# Combined Heat and Power Dynamic Economic Dispatch with Emission Limitations Using Hybrid DE-SQP Method

Authors: A. M. Elaiw; X. Xia; A. M. Shehata
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

Combined heat and power dynamic economic emission dispatch (CHPDEED) problem is
a complicated nonlinear constrained multiobjective optimization problem with nonconvex
characteristics. CHPDEED determines the optimal heat and power schedule of committed
generating units by minimizing both fuel cost and emission simultaneously under ramp rate
constraints and other constraints. This paper proposes hybrid differential evolution (DE) and
sequential quadratic programming (SQP) to solve the CHPDEED 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

Recently, combined heat and power (CHP) units, known as cogeneration or distributed generation, have played an increasingly important role in the utility industry. CHP units can provide not only electrical power but also heat to the customers. While the efficiency of the normal power generation is between 50% and 60%, the power and heat cogeneration increases the efficiency to around 90% [1]. Besides thier high efficiency, CHP units reduce the emission of gaseous pollutants (SO2, NOx, CO, and) by about 13–18% [2].In order to utilize the integrated CHP system more CO2 economically, combined heat and power economic dispatch (CHPED) problem is applied. 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 introduces a complication to the problem [3]. 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.In the literature, several optimization techniques have been used to solve the CHPED problem with complex objective functions or constraints such as Lagrangian relaxation (LR) [4, 5], semidefinite programming (SDP) [6], augmented Lagrange combined with Hopfield neural network [7], harmony search (HS) algorithm [1, 8], genetic algorithm (GA) [9], ant colony search algorithm (ACSA) [10], mesh adaptive direct search (MADS) algorithm [11], self adaptive real-coded genetic algorithm (SARGA) [3], particle swarm optimization (PSO) [2, 12], artificial immune system (AIS) [13], bee colony optimization (BCO) [14], differential evolution [15], and evolutionary programming (EP) [16]. In [2, 13–15], the valve point effects and the transmission line losses are incorporated into the CHPED problem.In the CHPED formulation the ramp rate limits of the units are neglected. 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. Combined heat and power dynamic economic dispatch (CHPDED) problem is an extension of CHPED problem where the ramp rate constraint is considered. The primary objective of the CHPDED problem is to determine the heat and power schedule of the committed units so as to meet the predicted heat and electricity load demands over a time horizon at minimum operating cost under ramp rate constraints and other constraints [17]. Since the ramp rate constraints couple the time intervals, the CHPDED problem is a difficult optimization problem. If the ramp rate constraints are not included in the optimization problem, the CHPDED problem is reduced to a set of uncoupled CHPED problems that can easily be solved. In the literature an overwhelming number of reported works deal with CHPED problem; however, the CHPDED problem has only been considered in [17].The traditional dynamic economic dispatch (DED) problem which considers only thermal units that provide only electric power has been studied by several authors (see the review paper [18]). The emission has been taken into the traditional (DED) formulation in three main approaches. The first approach is to minimize the fuel cost and treat the emission as a constraint with a permissible limit (see, e.g., [19–21]). This formulation, however, has a severe difficulty in getting the trade-off relations between cost and emission [22]. The second approach handles both fuel cost and emission simultaneously as competing objectives [23–25]. The third approach treats the emission as another objective in addition to fuel cost objective. However, the multiobjective optimization problem is converted to a single-objective optimization problem by linear combination of both objectives [19, 26–30]. In the second and third approaches, the dynamic dispatch problem is referred to as dynamic economic emission dispatch (DEED) which is a multiobjective optimization problem, which minimizes both fuel cost and emission simultaneously under ramp rate constraint and other constraints [19, 24]. In this paper, we incoroporate the CHP units into the DEED problem. Combined heat and power dynamic economic emission dispatch (CHPDEED) is formulated with the objective to determine the unit power and heat production so that the system’s production cost and emission are simultaneously minimized, while the power and heat demands and other constraints are met [17]. The emission has been taken into consideration in the CHPED and CHPDED in [17, 31], respectively. In [17], both fuel cost and emission are simultaneously handled as competing objectives and the multiobjective problem is solved using an enhanced firefly algorithm (FA). In the present paper, the multiobjective optimization problem is converted into a single-objective optimization using the weighting method. This approach yields meaningful result to the decision maker when solved many times for different values of the weighting factor. In [17], the simulation results for test system are shown, but the data of the heat demand is not explicitly tabulated; instead it is expressed graphically (see Figure 12 in [17]). In this case a comparison of our proposed method and FA cannot be performed. In our paper, all the data and the solutions of the test system are available for comparison.Differential evolution algorithm (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 applied to the CHPED and CHPDED problems with non-smooth and non-convex cost functions in [15, 33], respectively.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 seem to be good methods to solve the CHPDEED problem with non-smooth and non-convex 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 CHPDEED problems with valve point effects.The main contributions of the paper are as follows. (1) A multi-objective optimization problem is formulated using CHPDEED approach. The multi-objective optimization problem is converted into a single-objective optimization using the weighting method. (2) Hybrid DE-SQP method is proposed and validated for solving the CHPDEED 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. (3) The effectiveness of the proposed method is shown for test systems.


