This article was downloaded by: [Universidad De Concepcion] On: 30 May 2014, At: 03:18 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Electric Power Components and Systems Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/uemp20
A Lexicographic Optimization and Augmented ϵconstraint Technique for Short-term Environmental/ Economic Combined Heat and Power Scheduling a
b
Abdollah Ahmadi , Mohammad Reza Ahmadi & Ali Esmaeel Nezhad
c
a
Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Fars, Iran b
Department of Civil Engineering, Science and Research Branch, Islamic Azad University, Fars, Iran c
Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran Published online: 28 May 2014.
To cite this article: Abdollah Ahmadi, Mohammad Reza Ahmadi & Ali Esmaeel Nezhad (2014) A Lexicographic Optimization and Augmented ϵ-constraint Technique for Short-term Environmental/Economic Combined Heat and Power Scheduling, Electric Power Components and Systems, 42:9, 945-958, DOI: 10.1080/15325008.2014.903542 To link to this article: http://dx.doi.org/10.1080/15325008.2014.903542
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
Downloaded from http://www.elearnica.ir
Electric Power Components and Systems, 42(9):945–958, 2014 C Taylor & Francis Group, LLC Copyright ISSN: 1532-5008 print / 1532-5016 online DOI: 10.1080/15325008.2014.903542
A Lexicographic Optimization and Augmented ε-constraint Technique for Short-term Environmental/Economic Combined Heat and Power Scheduling Abdollah Ahmadi,1 Mohammad Reza Ahmadi,2 and Ali Esmaeel Nezhad3 1
Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Fars, Iran Department of Civil Engineering, Science and Research Branch, Islamic Azad University, Fars, Iran 3 Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
2
CONTENTS 1. Introduction 2. Multi-Objective Problem Description and Formulation 3. Mmp 4. Case Study and Simulation Results 5. Conclusion References Appendix A
Abstract—In recent years, combined heat and power units have become significant elements in conventional power stations due their numerous merits, including operational cost savings and reduced emissions. In this regard, this article proposes a short-term multi-objective framework for the combined heat and power economic/emission dispatch problem. In addition, to more precisely model the problem, the non-linear forms of fuel cost functions and valve-point loading along with power transmission loss are considered. The objectives of the problem are total cost minimization as well as minimization of pollutant emissions; lexicographic optimization and the augmented epsilon-constraint technique are employed to solve the multi-objective problem. Also, a fuzzy decision making technique has been used to select the most preferred solution among the Pareto solutions. Afterward, a comprehensive comparison is performed between the results obtained from the proposed method and those derived from the non-dominated sorting genetic algorithm II, strength Pareto evolutionary algorithm 2, and multi-objective line-up competition algorithm, verifying the superiority of the presented approach for lower execution time, total cost, and emission. Furthermore, the proposed model is implemented on a large-scale test system while the execution time is rational.
1. INTRODUCTION
Keywords: economic/emission dispatch, multi-objective mathematical programming, fuzzy decision-making technique, combined heat and power, valve-point loading Received 27 June 2013; accepted 8 March 2014 Address correspondence to Mr. Abdollah Ahmadi, Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Fars, P.O. Box 73481-13111, Iran. E-mail:
[email protected]
Cogeneration or combined heat and power (CHP) generation is considered to be a mature and well-established technology in the center of attention due to its high energy efficiency and environmental advantages compared to conventional forms of energy supply. However, economic dispatch (ED) should be implemented to attain the optimal use of such units [1, 2]. In most published studies, the objective considered for the CHP ED (CHPED) problem is to find the optimal point of power and heat generation along with minimum fuel cost in a way that the demands of heat and power, as well as other constraints, are 945
946
Electric Power Components and Systems, Vol. 42 (2014), No. 9
NOMENCLATURE
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
Indices h, i, j heat-only, thermal, and combined heat and power unit indices, respectively Constants cost coefficients of heat-only unit h a h , bh , ch cost coefficients of thermal unit i ai , bi , ci , di , ei a j , b j , c j , d j , e j , f j cost coefficients of combined heat and power unit j loss coefficient parameter (MW) B00 loss coefficient associated with B0i production of electricity power generating unit i loss coefficient relating to production Bi j of electricity power generating units i and j (1/MW) minimum and maximum heat output H Cj H P , H¯ jC H P of combined heat and power unit j, respectively (MWth) minimum and maximum heat output of H hH , H¯ hH heat-only unit h, respectively (MWth) number of combined heat and power, NC H P , N T U , N H thermal, and heat-only units, respectively Variables F1 , F2 total operation costs ($) and emissions (kg), respectively heat output of combined heat and power unit H jC H P , HhH j and heat-only unit h, respectively (MWth)
