# Hybrid DE-SQP Method for Solving Combined Heat and Power Dynamic Economic Dispatch Problem

Authors: A. M. Elaiw; X. Xia; A. M. Shehata
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Combined heat and power dynamic economic dispatch (CHPDED) plays a key role in economic operation of power systems. CHPDED determines the optimal heat and power schedule of committed generating units by minimizing the fuel cost under ramp rate constraints and other constraints. Due to complex characteristics, heuristic and evolutionary based optimization approaches have became effective tools to solve the CHPDED problem. This paper proposes hybrid differential evolution (DE) and sequential quadratic programming (SQP) to solve the CHPDED problem with nonsmooth and nonconvex cost function due to 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 proposed hybrid DE-SQP method has been tested and compared to demonstrate its effectiveness.


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## Body

## 1. Introduction

In the past decades, increasing demand for power and heat resulted in the existence of combined heat and power (CHP) units, known as cogeneration or distributed generation. It produces electricity and useful heat simultaneously. While the efficiency of the normal power generation is between50% and 60%, the power and heat cogeneration increases the efficiency around 90% [1]. Utilization of CHP units besides conventional thermal power generating units and heat-only units to satisfy heat and power load demands in an economical manner emphasizes the need to combined heat and power economic dispatch (CHPED). The objective of the CHPED problem is to determine both power generation and heat production from units by minimizing the fuel cost such that both heat and power demands are met while the combined heat and power units are operated in a bounded heat versus power plane. For most CHP units, the heat production capacities depend on the power generation. This mutual dependency of the CHP units introduce a complication to the problem [2]. In addition, considering valve point effects in the CHPED problem makes the problem nonsmooth with multiple local optimal point which makes finding the global optimal challenging.Over the past few years, a number of approaches have been developed for solving the CHPED problem with complex objective functions or constraints such as Lagrangian Relaxation (LR) [3, 4], Semidefinite Programming (SDP) [5], augmented Lagrange combined with Hopfield neural network [6], Harmony Search (HS) algorithm [1, 7], Genetic Algorithm (GA) [8], Ant Colony Search Algorithm (ACSA) [9], Mesh Adaptive Direct Search (MADS) algorithm [10], Self Adaptive Real-Coded Genetic Algorithm (SARGA) [2], Particle Swarm Optimization (PSO) [11, 12], Artificial Immune System (AIS) [13], and Evolutionary Programming (EP) [14]. In [11, 13], the valve point effects and the transmission line losses are incorporated into the CHPED problem.The main drawbacks of the CHPED is that it may fail to deal with the large variations of the heat and power load demands due to the ramp rate limits of the units; moreover, it does not have the look-ahead capability. To overcome these drawbacks, combined heat and power dynamic economic dispatch (CHPDED) problem is formulated with the objective to determine the optimal heat and power schedule of the committed units so as to meet the predicted heat and power load demands over a time horizon at minimum operating cost under ramp rate constraints and other constraints [15]. CHPDED has a look-ahead capability which is necessary to schedule the load beforehand so that the system can anticipate sudden changes in power and heat demands in the near future. The ramp rate constraint is a dynamic constraint which is important to maintain the life of the generators [16]. Since the ramp rate constraint couples the time intervals, the CHPDED problem is a difficult optimization problem. If the ramp rate constraint is not included in the optimization problem, the CHPDED problem is reduced to a set of uncoupled CHPED problems that can easily be solved. The traditional dynamic economic dispatch (DED) problem which considers only conventional thermal units that provide only electric power has been studied by several authors (see e.g., [17, 18] and the review paper [16]). However, the CHPDED problem has only been considered in [15, 19].Differential Evolution (DE) algorithm, which was proposed by Storn and Price [20] 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 [20]. Moreover, it has a simple structure and a good convergence property, and it requires a few robust control parameters [20]. DE has been applied to the CHPED problem with nonsmooth and nonconvex cost functions in [21].The DE shares many similarities with evolutionary computation techniques such as Genetic Algorithms (GA) techniques. The system is initialized with a population of random solutions and searches for optima by updating generations. DE has evolution operators such as crossover and mutation. Although DE seems to be a good method to solve the CHPDED problem with nonsmooth and nonconvex cost functions, solutions obtained are just near global optimum with long computation time. Therefore, hybrid methods such as DE-SQP can be effective in solving the CHPDED problem with valve-point effects. Hybrid DE-SQP method has been used for solving the DED problem in [22, 23].The aim of this paper is to propose a hybrid DE-SQP method for solving the CHPDED problem with nonsmooth and nonconvex objective function. 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. The effectiveness of the proposed method is shown for test system.