## 2. Problem Formulation

In this section we formulate the CHPDEED 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 CHPDEED problem is to simultaneously minimize the system’s production cost and emission 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 CHPDEED problem.


### 2.1. Objective Functions

In this section, we introduce the cost and emission functions of three types of generating units, conventional thermal units which produce power only, CHP units which produce both heat and power, and heat-only units which produce heat only.


#### 2.1.1. Conventional Thermal Units

Cost. The cost function curve of a conventional thermal unit can be approximated by a quadratic function [35]. 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 [18, 36]. 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, 24, 25, 37]. 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.Emission. The amount of emission of gaseous pollutants from conventional thermal units can be expressed as a combination of quadratic function and exponential function of the unit’s active power output [21]. The emission function is given by
(2)EiTU(Pi,tTU)=αi+βiPi,tTU+γi(Pi,tTU)2+ηiexp(δiPi,tTU),
where EiTU(Pi,tTU) is the amount of emission from unit i from producing power Pi,tTU. Constants αi, βi, γi, ηi, and δi are the coefficients of the ith unit emission characteristics [24].


#### 2.1.2. CHP Units

Cost. 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 [6, 17] the following:(3)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 heat Hj,tCHP. Constants a¯j,b¯j,c¯j,d¯j,e¯j, and f¯j are the fuel cost coefficients of CHP unit j.Emission. The emission of gaseous pollutants from CHP units is proportional to their active power output [17, 31]:
(4)EjCHP(Pj,tCHP)=(α¯j+β¯j)Pj,tCHP,
where α¯j and β¯j are the emission coefficients of CHP unit j.


#### 2.1.3. Heat-Only Units

Cost. The cost function of heat-only units can take the following form [6, 17]:
(5)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.Emission. The emission of gaseous pollutants from CHP units is proportional to their heat output [17, 31]:
(6)EkH(HtH)=(α~k+β~k)Hk,tH,
where α~k and β~k are the emission coefficients of heat-only unit k.LetN be the number of dispatch intervals and Np+Nc+Nh 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 and amount of emission over the dispatch period [0,N] are given, respectively, by
(7)C(PH)=∑t=1N(∑i=1NpCiTU(Pi,tTU)+∑j=1NcCjCHP(Pj,tCHP,Hj,tCHP)0000‍+∑k=1NhCkH(Hk,tH)),E(PH)=∑t=1N(∑i=1NpEiTU(Pi,tTU)+∑j=1NcEjCHP(Pj,tCHP)‍‍0000+∑k=1NhEkH(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)′.