satisfied, while CHP units operate in a restricted heat versus power plane. Different optimization techniques have been proposed to solve the CHPED problem. A decomposition-based algorithm to deal with the ED problem for cogeneration systems was presented in [3]. A self-adaptive real-coded genetic algorithm (SARGA) was applied to the CHPED problem in [4], wherein the self-adaptation was obtained using tournament selection together with simulated binary crossover. In [5], a two-level framework was proposed to solve the CHPED; an artificial immune system algorithm is used in [6] to deal with the CHPED problem. An optimization model for a system comprising both CHP units and wind turbines was proposed in [7], in which ED was analyzed and the probability of stochastic
j’
linear inequality constraint index of combined heat and power feasible operation region
Nlin
PD P Cj H P , P¯ jC H P
P iT U , P¯iT U x j , j , y j , j , z j , j
αh , βh , γh αi , βi , γi , ξi , λi , τi αj, βj, γj
CHp
H Cj H P , H j
PLoss PiT U , P jC H P
number of linear inequality constraints of combined heat and power feasible operation region electricity load demand (MW) minimum and maximum power output of combined heat and power unit j, respectively (MW) minimum and maximum power output of thermal unit i, respectively (MW) coefficients of power heat feasible operation region of linear inequality equation j for combined heat and power unit j emission coefficients of heat-only unit h emission coefficients of thermal unit i emission coefficients of combined heat and power unit j
lower and upper limits of jth combined heat and power unit output heat, respectively (MWth) real power loss (MW) power output of thermal unit i and combined heat and power unit j, respectively (MW)
wind power generation taken into account as one of the constraints of the problem. The authors of [8] considered a model for CHP reliability and availability based on a state-space and continuous Markov approach with electricity-generation, fuel-distribution, and heat-generation subsystems. The CHPED problem was solved in [9] utilizing direct search method. The non-convex operating region in the CHPED problem was taken into consideration more precisely using a technique proposed in [10]; in this method, the non-convex operating region was divided into convex operating sub-regions utilizing two binary variables introduced to the problem representing the searching region. Thermo economic analysis was performed in [11] with the purpose of providing cost-based information and suggesting a plausible site to enhance the CHP system,
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
Ahmadi et al.: Short-term Environmental/Economic Combined Heat and Power Scheduling
where the specific exergy costing (SPECO) method was used. The authors of [12] dealt with the problem of ED for a smart home integrated energy system, including CHP and the battery connected to the utility. Numerous optimization algorithms have been proposed based on stochastic searching techniques to derive accurate dispatch results, including differential evolution (DE) [13], semi-definite programming (SDP) [14], evolutionary programming (EP) [15], the ant colony search algorithm (ACSA) [16], the genetic algorithm (GA) [17], and harmony search algorithm (HSA) [18, 19], which can be developed to deal with the highly non-linear CHP dispatch problem without any specific limitation on the shape of fuel cost functions. A comprehensive review on the exergoeconomic analysis and optimization of CHP units was represented in [20]. After the Clean Air Act amendments was passed in 1990 [21], the requirements of the generation scheduling problem are no longer met through conventional pure economic scheduling, and emission concerns must be unavoidably taken into consideration. An appropriate method to model these pollutants is to use a polynomial function for SO2 and an exponential function for NOx [22]. In addition, a linear function of power and heat generation is used to model CO2 emission [23]. A multi-objective optimization framework based on the non-dominated sorting GA II (NSGA-II) was proposed in [22] to solve the CHP economic emission dispatch problem. In [23], the multi-objective line-up competition algorithm (MLCA) was applied to the multi-objective CHP economic/emission dispatch (MCHPEED) problem. The main focus of [24] was on the comparison between a meta-heuristic method, i.e., the HSA, and an analytical solution method; that study also presented a single-objective optimization framework that is straightforward to solve, while the present article proposes an efficient multi-objective optimization framework, i.e., augmented ε-constraint method, with lexicographic optimization in addition to a fuzzy decision-making procedure. It is noted that emission issues are considered herein, while the authors of [24] discarded the emission generation. Furthermore, a precise model has been proposed for emission generation; an important item, i.e., losses, was ignored in [24], while the present work has well modeled the inseparable issues of power systems. The main contributions of this artice can be summarized as follows: • an optimal generation strategy for conventional, cogeneration, and heat-only units is proposed; • a fuzzified ε-constraint technique to minimize the total cost and the emission is employed; and
947
• a comprehensive comparison between the results obtained from the proposed method and those derived from NSGA-II, strength Pareto evolutionary algorithm 2 (SPEA-2), and MLCA is provided. The remainder of this article is organized as follows. Section 2 presents a detailed formulation of the proposed multiobjective problem. Section 3 includes the major concept in the multi-objective mathematical programming (MMP) problem. Numerical results are given and discussed in Section 4, and some relevant conclusions are provided in Section 5.