## 2. Problem Formulation

In this section, we formulate the CHPDED problem. The system under consideration has three types of generating units, conventional thermal units (TU), CHP units, and heat-only units (H). The power is generated by conventional thermal units and CHP units, while the heat is generated by CHP units and heat-only units. The objective of the CHPDED problem is to minimize the system’s production cost so as to meet the predicted heat and power load demands over a time horizon under ramp rate and other constraints. The following objectives and constraints are taken into account in the formulation of the CHPDED problem.


### 2.1. Objective Functions

In this section, we introduce the cost function of three types of generating units, conventional thermal units, CHP units, and heat-only units.Conventional Thermal Units. The cost function curve of a conventional thermal unit can be approximated by a quadratic function [24, 25]. Power plants commonly have multiple valves which are used to control the power output of the unit. When steam admission valves in conventional thermal units are first open, a sudden increase in losses is registered which results in ripples in the cost function [16, 26]. This phenomenon is called as valve-point effects. The generator with valve-point effects has very different input-output curve compared with smooth cost function. Taking the valve-point effects into consideration, the fuel cost is expressed as the sum of a quadratic and sinusoidal functions [17, 19, 27]. Therefore, the fuel cost function of the conventional thermal units is given by
(1)CiTU(Pi,tTU)=ai+biPi,tTU+ci(Pi,tTU)2+|eisin(fi(Pi,minTU-Pi,tTU))|,
where ai,bi, and ci are positive constants, ei and fi are the coefficients of conventional thermal unit i reflecting valve-point effects, Pi,tTU is the power generation of conventional thermal unit i during the tth time interval [t-1,t), Pi,minTU is the minimum capacity of conventional thermal unit i, and CiTU(Pi,tTU) is the fuel cost of conventional thermal unit i to produce Pi,tTU.CHP Units. A CHP unit has a convex cost function in both power and heat. The form of the fuel cost function of CHP units can be given by [5, 19]
(2)CjCHP(Pj,tCHP,Hj,tCHP)=a¯j+b¯jPj,tCHP+c¯j(Pj,tCHP)2+d¯jHj,tCHP+e¯j(Hj,tCHP)2+f¯jPj,tCHPHj,tCHP,
where CjCHP(Pj,tCHP,Hj,tCHP) is the generation fuel cost of CHP unit i to produce power Pj,tCHPand heatHj,tCHP. Constants a¯j,b¯j,c¯j,d¯j,e¯j, and f¯j are the fuel cost coefficients of CHP unit j.Heat-Only Units. Cost: The cost function of heat-only units can take the following form [5, 19]:
(3)CkH(Hk,tH)=a~k+b~kHk,tH+c~k(Hk,tH)2,