### 2.2. Constraints

There are three kinds of constraints considered in the CHPDEED problem, that is, the equilibrium constraints of power and heat production, the capacity limits of each unit, and the ramp rate limits.(i) Power Production and Demand Balance
(8)
∑
i
=
1
N
p
P
i
,
t
TU
‍
+
∑
j
=
1
N
c
P
j
,
t
CHP
‍
=
P
D
,
t
+
Loss
t
,
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 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:(9)Losst=∑i=1Np+Nc‍∑j=1Np+Nc‍PLi,tBijPLj,t,t=1,…,N,
where
(10)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.(ii) Heat Production and Demand Balance
(11)
∑
j
=
1
N
c
‍
H
j
,
t
CHP
+
∑
k
=
1
N
h
‍
H
k
,
t
H
=
H
D
,
t
,
t
=
1
,
…
,
N
,
where HD,t is the system heat demand at time t.(iii) Capacity Limits of Conventional Thermal Units
(12)
P
i
,
min
TU
≤
P
i
,
t
TU
≤
P
i
,
max
TU
,
i
=
1
,
…
,
N
p
,
t
=
1
,
…
,
N
,
where Pi,minTU and Pi,maxTU are the minimum and maximum power capacity of conventional thermal unit i, respectively.(iv) Capacity Limits of CHP Units
(13)
P
j
,
min
CHP
(
H
j
,
t
CHP
)
≤
P
j
,
t
CHP
≤
P
j
,
max
CHP
(
H
j
,
t
CHP
)
,
000000000
j
=
1
,
…
,
N
c
,
t
=
1
,
…
,
N
,
H
j
,
min
CHP
(
P
j
,
t
CHP
)
≤
H
j
,
t
CHP
≤
H
j
,
max
CHP
(
P
j
,
t
CHP
)
,
000000000
j
=
1
,
…
,
N
c
,
t
=
1
,
…
,
N
,
where Pj,minCHP(Hj,tCHP) and Pi,maxCHP(Hj,tCHP) are the minimum and maximum power limit 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).(v) Capacity Limits of Heat-Only Units
(14)
H
k
,
min
H
≤
H
k
,
t
H
≤
H
k
,
max
H
,
k
=
1
,
…
,
N
h
,
t
=
1
,
…
,
N
,
where Hk,minH and Hk,maxH are the minimum and maximum heat capacity of heat-only unit k, respectively.(vi) Upper/Down Ramp Rate Limits of Conventional Thermal Units
(15)
-
D
R
i
TU
≤
P
i
,
t
+
1
TU
-
P
i
,
t
TU
≤
U
R
i
TU
,
0
i
=
1
,
…
,
N
p
,
t
=
1
,
…
,
N
-
1
,
where URiTU and DRiTU are the maximum ramp up/down rates for conventional thermal unit i [18].(vii) Upper/Down Ramp Rate Limits of CHP Units
(16)
-
D
R
j
CHP
≤
P
j
,
t
+
1
CHP
-
P
j
,
t
CHP
≤
U
R
j
CHP
,
000
j
=
1
,
…
,
N
c
,
t
=
1
,
…
,
N
-
1
,
where URjCHP and DRjCHP are the maximum ramp up/down rates for CHP unit j [17].


### 2.3. The Optimization Problem

Aggregating the objectives and constraints, the CHPDEED problem can be mathematically formulated as a nonlinear constrained multi-objective optimization problem which can be converted into a single-objective optimization using the weighting method as(17)00000minPHF(PH)=wC(PH)+(1-w)E(PH),subjecttoconstraints(8)–(16),
where w∈[0,1] is a weighting factor. It will be noted that, when w=1, problem (17) determines the optimal amount of the generated heat and power by minimizing the fuel cost regardless of emission and the problem will be referred to as combined heat and power dynamic economic dispatch (CHPDED) problem. If w=0, then problem (17) determines the optimal amount of the generated power by minimizing the emission regardless of cost and the problem will be referred to as combined heat and power pure dynamic emission dispatch (CHPPDED).


## 2.1. Objective Functions

In this section, we introduce the cost and emission functions of three types of generating units, conventional thermal units which produce power only, CHP units which produce both heat and power, and heat-only units which produce heat only.


### 2.1.1. Conventional Thermal Units

Cost. The cost function curve of a conventional thermal unit can be approximated by a quadratic function [35]. 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 [18, 36]. 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, 24, 25, 37]. 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.Emission. The amount of emission of gaseous pollutants from conventional thermal units can be expressed as a combination of quadratic function and exponential function of the unit’s active power output [21]. The emission function is given by
(2)EiTU(Pi,tTU)=αi+βiPi,tTU+γi(Pi,tTU)2+ηiexp(δiPi,tTU),
where EiTU(Pi,tTU) is the amount of emission from unit i from producing power Pi,tTU. Constants αi, βi, γi, ηi, and δi are the coefficients of the ith unit emission characteristics [24].


### 2.1.2. CHP Units

Cost. 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 [6, 17] the following:(3)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 heat Hj,tCHP. Constants a¯j,b¯j,c¯j,d¯j,e¯j, and f¯j are the fuel cost coefficients of CHP unit j.Emission. The emission of gaseous pollutants from CHP units is proportional to their active power output [17, 31]:
(4)EjCHP(Pj,tCHP)=(α¯j+β¯j)Pj,tCHP,
where α¯j and β¯j are the emission coefficients of CHP unit j.