2.
MULTI-OBJECTIVE PROBLEM DESCRIPTION AND FORMULATION
The multi-objective optimization problem in the form of nonlinear programming (NLP) is indicated as F1 Cost Minimization , Multi−objective functions = F2 Emission Minimization (1) where the main objective of the problem includes production cost, emission concerns pertaining to conventional (thermal) generation units, as well as heat generation, taking into consideration three different options of generation units as poweronly units, CHP units, and heat-only units. 2.1.
Objective Functions
The objective functions considered in this paper can be stated as follows: F1 =
NT U
CHP N C T Ui PiT U + CC H P j P jCHP
i=1
j=1
+
NH
C Hh HhH ,
(2)
h=1
F2 =
NT U
CHP N E T Ui PiT U + E C H P j P jCHP
i=1
j=1
+
NH
E Hh HhH ,
(3)
h=1
where cost and emission functions of generation units and heat generation of CHP and heat-only units under three different operational options are denoted by functions G(), E(), and H(), respectively. Detailed statements of G() and E() are separately given below regarding each operational condition. 2.1.1.
Thermal Units (TUs)
The first parts of the objective functions above indicate the cost of power generation and emission functions, respectively.
948
Electric Power Components and Systems, Vol. 42 (2014), No. 9
Equations (4) and (5)represent the detailed formulation of these functions: NT U 2 ai + bi PiT U + ci PiT U G1 P T U = + di sin ei P iT U − PiT U , NT U 2 TU E 1 (P ) = αi + βi PiT U + γi PiT U i=1
(4)
i=1
NT U TU + + ξi exp λi Pi τi PiT U .
(5)
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
i=1
It is worth mentioning that the theoretical cost function pertaining to each TU that is presented as a quadratic function with a smooth manner is relatively different from what really exists in practical cases. In fact, the fuel-cost curve is exposed to a sharp increase in fuel loss because of wire drawing effects caused by opening the steam admission valve. This phenomenon is identified as the valve point effect, indicated by the superposition of quadratic and sinusoidal functions being similar to the statement in Eq. (4). SO2 and NOx emission generation could be observed in Eq. (5)as the first summation, while CO2 emission caused by TUs is depicted by the second summation. 2.1.2.
FIGURE 1. Power heat FOR for CHP unit.
+
CHP or cogeneration is defined as the simultaneous generation of electricity and heat using a single part of the plant so that the electric power demands, as well as the district heat demands, are supplied. As depicted in Figure 1, a power heat feasible operation region (FOR) exists for each CHP unit that can be declared using a set of Nlin linear inequality constraints as follows:
j=1
+
d j H jCHP
+ ej
The purpose of using heat-only units as an ancillary option is to raise the flexibility of CHP units to meet high heat load demands. Equations (9) and (10)represent cost and emission functions of heat-only units, respectively: G 3 (H H ) =
(6)
2 H jCHP
+ f j P jCHP H jCHP , E2
PtCHP
=
N CHP j=1
α j + β j P jCHP
(7)
Heat-only Units
E 3 (H H ) =
The second parts of Eqs. (2) and (3)denoting the operational cost and emission functions of CHP units can be stated in detail as CHP CHP CHP N 2 a j + b j P jCHP + ci P jCHP = ,H G2 P
(8)
SO2 and NOx emission generation are shown by the first summation in Eq. (8), while CO2 emission caused by CHP units appears in the second summation.
x f, j H jCHP + y f, j P jCHP ≥ z f, j j = 1, ..., Nlin ; j = 1, ..., NCHP .
γ j P jCHP ;
j=1
2.1.3.
CHP Units
N CHP
NH ah + bh HhH + ch (HhH )2 , h=1 NH h=1
NH (αh + βh )HhH + γh HhH ;
(9)
(10)
h=1
SO2 and NOx emission generation is indicated by the first summation in Eq. (10), while CO2 emission generated by heatonly units is represented by the second summation. 2.2.