where a~k, b~k, and c~k, are the fuel cost coefficients of heat-only unit k and they are constants.LetN be the number of dispatch intervals and let Np+Nc+Nh be the number of committed units, where Np is the number of conventional thermal units, Nc is the number of the CHP units, and Nh is the number of the heat-only units. Then the total fuel cost over the dispatch period [0,N] is given by
(4)C(PH)=∑t=1N‍(∑i=1Np‍CiTU(Pi,tTU)+∑j=1Nc‍CjCHP(Pj,tCHP,Hj,tCHP)+∑k=1Nh‍CkH(Hk,tH)),
where PH=(PH1,PH2,…,PHt,…,PHN)′, PHt=(PtTU,PtCHP,HtCHP,HtH)′,PtTU=(P1,tTU,P2,tTU,…,PNp,tTU)′, PtCHP=(P1,tCHP,P2,tCHP,…,PNc,tCHP)′, HtCHP=(H1,tCHP,H2,tCHP,…,HNc,tCHP)′, and HtH=(H1,tH,H2,tH,…,HNh,tH)′.The CHPDED problem can be mathematically formulated as a nonlinear constrained optimization problem as(5)minPHC(PH)
subject to the constraints.Power production and demand balance:(6)∑i=1Np‍Pi,tTU+∑j=1Nc‍Pj,tCHP=PD,t+Losst,t=1,…,N,
where PD,t and Losst are the system power demand and transmission line losses at time t (i.e., the tth time interval), respectively. The B-coefficient method is one of the most commonly used methods 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 in the following:
(7)Losst=∑i=1Np+Nc‍∑j=1Np+Nc‍PLi,tBijPLj,tt=1,…,N,
where
(8)PLi,t={Pi,tTU,i=1,…,Np,Pi-Np,tCHP,i=Np+1,…,Np+Nc,
and Bij is the ijth element of the loss coefficient square matrix of size Np+Nc.Heat production and demand balance:(9)∑j=1Nc‍Hj,tCHP+∑k=1Nh‍Hk,tH=HD,t,t=1,…,N,
where HD,t is the system heat demand at time t.Capacity limits of conventional thermal units:(10)Pi,minTU≤Pi,tTU≤Pi,maxTU,i=1,…,Np,t=1,…,N,
where Pi,minTU and Pi,maxTU are the minimum and maximum power capacities of conventional thermal unit i, respectively.Capacity limits of CHP units:(11)Pj,minCHP(Hj,tCHP)≤Pj,tCHP≤Pj,maxCHP(Hj,tCHP),j=1,…,Nc,t=1,…,N,Hj,minCHP(Pj,tCHP)≤Hj,tCHP≤Hj,maxCHP(Pj,tCHP),j=1,…,Nc,t=1,…,N,
where Pj,minCHP(Hj,tCHP) and Pj,maxCHP(Hj,tCHP) are the minimum and maximum power limits of CHP unit j, respectively, and they are functions of generated heat (Hj,tCHP). Hj,minCHP(Pj,tCHP) and Hj,maxCHP(Pj,tCHP) are the heat generation limits of CHP unit j which are functions of generated power (Pj,tCHP).Capacity limits of heat-only units:(12)Hk,minH≤Hk,tH≤Hk,maxH,k=1,…,Nh,t=1,…,N,
where Hk,minH and Hk,maxH are the minimum and maximum heat capacities of heat-only unit k, respectively.Upper/down ramp rate limits of conventional thermal units:(13)-DRiTU≤Pi,t+1TU-Pi,tTU≤URiTU,i=1,…,Np,t=1,…,N-1,
where URiTU and DRiTU are the maximum ramp up/down rates for conventional thermal unit i [16].Upper/down ramp rate limits of CHP units:(14)-DRjCHP≤Pj,t+1CHP-Pj,tCHP≤URjCHP,j=1,…,Nc,t=1,…,N-1,
where URjCHP and DRjCHP are the maximum ramp up/down rates for CHP unit j [19].