### 2.1.3. Heat-Only Units

Cost. The cost function of heat-only units can take the following form [6, 17]:
(5)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.Emission. The emission of gaseous pollutants from CHP units is proportional to their heat output [17, 31]:
(6)EkH(HtH)=(α~k+β~k)Hk,tH,
where α~k and β~k are the emission coefficients of heat-only unit k.LetN be the number of dispatch intervals and Np+Nc+Nh 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 and amount of emission over the dispatch period [0,N] are given, respectively, by
(7)C(PH)=∑t=1N(∑i=1NpCiTU(Pi,tTU)+∑j=1NcCjCHP(Pj,tCHP,Hj,tCHP)0000‍+∑k=1NhCkH(Hk,tH)),E(PH)=∑t=1N(∑i=1NpEiTU(Pi,tTU)+∑j=1NcEjCHP(Pj,tCHP)‍‍0000+∑k=1NhEkH(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)′.


## 2.1.1. Conventional Thermal Units

Cost. The cost function curve of a conventional thermal unit can be approximated by a quadratic function [35]. 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 [18, 36]. 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, 24, 25, 37]. 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.Emission. The amount of emission of gaseous pollutants from conventional thermal units can be expressed as a combination of quadratic function and exponential function of the unit’s active power output [21]. The emission function is given by
(2)EiTU(Pi,tTU)=αi+βiPi,tTU+γi(Pi,tTU)2+ηiexp(δiPi,tTU),
where EiTU(Pi,tTU) is the amount of emission from unit i from producing power Pi,tTU. Constants αi, βi, γi, ηi, and δi are the coefficients of the ith unit emission characteristics [24].


## 2.1.2. CHP Units

Cost. 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 [6, 17] the following:(3)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 heat Hj,tCHP. Constants a¯j,b¯j,c¯j,d¯j,e¯j, and f¯j are the fuel cost coefficients of CHP unit j.Emission. The emission of gaseous pollutants from CHP units is proportional to their active power output [17, 31]:
(4)EjCHP(Pj,tCHP)=(α¯j+β¯j)Pj,tCHP,
where α¯j and β¯j are the emission coefficients of CHP unit j.


## 2.1.3. Heat-Only Units

Cost. The cost function of heat-only units can take the following form [6, 17]:
(5)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.Emission. The emission of gaseous pollutants from CHP units is proportional to their heat output [17, 31]:
(6)EkH(HtH)=(α~k+β~k)Hk,tH,
where α~k and β~k are the emission coefficients of heat-only unit k.LetN be the number of dispatch intervals and Np+Nc+Nh 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 and amount of emission over the dispatch period [0,N] are given, respectively, by
(7)C(PH)=∑t=1N(∑i=1NpCiTU(Pi,tTU)+∑j=1NcCjCHP(Pj,tCHP,Hj,tCHP)0000‍+∑k=1NhCkH(Hk,tH)),E(PH)=∑t=1N(∑i=1NpEiTU(Pi,tTU)+∑j=1NcEjCHP(Pj,tCHP)‍‍0000+∑k=1NhEkH(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)′.


