the DED problem is over, for example, 24h(T=24), and is given in Table 2.Table 2
Hourly generation (MW) schedule of 10-unit system without losses using DE-SQP.
Hour
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
1
226.2680
135.0000
73.3721
64.0280
123.9052
123.9200
129.7852
84.7215
20.0000
55
2
223.7127
137.2833
75.0068
114.0280
173.5295
126.5124
129.8165
55.0235
20.0871
55
3
301.2838
217.1962
148.4157
64.4870
123.8118
120.6932
129.5217
77.5907
20.0000
55
4
302.6657
225.8898
197.9257
114.4870
122.4412
160.0000
130.0000
47.5907
50.0000
55
5
302.9885
305.7038
217.5037
119.5778
122.8379
159.9003
129.4879
47.0000
20.0000
55
6
379.2909
308.5845
297.3863
125.2933
172.8379
122.3651
99.4879
47.7542
20.0000
55
7
376.1425
313.2181
299.8249
175.2933
124.3167
160.0000
100.2252
77.7542
20.2252
55
8
455.9173
392.9929
246.5742
130.5711
170.4569
119.9589
130.0000
54.5287
20.0000
55
9
456.3707
401.4499
326.5742
180.5711
173.0766
124.5330
101.8959
84.5287
20.0000
55
10
383.4956
400.8323
328.1355
230.5711
221.4583
160.0000
127.9784
114.5287
50.0000
55
11
455.4912
460.0000
302.9254
245.0551
173.4924
160.0000
128.7601
85.2758
80.0000
55
12
457.9600
460.0000
339.5183
237.7736
223.4387
160.0000
129.3094
77.0000
80.0000
55
13
454.8727
460.0000
300.1122
187.7918
229.2453
160.0000
127.9416
47.0182
50.0182
55
14
378.7648
389.8503
298.0170
226.9459
218.6326
160.0000
129.6473
47.0711
20.0711
55
15
380.4469
309.8503
284.0959
176.9459
221.4171
122.4799
128.8134
76.9505
20.0000
55
16
300.6291
309.9676
204.7668
187.0835
173.1144
126.1948
130.0000
47.1326
20.1112
55
17
305.6309
309.8471
185.3632
186.0791
123.1144
118.3037
129.2531
47.4085
20.0000
55
18
305.1398
311.8823
216.0384
236.0227
124.3315
122.3716
129.9350
77.3435
49.9350
55
19
383.3568
313.4480
293.2465
233.9297
173.3850
121.5403
128.3334
53.7371
20.0233
55
20
385.2071
393.3921
339.9674
281.5235
223.3292
159.9674
129.9320
83.6812
20.0000
55
21
383.3007
395.5746
289.7491
231.5539
222.3756
119.5615
129.8692
76.9978
20.0176
55
22
303.9705
385.3051
209.7887
181.5935
172.4152
122.9631
99.9088
47.0374
50.0176
55
23
224.3924
305.3051
175.5064
131.5960
122.4177
124.8285
95.9138
77.0000
20.0399
55
24
150.0000
225.3051
186.0551
180.6403
73.3908
121.6506
124.9581
47.0000
20.0000
55Comparisons between our proposed method (DE-SQP) and other methods for both examples (five units with losses and ten units without losses) are given in Table3. It is observed that the proposed method reduces the total generation cost better than the other methods reported in the literature. These methods can be classified into (1) heuristic methods such as pattern search [23], particle swarm optimization [17], differential evolution [18], evolutionary programming [9], and modified differential evolution [20], (2) mathematical programming-based methods such as sequential quadratic programming [9], and (3) hybrid methods such as hybrid evolutionary programming and sequential quadratic programming [9], and hybrid particle swarm optimization and sequential quadratic programming [30]. Moreover, it is observed that the transmission line losses calculated by our method are smaller than those of other methods. For more details about these methods and their way of working we refer the reader to the review paper [6].Table 3
Comparison of the results with other methods.
Optimization technique
5-unit system with losses
10-unit system without losses
Cost ($)
Total losses (MW)
Cost ($)
Pattern search [23]
46530
192.21
—
Particle swarm optimization [17]
47852
—
—
Differential evolution [18]
45800
194.35
—
Sequential quadratic programming [9]
—
—
1051163
Evolutionary programming [9]
—
—
1048638
Hybrid evolutionary programming and
Sequential quadratic programming [9]
—
—
1031746
Modified differential evolution [20]
—
—
1031612
Hybrid particle swarm optimization and
Sequential quadratic programming [30]
—
—
1030773
Proposed hybrid differential evolution and
Sequential quadratic programming
43231
193.81
1030500


## 6. Conclusion

This paper presents hybrid method, combining differential evolution (DE), and sequential quadratic programming (SQP) for solving the DED problem with valve-point effects. At first we, applied DE to find the best solution, then this best solution is given to SQP as an initial condition to fine-tune the optimal solution at the final. The feasibility and efficiency of the DE-SQP method are illustrated by conducting two examples consisting of five and ten units with valve-point effects, respectively. Our results are compared with other methods. It has been shown that our proposed methods give less cost than other methods reported in the literature.



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*Source: 184986-2012-09-12.xml*