## 2.1. Objective Functions

In this section, we introduce the cost function of three types of generating units, conventional thermal units, CHP units, and heat-only units.Conventional Thermal Units. The cost function curve of a conventional thermal unit can be approximated by a quadratic function [24, 25]. Power plants commonly have multiple valves which are used to control the power output of the unit. When steam admission valves in conventional thermal units are first open, a sudden increase in losses is registered which results in ripples in the cost function [16, 26]. This phenomenon is called as valve-point effects. The generator with valve-point effects has very different input-output curve compared with smooth cost function. Taking the valve-point effects into consideration, the fuel cost is expressed as the sum of a quadratic and sinusoidal functions [17, 19, 27]. Therefore, the fuel cost function of the conventional thermal units is given by
(1)CiTU(Pi,tTU)=ai+biPi,tTU+ci(Pi,tTU)2+|eisin(fi(Pi,minTU-Pi,tTU))|,
where ai,bi, and ci are positive constants, ei and fi are the coefficients of conventional thermal unit i reflecting valve-point effects, Pi,tTU is the power generation of conventional thermal unit i during the tth time interval [t-1,t), Pi,minTU is the minimum capacity of conventional thermal unit i, and CiTU(Pi,tTU) is the fuel cost of conventional thermal unit i to produce Pi,tTU.CHP Units. A CHP unit has a convex cost function in both power and heat. The form of the fuel cost function of CHP units can be given by [5, 19]
(2)CjCHP(Pj,tCHP,Hj,tCHP)=a¯j+b¯jPj,tCHP+c¯j(Pj,tCHP)2+d¯jHj,tCHP+e¯j(Hj,tCHP)2+f¯jPj,tCHPHj,tCHP,
where CjCHP(Pj,tCHP,Hj,tCHP) is the generation fuel cost of CHP unit i to produce power Pj,tCHPand heatHj,tCHP. Constants a¯j,b¯j,c¯j,d¯j,e¯j, and f¯j are the fuel cost coefficients of CHP unit j.Heat-Only Units. Cost: The cost function of heat-only units can take the following form [5, 19]:
(3)CkH(Hk,tH)=a~k+b~kHk,tH+c~k(Hk,tH)2,
where a~k, b~k, and c~k, are the fuel cost coefficients of heat-only unit k and they are constants.LetN be the number of dispatch intervals and let Np+Nc+Nh be the number of committed units, where Np is the number of conventional thermal units, Nc is the number of the CHP units, and Nh is the number of the heat-only units. Then the total fuel cost over the dispatch period [0,N] is given by
(4)C(PH)=∑t=1N‍(∑i=1Np‍CiTU(Pi,tTU)+∑j=1Nc‍CjCHP(Pj,tCHP,Hj,tCHP)+∑k=1Nh‍CkH(Hk,tH)),
where PH=(PH1,PH2,…,PHt,…,PHN)′, PHt=(PtTU,PtCHP,HtCHP,HtH)′,PtTU=(P1,tTU,P2,tTU,…,PNp,tTU)′, PtCHP=(P1,tCHP,P2,tCHP,…,PNc,tCHP)′, HtCHP=(H1,tCHP,H2,tCHP,…,HNc,tCHP)′, and HtH=(H1,tH,H2,tH,…,HNh,tH)′.The CHPDED problem can be mathematically formulated as a nonlinear constrained optimization problem as(5)minPHC(PH)
subject to the constraints.Power production and demand balance:(6)∑i=1Np‍Pi,tTU+∑j=1Nc‍Pj,tCHP=PD,t+Losst,t=1,…,N,
where PD,t and Losst are the system power demand and transmission line losses at time t (i.e., the tth time interval), respectively. The B-coefficient method is one of the most commonly used methods 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 in the following:
(7)Losst=∑i=1Np+Nc‍∑j=1Np+Nc‍PLi,tBijPLj,tt=1,…,N,
where
(8)PLi,t={Pi,tTU,i=1,…,Np,Pi-Np,tCHP,i=Np+1,…,Np+Nc,
and Bij is the ijth element of the loss coefficient square matrix of size Np+Nc.Heat production and demand balance:(9)∑j=1Nc‍Hj,tCHP+∑k=1Nh‍Hk,tH=HD,t,t=1,…,N,
where HD,t is the system heat demand at time t.Capacity limits of conventional thermal units:(10)Pi,minTU≤Pi,tTU≤Pi,maxTU,i=1,…,Np,t=1,…,N,
where Pi,minTU and Pi,maxTU are the minimum and maximum power capacities of conventional thermal unit i, respectively.Capacity limits of CHP units:(11)Pj,minCHP(Hj,tCHP)≤Pj,tCHP≤Pj,maxCHP(Hj,tCHP),j=1,…,Nc,t=1,…,N,Hj,minCHP(Pj,tCHP)≤Hj,tCHP≤Hj,maxCHP(Pj,tCHP),j=1,…,Nc,t=1,…,N,
where Pj,minCHP(Hj,tCHP) and Pj,maxCHP(Hj,tCHP) are the minimum and maximum power limits of CHP unit j, respectively, and they are functions of generated heat (Hj,tCHP). Hj,minCHP(Pj,tCHP) and Hj,maxCHP(Pj,tCHP) are the heat generation limits of CHP unit j which are functions of generated power (Pj,tCHP).Capacity limits of heat-only units:(12)Hk,minH≤Hk,tH≤Hk,maxH,k=1,…,Nh,t=1,…,N,
where Hk,minH and Hk,maxH are the minimum and maximum heat capacities of heat-only unit k, respectively.Upper/down ramp rate limits of conventional thermal units:(13)-DRiTU≤Pi,t+1TU-Pi,tTU≤URiTU,i=1,…,Np,t=1,…,N-1,
where URiTU and DRiTU are the maximum ramp up/down rates for conventional thermal unit i [16].Upper/down ramp rate limits of CHP units:(14)-DRjCHP≤Pj,t+1CHP-Pj,tCHP≤URjCHP,j=1,…,Nc,t=1,…,N-1,
where URjCHP and DRjCHP are the maximum ramp up/down rates for CHP unit j [19].