## 2.2. Constraints

There are three kinds of constraints considered in the CHPDEED problem, that is, the equilibrium constraints of power and heat production, the capacity limits of each unit, and the ramp rate limits.(i) Power Production and Demand Balance
(8)
∑
i
=
1
N
p
P
i
,
t
TU
‍
+
∑
j
=
1
N
c
P
j
,
t
CHP
‍
=
P
D
,
t
+
Loss
t
,
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 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:(9)Losst=∑i=1Np+Nc‍∑j=1Np+Nc‍PLi,tBijPLj,t,t=1,…,N,
where
(10)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.(ii) Heat Production and Demand Balance
(11)
∑
j
=
1
N
c
‍
H
j
,
t
CHP
+
∑
k
=
1
N
h
‍
H
k
,
t
H
=
H
D
,
t
,
t
=
1
,
…
,
N
,
where HD,t is the system heat demand at time t.(iii) Capacity Limits of Conventional Thermal Units
(12)
P
i
,
min
TU
≤
P
i
,
t
TU
≤
P
i
,
max
TU
,
i
=
1
,
…
,
N
p
,
t
=
1
,
…
,
N
,
where Pi,minTU and Pi,maxTU are the minimum and maximum power capacity of conventional thermal unit i, respectively.(iv) Capacity Limits of CHP Units
(13)
P
j
,
min
CHP
(
H
j
,
t
CHP
)
≤
P
j
,
t
CHP
≤
P
j
,
max
CHP
(
H
j
,
t
CHP
)
,
000000000
j
=
1
,
…
,
N
c
,
t
=
1
,
…
,
N
,
H
j
,
min
CHP
(
P
j
,
t
CHP
)
≤
H
j
,
t
CHP
≤
H
j
,
max
CHP
(
P
j
,
t
CHP
)
,
000000000
j
=
1
,
…
,
N
c
,
t
=
1
,
…
,
N
,
where Pj,minCHP(Hj,tCHP) and Pi,maxCHP(Hj,tCHP) are the minimum and maximum power limit 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).(v) Capacity Limits of Heat-Only Units
(14)
H
k
,
min
H
≤
H
k
,
t
H
≤
H
k
,
max
H
,
k
=
1
,
…
,
N
h
,
t
=
1
,
…
,
N
,
where Hk,minH and Hk,maxH are the minimum and maximum heat capacity of heat-only unit k, respectively.(vi) Upper/Down Ramp Rate Limits of Conventional Thermal Units
(15)
-
D
R
i
TU
≤
P
i
,
t
+
1
TU
-
P
i
,
t
TU
≤
U
R
i
TU
,
0
i
=
1
,
…
,
N
p
,
t
=
1
,
…
,
N
-
1
,
where URiTU and DRiTU are the maximum ramp up/down rates for conventional thermal unit i [18].(vii) Upper/Down Ramp Rate Limits of CHP Units
(16)
-
D
R
j
CHP
≤
P
j
,
t
+
1
CHP
-
P
j
,
t
CHP
≤
U
R
j
CHP
,
000
j
=
1
,
…
,
N
c
,
t
=
1
,
…
,
N
-
1
,
where URjCHP and DRjCHP are the maximum ramp up/down rates for CHP unit j [17].


## 2.3. The Optimization Problem

Aggregating the objectives and constraints, the CHPDEED problem can be mathematically formulated as a nonlinear constrained multi-objective optimization problem which can be converted into a single-objective optimization using the weighting method as(17)00000minPHF(PH)=wC(PH)+(1-w)E(PH),subjecttoconstraints(8)–(16),
where w∈[0,1] is a weighting factor. It will be noted that, when w=1, problem (17) determines the optimal amount of the generated heat and power by minimizing the fuel cost regardless of emission and the problem will be referred to as combined heat and power dynamic economic dispatch (CHPDED) problem. If w=0, then problem (17) determines the optimal amount of the generated power by minimizing the emission regardless of cost and the problem will be referred to as combined heat and power pure dynamic emission dispatch (CHPPDED).