Constraints
In general, two main categories of major constraints are considered in the MCHPEED problem. The first category relates to the generation constraints, such as system real power and heat balance (Eqs. (11) and (12), bounds on real power generation (Eqs. (13) and (14)), as well as limitations on heat production (Eqs. (15) and (16)). These constraints are presented in mathematical form in what follows.
Ahmadi et al.: Short-term Environmental/Economic Combined Heat and Power Scheduling
2.2.1.
Real Power Balance
are taken as constraints:
The equality of total generated power by TUs and CHP units with the total electric power demand at each time of scheduling is ensured through real power balance constraint as NT U
PiT U +
N CHP
i=1
P jCHP = PD + PLoss ,
j=1
(11)
where the power loss is denoted by PLoss , which is a function of generation unit power output and B-loss coefficients and can be stated in detail as
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
PLoss =
NT U NT U
(PiT U )T Bi j PiT U +
i=1 j=1
+
NT U N CHP
(P jCHP )T Bi j P jCHP +
i=1 j=1
+
N CHP
(PiT U )T Bi j P jCHP
i=1 j=1
N CHP N CHP
NT U
(B0i )T PiT U
i=1
(B0i )T PiCHP + B00 .
(12)
i=1
2.2.2.
Power Generation Limits
The lower and upper bounds on the generation of thermal and CHP units can be indicated as follows: P iT U ≤ PiT U ≤ P¯iT U
i = 1, ..., N T U ,
P jCHP ≤ P jCHP ≤ P¯ jCHP
2.2.3.
j = 1, ..., NCHP .
(13) (14)
Limits of Heat Generation
The following equations express the lower and upper bounds on the heat generation of CHP and heat-only units, respectively, CHP
(P jCHP ) ≤ H jCHP ≤ H j H CHP j
(P jCHP )
j = 1, ..., NCHP , (15)
H hH 3.
≤
H Hh,t
≤
H Hh
949
h = 1, ..., N H .
(16)
MMP
The concept of MMP involves more than one objective function, and as a result, a single optimal solution concurrently optimizing all objective functions is no longer valid [25]. In this regard, a ε-constraint method would be a well-organized technique to deal with MMP problems. This method takes into consideration one of the objective functions as the main one. Generally, the ε-constraint approach [26, 27] optimizes this main objective function f 1 , while all other objective functions
¯ minimi ze f 1 (x) ¯ ≤ e2 subject to f 2 (x)
¯ ≤ e3 · · · f p (x) ¯ ≤ ep, f 3 (x) (17) where the number of conflicting and competing objective functions pertaining to the MMP problem and the vector of decision ¯ respectively. In Eq. (17), all variables are denoted by p and x, p objective functions are supposed to be minimized. To well implement the ε-constraint technique, at least the ranges of p – 1 objective functions must be known, which will be considered extra objective function constraints. A payoff table would be the most common method to calculate these ranges. The first step in constructing the payoff table for an MMP problem including p competing objective functions is to calculate the individual optima of objective functions fi . f i∗ (x¯i∗ ) denotes the optimum value of fi , while the vector of decision variables optimizing objective functions are indicated by x¯i∗ . Afterward, the values of other objective functions f 1 , f 2 ,. . ., fi – 1 , fi + 1 ,. . ., fp are calculated using the solution optimizing the objective function fi . These values are denoted by f 1 (x¯i∗ ), f 2 (x¯i∗ ), . . ., f i−1 (x¯i∗ ), f i+1 (x¯i∗ ), . . ., f p (x¯i∗ ). The ith row of the payoff table represents f 1 (x¯i∗ ), f 2 (x¯i∗ ), . . ., f i∗ (x¯i∗ ), . . ., f p (x¯i∗ ). All other rows of the payoff table are calculated in the same way; ⎞ ⎛ ∗ ∗ f 1 (x¯1 ) · · · f i (x¯1∗ ) · · · f p (x¯1∗ ) .. ⎟ ⎜ .. .. ⎜ . . . ⎟ ⎟ ⎜ ∗ ∗ ∗ ⎜ (18) = ⎜ f 1 (x¯i ) · · · f i (x¯i ) · · · f p (x¯i∗ ) ⎟ ⎟. ⎟ ⎜ . . . .. .. ⎠ ⎝ .. ∗ ∗ ∗ ∗ f 1 (x¯ p ) · · · f i (x¯ p ) · · · f p (x¯ p ) The payoff table includes p rows and columns. The obtained values for objective function fj are included in the jth column of this table; among them, the minimum and maximum values represent the range of objective function fj for the ε-constraint technique. To properly apply the ε-constraint method to the presented MMP problem, a few concepts must first be introduced. Without losing generality, it is again assumed that all objective functions are intended to be minimized. The first concept relates to the utopia point, which is a specific point usually outside the feasible region and corresponds to all objective functions that concurrently have their best plausible values. The mathematical presentation of the utopia point would be as follows: f U = f 1U , ..., f iU , ..., f pU = f 1∗ (x¯1∗ ), ..., f i∗ (x¯i∗ ), ..., f p∗ (x¯ ∗p ) .