## 3. Differential Evolution Method

DE is a simple yet powerful heuristic method for solving nonlinear, nonconvex, 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 [20]. In the initialization, a population of NP target vectors (parents) Xi={x1i,x2i,…,xDi}, i=1,2,…,NP is randomly generated within user-defined bounds, where Dis the dimension of the optimization problem. Let XiG={x1iG,x2iG,…,xDiG} be the individual i at the current generation G. A mutant vector ViG+1=(v1iG+1,v2iG+1,…,vDiG+1) is generated according to
(15)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}. Here ℱ is the mutation factor.According to the target vectorXiG and the mutant vector ViG+1, a new trial vector (offspring) UiG+1={u1iG+1,u2iG+1,…,uDiG+1} is created with
(16)ujiG+1={vjiG+1if(rand(j)≤CR)orj=rnb(i)xjiGotherwise,
where j=1,2,…,D, i=1,2,…,NP, and rand(j) is the jth evaluation of a uniform random number between [0,1]. CR∈[0,1] is the crossover constant which has to be determined by the user. rnb(i) is a randomly chosen index from 1,2,…,D which ensures that UiG+1 gets at least one parameter from ViG+1 [20].The selection process determines which of the vectors will be chosen for the next generation by implementing one-to-one competition between the offsprings and their corresponding parents. Iff denotes the function to be minimized, then
(17)XiG+1={UiG+1iff(UiG+1)≤f(XiG)XiGotherwise,
where i=1,2,…,NP. The value of f of each trial vector UiG+1 is compared with that of its parent target vector XiG. The above iteration process of reproduction and selection will continue until a user-specified stopping criteria is met.In this paper, we define the evaluation function for evaluating the fitness of each individual in the population in DE algorithm as follows:(18)f=C+λ1∑t=1N(∑i=1Np‍Pi,tTU+∑j=1Nc‍Pj,tCHP-(PD,t+Losst))2‍+λ2∑t=1N‍(∑j=1Nc‍Hj,tCHP+∑k=1Nh‍Hk,tH-HD,t)2,
where λ1 and λ2 are penalty values. 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 constraints. Once the minimum of f is reached, the equality constraints are satisfied.


## 4. Sequential Quadratic Programming Method

SQP method can be considered as one of the best nonlinear programming method for constrained optimization problems [28]. 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 is made of the Hessian of the Lagrangian function 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 of the CHPDED problem is nonconvex and nonsmooth, SQP ensures a local minimum for an initial solution. In this paper, DE is used as a global search and finally the best solution obtained from DE is given as initial condition for SQP method as a local search to fine-tune the solution. SQP simulations can be computed by the fmincon code of the MATLAB Optimization Toolbox.