## 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 [32]. 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
(18)000ViG+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
(19)ujiG+1={vjiG+1,if(rand(j)≤CR)orj=rnb(i),xjiG,otherwise,
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 [32].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
(20)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:(21)f=F+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 methods for constrained optimization problems [38]. 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 CHPDEED problem is non-convex and non-smooth, 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 two examples. The first example shows the efficiency of the proposed DE-SQP method for the DED problem. In the second example, the hybrid DE-SQP method is applied to the CHPDEED problem. In DE-SQP method, the control parameters are chosen asNP=80,  ℱ=0.423 and CR=0.885. The maximum number of iterations are selected as 20,000. The results represent the average of 30 runs of the proposed method. All computations are carried out by MATLAB program.Example 1.
This example consists of ten conventional thermal units to investigate the effectiveness of the proposed DE-SQP technique in solving the DED problem with valve point effects and transmission line losses. The technical data of the units as well as the demand for the 10-unit system are taken from [24]. The best solution of the DED problem is given in Table 1. Comparison between our proposed method (DE-SQP) and other methods is given in Table 2. It is observed that the proposed method reduces the total generation cost better than the other methods reported in the literature.Table 1
Hourly generation (MW) schedule obtained from DED using DE-SQP for 10-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
9
TU
P
10
TU
Loss
1
150.0000
135.0000
73.0000
70.3333
222.9974
155.1682
99.2918
120.0000
20.0000
10.0000
19.7912
2
150.0000
135.0000
101.9485
120.3333
222.6154
123.7029
129.2918
90.0000
48.7980
10.7150
22.4058
3
150.0000
135.0000
181.9485
170.3333
174.2621
130.9190
129.6896
120.0000
53.5785
40.7150
28.4468
4
150.0000
135.0000
183.1516
218.2899
223.5485
160.0000
129.3947
120.0000
80.0000
42.0564
35.4415
5
150.0000
135.0000
258.8414
249.7412
224.0147
160.0000
128.5373
120.0000
80.0000
13.2136
39.3484
6
150.0000
135.0000
315.1962
299.7412
243.0000
160.0000
129.8624
120.0000
80.0000
43.2136
48.0136
7
150.0000
176.9470
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
55.0000
52.9470
8
178.2448
228.3049
340.0000
300.0000
243.0000
160.0000
129.9436
120.0000
80.0000
54.9118
58.4054
9
258.2448
308.3049
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
55.0000
70.5500
10
289.0490
384.5331
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
55.0000
79.5821
11
368.7363
397.1230
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
55.0000
87.8595
12
374.8564
439.5807
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
55.0000
92.4378
13
342.1737
386.2429
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
55.0000
84.4166
14
262.1737
306.2429
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
53.1527
70.5693
15
182.1737
226.2429
340.0000
299.9639
243.0000
160.0000
130.0000
120.0000
80.0000
53.0342
58.4148
16
150.0000
146.2429
294.7660
249.9639
223.6700
160.0000
129.6353
120.0000
80.0000
43.3613
43.6398
17
150.0000
135.0000
258.1720
249.5279
223.9121
160.0000
128.8682
120.0000
80.0000
13.8650
39.3459
18
150.0000
151.6366
298.4749
299.5279
243.0000
160.0000
129.7933
120.0000
80.0000
43.6183
48.0511
19
227.2425
231.6366
299.3393
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
43.5728
58.7914
20
307.2425
311.6366
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
55.0000
74.8793
21
265.4293
301.1183
340.0000
300.0000
243.0000
160.0000
130.0000
120.0000
80.0000
55.0000
70.5476
22
185.4293
221.1183
263.3759
250.0000
225.8767
160.0000
129.8685
120.0000
80.0000
41.1109
48.7801
23
150.0000
141.1183
183.3759
200.0000
223.4887
155.9437
128.7427
120.0000
50.0000
11.1109
31.7806
24
150.0000
135.0000
173.1056
180.5739
173.7249
118.1382
128.6826
120.0000
20.0000
10.0000
25.2260Table 2
Comparison results of 10-thermal-unit system (cost ×106 $) for the DED problem.
Method
EP [34]
PSO [34]
AIS [34]
NSGA-II [24]
IBFA [30]
DE-SQP
cost ($)
2.5854
2.5722
2.5197
2.5168