(19)
The next concept concerns the nadir point, which is a point in the objective space where all objective functions
950
Electric Power Components and Systems, Vol. 42 (2014), No. 9
simultaneously have their worst plausible values: f N = f 1N , ..., f i N , ..., f pN ,
(20)
where ¯ f iN = max f i (x)
subject to x¯ ∈ .
x¯
(21)
The feasible region is denoted by . The pseudo-nadir point is another point having a close concept to the nadir point. This point can be stated as (22) f S N = f 1S N , ..., f iS N , ..., f pS N ,
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
f iS N = max f i (x¯1∗ ), ..., f i∗ (x¯i∗ ), ..., f i (x¯ ∗p ) .
(23)
Note that all three of these points are defined in the objective space, which is a vector space with the objective functions as its dimensions. The range of each objective function in the payoff table is calculated using the utopia and pseudo-nadir points, as in the following: ¯ ≤ f iS N . f iU ≤ f i (x)
(24)
The purpose of solving multi-objective optimization problems is to determine the set of Pareto optimal solutions. In a general MMP problem, The Pareto optimal (Pareto efficient) solution is denoted by a point x¯ ∗ ∈ , if and only if there is ¯ ≤ f i (x¯ ∗ ) for all i = 1, 2, . . ., p with no x¯ ∈ such that f i (x) at least one strict inequality. In the next step, after the ranges of all objective functions are identified, the ε-constraint method divides the range of p – 1 objective functions f 2 , . . ., fp to q2 , . . ., qp into equal intervals utilizing (q2 – 1), . . ., (qp – 1) intermediate equidistant grid points, respectively. There are a total of (q2 + 1), . . ., (qp + 1) grid points for f 2 , . . ., fp , respectively, taking into consideration the minimum and maximum values of the p range. Thus, i=2 (qi + 1) optimization sub-problems must be solved, wherein the sub-problem (n2, . . ., np) takes the following form: ¯ Minimi ze f 1 (x) ¯ ≤ e2,n2 , subject to f 2 (x) e2,n2 = f 2S N −
f 2S N − f 2U q2
· · ·,
¯ ≤ e p,np , f p (x)
(25)
× n2
n2 = 0, 1, ..., q2 , (26)
e2,np =
f pS N
−
f pS N − f pU qp
× np
np = 0, 1, ..., q p ,
(27) where the values of the objective function in the utopia and pseudo-nadir points are denoted by superscripts U and SN, as
depicted in Eqs. (19) and (23), respectively. It is noted that each of these optimization sub-problems takes into consideration the constraints of the MMP problem in addition to the objective function constraints described in Eq. (25). Solving each of the optimization sub-problems leads to one Pareto optimal solution in the ε-constraint method. Optimization sub-problems with infeasible solution space are all discarded through this technique. The decision maker chooses the most desired solution among the derived Pareto optimal solutions. This method has the following advantages over the weighting method: • only efficient extreme solutions can be generated through weighting methods for linear problems, while the εconstraint technique has the ability to generate nonextreme efficient solutions [26, 27]; • unlike the weighting method, the ε-constraint approach is able to generate unsupported efficient solutions in multiobjective integer and mixed-integer programming (MIP) problems [26, 27]; • the weighting method requires scaling objective functions; on the contrary, it is not needed in the ε-constraint method [26, 27]; • the number of generated efficient solutions can be controlled in the ε-constraint approach by appropriately adjusting the number of grid points in each objective function’s range [26, 27]. In spite of aforementioned advantages of the ε-constraint technique, there are two issues with this method that need attention. 1. The first issue is with the ranges of objective functions over the efficient sets that are not optimized. To overcome this deficit, this study employed lexicographic optimization. 2. The second problem with this technique is that the generated Pareto optimal solutions using this method may be dominated or inefficient; therefore, an augmented εconstraint technique is utilized herein to eliminate this shortcoming [28]. During construction of the payoff table, it should be ensured that the solutions derived from the individual optimization of the objective functions are all indeed Pareto optimal (Pareto efficient) solutions. If alternative optima exist, the optimal solution is no longer a certain efficient solution. However, this study uses lexicographic optimization to form a payoff table consisting of only efficient solutions [28]. Generally, in the case of a series of objective functions, the first is optimized