## 5. Simulation Results

In this section, we present an eleven-unit test system. The hybrid DE-SQP method is applied to the CHPDED problem, where three types of generating units, conventional thermal units, CHP units, and heat-only units, are considered. In DE-SQP method, the control parameters are chosen asNP=80,ℱ=0.423, and CR=0.885. The maximum number of iterations is selected as 20,000. The results represent the average of 30 runs of the proposed method. All computations are carried out by MATLAB program.Eleven-Unit System. This system consists of eight conventional thermal units, two CHP units, and one heat-only unit. The CHPDED problem is solved by hybrid DE-SQP method. The technical data of conventional thermal units, the matrix B, and the power demand are taken from the ten-unit system presented in [27]. The 5th and 8th conventional units in [27] were replaced by two CHP units. The technical data of the two CHP units and the heat-only unit are taken from [19] and are given in Table 1. The heat demand for 24 hours is given in Table 2. The feasible operating regions of the two CHP units are taken from [3] and are given in Figures 1 and 2.Table 1
Data of the CHP units and heat-only unit of the eleven-unit system.
CHP units
a
¯
j
b
¯
j
c
¯
j
d
¯
j
e
¯
j
f
j
¯
D
R
j
CHP
=
U
R
j
CHP
j
=
1
2650
14.5
0.0345
4.2
0.030
0.031
70
j
=
2
1250
36
0.0435
0.6
0.027
0.011
50
Heat-only units
H
k
,
max
H
H
k
,
min
H
a
~
k
b
~
k
c
~
k
k
=
1
2695.2
0
950
2.0109
0.038Table 2
Heat load demand of the eleven-unit system.
Time (h)
Demand (MWth)
1
390
2
400
3
410
4
420
5
440
6
450
7
450
8
455
9
460
10
460
11
470
12
480
13
470
14
460
15
450
16
450
17
420
18
435
19
445
20
450
21
445
22
435
23
400
24
400Figure 1
Heat-power feasible operating region for CHP unit 1 of the eleven-unit system.Figure 2
Heat-power feasible operating region for CHP unit 2 of the eleven-unit system.The best solution of the CHPDED problem obtained by DE-SQP algorithm is given in Table3. The best cost and transmission line losses are also given in Table 3.Table 3
Hourly heat and power schedule obtained from CHPDED using DE-SQP for eleven-unit system.
H
P
1
TU
P
2
TU
P
3
TU
P
4
TU
P
5
TU
P
6
TU
P
7
TU
P
8
TU
P
1
CHP
P
2
CHP
Loss
H
1
CHP
H
2
CHP
H
1
H
1
150.0000
135.0000
74.5372
72.0784
124.5129
124.4302
20.0000
10.0000
236.8041
110.1974
21.5630
57.3450
135.5994
197.0556
2
150.0000
135.0000
98.1135
122.0784
122.2113
101.6179
48.2025
10.0000
236.8011
110.1974
24.2248
57.3614
135.5994
207.0392
3
150.0000
135.0000
178.1135
172.0784
120.7640
98.7468
78.2025
10.0000
235.3275
110.1974
30.4319
65.6496
135.5994
208.7509
4
150.0000
135.0000
188.0106
218.5077
160.0000
126.3142
80.0000
40.0000
235.2182
110.1974
37.2496
66.2643
135.5994
218.1363
5
150.0000
135.0000
268.0106
244.7145
128.0292
129.9179
80.0000
42.2707
233.2313
110.1974
41.3736
77.4390
135.5994
226.9616
6
150.0000
135.0000
334.4706
294.7145
160.0000
130.0000
80.0000
48.0931
235.6609
110.1974
50.1383
63.7746
135.5994
250.6260
7
150.0000
199.1593
340.0000
300.0000
160.0000
130.0000
80.0000
49.7990
238.0991
110.1974
55.2549
50.0614
135.5994
264.3392
8
189.7336
229.5497
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
242.2569
110.1974
60.7377
26.6766
135.5994
292.7240
9
265.3596
309.5497
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
110.1974
73.1068
0.0
135.5994
324.4006
10
303.6024
378.5162
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
246.9410
110.1974
82.2580
0.3317
135.5994
324.0689
11
368.8317
405.6648
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
110.1974
90.6945
0.0
135.5994
334.4006
12
367.7179
455.4472
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
110.1974
95.3624
0.0
135.5994
344.4006
13
352.0071
385.0034
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
110.1974
87.2079
0.0
135.5994
334.4006
14
272.0071
305.0034
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
244.9090
110.1974
73.1169
11.7604
135.5994
312.6402
15
193.6233
225.0034
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
242.9121
110.1974
60.7362
22.9917
135.5994
291.4089
16
150.0000
145.0034
296.8330
250.8703
160.0000
129.9573
80.0000
43.4626
233.2660
110.1974
45.5900
77.2439
135.5994
237.1567
17
150.0000
135.0000
260.0109
250.0000
160.0000
100.0000
80.0000
40.9143
235.3888
110.1974
41.5121
65.3046
135.5994
219.0959
18
150.0000
151.0646
319.4485
300.0000
160.0000
130.0000
80.0000
40.0577
237.4722
110.1974
50.2419
53.5869
135.5994
245.8137
19
229.4141
231.0646
313.3779
300.0000
160.0000
130.0000
80.0000
46.0360
237.0065
110.1974
61.0988
56.2062
135.5994
253.1943
20
309.4141
311.0646
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
116.9757
77.4552
0.0
90.7694
359.2306
21
272.4577
300.8037
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
111.8344
73.0959
0.0
124.7723
320.2277
22
192.4577
220.8037
260.6669
250.0000
160.0000
124.1397
80.0000
45.9763
234.6724
110.1974
50.9154
69.3338
135.5994
230.0668
23
150.0000
140.8037
180.6669
200.0000
127.6584
130.0000
50.0000
40.0000
236.4213
110.1974
33.7482
59.4980
135.5994
204.9026
24
150.0000
135.0000
100.6669
177.0362
123.2649
128.6636
42.3316
10.0000
234.6572
109.5624
27.1834
69.4196
135.0513
195.5291
Cost ($)=2.5257×106; total loss (MW) =1.3443×103.


## 6. Conclusion

This paper presents a hybrid method combining DE and SQP for solving the CHPDED problem with valve-point effects. In this paper, DE is first applied to find the best solution. 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 were illustrated by conducting case study with eleven-unit test system.



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*Source: 982305-2013-09-21.xml*