2.4817
2.4659Example 2.
This example is 11-unit system (eight conventional thermal units, two CHP units, and one heat-only unit) for solving the CHPDED, CHPDEED, and CHPPDED problems using DE-SQP method. We shall solve the CHPDEED problem whenw=0.5, in addition to the CHPDED and CHPPDED problems which correspond to w=1 and w=0, respectively. The technical data of conventional thermal units, the matrix B, and the demand are taken from the 10-unit system presented in [24]. The 5th and 8th conventional units in [24] were replaced by two CHP units. The technical data of the two CHP units and the heat-only unit are taken from [17] and are given in Table 3. The heat demand for 24 hours is given in Table 4. The feasible operating regions of the two CHP units are given in Figures 1 and 2 (see [4, 14]).Table 3
Data of the CHP units and heat-only unit system.
CHP units
a
¯
j
b
¯
j
c
¯
j
d
¯
j
e
¯
j
f
j
¯
α
¯
j
β
¯
j
D
R
j
CHP
=
U
R
j
CHP
j
=
1
2650
14.5
0.0345
4.2
0.030
0.031
0.00015
0.0015
70
j
=
2
1250
36
0.0435
0.6
0.027
0.011
0.00015
0.0015
50
Heat-only units
H
k
,
max
H
H
k
,
min
H
a
~
k
b
~
k
c
~
k
α
~
j
β
~
j
k
=
1
2695.2
0
950
2.0109
0.038
0.0008
0.0010Table 4
Heat load demand of the three-unit system for 24 hours.
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.Figure 2
Heat-power feasible operating region for CHP unit 2.The best solutions of the CHPDED, CHPDEED, and CHPPDED problems for DE-SQP algorithm are given in Tables5, 6, and 7, respectively. The best cost, the amount of emission, and the transmission line losses are also given in Tables 5–7. It is seen that the cost is 2.5257×106$ under CHPDED, but it increases to 2.6945×106$ under CHPPDED. The emission obtained from CHPDED is 2.8287×105lb, but it decreases to 2.4195×105lb under CHPPDED. Under the CHPDEED problem, the cost is 2.5295×106$ which is more than 2.5257×106$ and less than 2.6945×106$. Moreover, the emission is 2.7209×105lb which is less than 2.8287×105lb and more than 2.4195×105lb.Table 5
Hourly heat and power schedule obtained from CHPDED.
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. Emission (lb) = 2.8287×105. Total loss (MW) = 1.3443×103.Table 6
Hourly heat and power schedule obtained from CHPDEED (w=0.5).
t
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
77.5875
65.0188
122.5177
129.0996
20.0000
10.0000
238.1722
110.1974
21.5935
49.6499
135.5994
204.7506
2
150.0000
135.0000
73.0000
115.0188
123.4971
126.6027
50.0000
13.3209
237.5744
110.1974
24.2115
53.0122
135.5994
211.3884
3
150.0000
135.0000
135.6390
143.2389
123.5028
130.0000
80.0000
43.3209
237.2689
110.1974
30.1680
54.7308
135.5994
219.6698
4
150.0000
135.0000
197.2495
193.2389
160.0000
130.0000
80.0000
46.3828
241.1942
110.1974
37.2630
32.6533
135.5994
251.7473
5
150.0000
135.0000
227.7945
243.2389
160.0000
130.0000
80.0000
46.9362
238.0276
110.1974
41.1946
50.4634
135.5994
253.9371
6
150.0000
148.0006
307.4622
293.2389
160.0000
130.0000
80.0000
55.0000
244.2131
110.1974
50.1122
15.6745
135.5994
298.7261
7
153.7715
216.0682
309.3974
300.0000
160.0000
130.0000
80.0000
55.0000
242.8740
110.1974
55.3090
23.2061
135.5994
291.1945
8
204.6091
224.9723
327.2185
300.0000
160.0000
130.0000
80.0000
55.0000
244.8130
110.1974
60.8102
12.3003
135.5994
307.1003
9
269.9376
304.9723
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
110.1974
73.1072
0.0
135.5994
324.4006
10
302.8816
379.1708
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
110.2066
82.2590
0.0
135.5384
324.4616
11
374.8455
398.1777
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
111.6628
90.6861
0.0
125.9076
344.0924
12
396.3649
416.9874
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
119.8750
95.2273
0.0
71.5941
408.4059
13
353.1036
382.7187
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
111.3732
87.1956
0.0
127.8228
342.1772
14
273.1036
302.7187
339.8378
300.0000
160.0000
130.0000
80.0000
55.0000
246.2562
110.1974
73.1138
4.1831
135.5994
320.2175
15
213.0095
222.7187
321.3321
300.0000
160.0000
130.0000
80.0000
55.0000
244.5974
110.1974
60.8552
13.5127
135.5994
300.8879
16
150.0000
142.7187
291.8181
250.0000
160.0000
130.0000
80.0000
46.1485
238.7061
110.1974
45.5889
46.6476
135.5994
267.7530
17
150.0000
135.0000
228.5656
240.9760
160.0000
130.0000
80.0000
45.7659
240.7225
110.1974
41.2275
35.3065
135.5994
249.0941
18
150.0000
207.5152
294.3486
250.0000
160.0000
130.0000
80.0000
55.0000
241.3976
110.1974
50.4588
31.5093
135.5994
267.8913
19
227.0251
235.5649
297.1019
300.0000
160.0000
130.0000
80.0000
55.0000
242.1587
110.1974
61.0481
27.2289
135.5994
282.1716
20
307.0251
315.5649
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
114.8749
77.4649
0.00
104.6636
345.3364
21
270.9950
301.2766
340.0000
300.0000
160.0000
130.0000
80.0000
55.0000
247.0000
112.8155
73.0871
0.00
118.2834
326.7166
22
190.9950
221.2766
260.0000
250.0000
157.3134
126.2505
80.0000
43.7439
239.1773
110.1974
50.9541
43.9973
135.5994
255.4033
23
150.0000
141.2766
180.0000
200.0000
154.2635
126.4607
51.1156
13.7439
238.8386
110.1974
33.8966
45.9019
135.5994
218.4987
24
150.0000
135.0000
100.0000
150.0000
118.3525
129.7054
78.8178
10.0000
235.4040
103.7586
27.0385
65.2191
130.0411
204.7398
Cost ($) =2.5295×106. Emission (lb) = 2.7209×105. Total loss (MW) = 1.3439×103.Table 7
Hourly heat and power schedule obtained from CHPPDED.
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
73.0000
60.0000
84.3406
63.6438
64.0384
55
247
125.8
21.8228
0.0
31.4722
358.5278
2
150.0000
135.0000
75.2831
75.5559
107.4058
83.4606
80.0000
55
247
125.8
24.5054
0.0
32.4074
367.5926
3
150.0000
146.7217
108.1787
108.2058
154.2362
113.4606
80.0000
55
247
125.8
30.6030
0.0
18.2661
391.7339
4
187.3290
187.7229
135.7677
135.7127
160.0000
130.0000
80.0000
55
247
125.8
38.3323
0.0
32.4074
387.5926
5
209.9448
210.5929
152.0706
152.2866
160.0000
130.0000
80.0000
55
247
125.8
42.6949
0.0
32.4074
407.5926
6
252.6588
252.9491
188.3610
188.4287
160.0000
130.0000
80.0000
55
247
125.8
52.1977
0.0
25.5244
424.4756
7
272.2261
272.7171
208.2382
208.3486
160.0000
130.0000
80.0000
55
247
125.8
57.3300
0.0
26.7637
423.2363
8
290.5854
291.0583
229.5277
229.7367
160.0000
130.0000
80.0000
55
247
125.8
62.7082
0.0
32.4074
422.5926
9
323.8400
324.1415
276.1324
276.3023
160.0000
130.0000
80.0000
55
247
125.8
74.2162
0.0
25.5487
434.4513
10
346.7105
346.8973
313.1106
300.0000
160.0000
130.0000
80.0000
55
247
125.8
82.5184
0.0
32.4074
427.5926
11
379.2210
379.5185
340.0000
300.0000
160.0000
130.0000
80.0000
55
247
125.8
90.5395
0.0
29.4012
440.5988
12
403.5504
403.8291
340.0000
300.0000
160.0000
130.0000
80.0000
55
247
125.8
95.1796
0.0
31.9845
448.0155
13
361.7512
362.0812
337.4700
300.0000
160.0000
130.0000
80.0000
55
247
125.8
87.1023
0.0
32.0189
437.9811
14
323.7805
324.1252
276.7607
275.7492
160.0000
130.0000
80.0000
55
247
125.8
74.2157
0.0
25.5863
434.4137
15
291.7264
292.3796
231.0966
225.7492
160.0000
130.0000
80.0000
55
247
125.8
62.7519
0.0
31.8710
418.1290
16
229.7976
230.1379
167.7688
175.7492
160.0000
130.0000
80.0000
55
247
125.8
47.2535
0.0
31.3306
418.6694
17
210.0699
210.4074
152.1822
152.2351
160.0000
130.0000
80.0000
55
247
125.8
42.6946
0.0
32.3578
387.6422
18
252.7542
253.2318
188.2091
188.2081
160.0000
130.0000
80.0000
55
247
125.8
52.2031
0.0
29.7791
405.2209
19
288.2429
288.7410
226.6332
237.2113
160.0000
130.0000
80.0000
55
247
125.8
62.6285
0.0
27.4724
417.5276
20
335.1319
335.4397
294.6392
287.2113
160.0000
130.0000
80.0000
55
247
125.8
78.2222
0.0
31.3390
418.6610
21
332.6192
333.0535
282.3523
252.7233
160.0000
130.0000
80.0000
55
247
125.8
74.5483
0.0
32.3100
412.6900
22
252.6192
253.0535
202.3523
202.7233
149.4115
112.3565
80.0000
55
247
125.8
52.3163
0.0
27.3503
407.6497
23
172.6192
173.0535
122.3523
152.7233
135.8629
102.0552
80.0000
55
247
125.8
34.4664
0.0
25.1547
374.8453
24
150.0000
135.0000
90.3380
102.7233
128.7354
96.7805
80.0000
55
247
125.8
27.3771
0..0
31.5331
368.4669
Cost ($) =2.6945×106. Emission (lb) = 2.4195×105. Total loss (MW) =1.3684×103.


## 6. Conclusion

This paper presents a hybrid method combining differential evolution (DE) and sequential quadratic programming (SQP) for solving dynamic dispatch (CHPDED, CHPDEED, and CHPPDED) problems with valve-point effects including generator ramp rate limits. In this paper, DE is first applied to find the best solution. This best solution is given to SQP as an initial condition that fine tunes the optimal solution at the final. The feasibility and efficiency of the DE-SQP were illustrated by conducting case studies with system consisting of eight conventional thermal units, two CHP units, and one heat-only unit.



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*Source: 120849-2013-11-13.xml*