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
Ahmadi et al.: Short-term Environmental/Economic Combined Heat and Power Scheduling
in lexicographic optimization. In general, the lexicographic optimization of a series of objective functions is to optimize the first objective function, and then, among the possible alternative optima, optimize for the second objective function, and so forth. In practical cases, lexicographic optimization is implemented as explained next. At the beginning, the first objective function is optimized, leading to min f 1 = z1 ∗ . Afterward, the second objective function is optimized by adding the constraint f 1 = z1 ∗ to maintain the optimal solution obtained from the first optimization. Hence, the optimal solution will be min f 2 = z2 ∗ subject to f 1 = z1 ∗ . Then, the third objective function should be optimized taking into account the constraints f 1 = z1 ∗ and f 2 = z2 ∗ to retain the previous optimal solutions and so on up to the time that the first row of payoff table in Eq. (18) is constructed. Subsequently, this procedure must be iterated until the remaining rows of become complete. For example, during the construction of the second row of payoff table , at first, the second objective function, i.e., min f 2 = z2 ∗ must be optimized. Afterward, the first (or third) objective function will be optimized considering the constraint f 2 = z2 ∗ , etc. The results derived from the lexicographic optimization are all non-dominated or efficient solutions [28]. Lexicographic optimization and the augmented ε-constraint method produce only efficient solutions (no weakly efficient solutions) and also avoid redundant iterations, as early exit from relevant loops (that lead to infeasible solutions) can be performed, accelerating the whole process [27]; therefore, lexicographic optimization and the augmented ε-constraint method are used for solving the proposed model.
FIGURE 2. Flowchart of ε-constraint optimization method for the MMP problem.
μrn n=2
⎧ 1 ⎪ ⎪ ⎨ f SN − f r n n = ⎪ f nS N − f nU ⎪ ⎩ 0
Fuzzy Decision Maker
μ = r
A fuzzy decision maker is proposed in this article to softly select the most desired solution among the Pareto solutions obtained from the ε-constraint method [27, 28]. In this regard, this method first calculates a linear membership function for each objective function in each Pareto solution, which measures the relative distance between the values of the objective function in the Pareto optimal solution from its values in the corresponding utopia and pseudo-nadir points. The mathematical presentation of these membership functions can be stated as
f nr ≤ f nS N f nS N ≤ f nr ≤ f nU , f nr ≥ f nU
(28)
f nU ≤ f nr ≤ f nS N . f nr
≥
(29)
f nS N
wn .μrn
n=1 p
.
(30)
wn
n=1
The most desired solution gives the Pareto solution with the highest value of μr or the highest preference for the MMP problem. This solution more optimizes the objective functions of the MMP problem by consider their relative importance over the other Pareto solutions [27, 28]. Figure 2 demonstrates the presented optimization framework for the ε-constraint method for the MMP problem [26, 28]. 4.
⎧ 0 ⎪ ⎪ ⎨ f r − f SN n n μrn = ⎪ f nU − f nS N n=1 ⎪ ⎩ 1
f nr ≤ f nU
Equation (28) is used for the objective functions intended to be maximized, while the linear membership function expressed in Eq. (29) is used for objective functions that should be minimized. Then, having taken into consideration the individual membership functions and the relative importance of the objective functions (wn values), the total membership function (total degree of optimality) is calculated for each Pareto optimal solution as p
3.1.
951
CASE STUDY AND SIMULATION RESULTS
Three different cases are simulated in this section to verify the performance of the proposed method in solving MMP problems. The scheduling horizon is considered to be 24 hr on the hourly basis for all case studies. The MMP problem is
952
Electric Power Components and Systems, Vol. 42 (2014), No. 9
solved using a PC with a 2.6-GHz Pentium IV CPU and 4 GB RAM, while solver CONOPT 3 ver. 3.14S is employed under GAMS [29]. The next section includes the results obtained from solving the proposed problem taking into consideration three case studies.
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
4.1.
Case 1
This case study is the same as that in [22], where the system is comprised of four conventional thermal generators, two CHP units, and one heat-only unit. This case study includes power transmission loss. Detailed data on cost functions of units, Bloss coefficients, and heat power FORs are available in [22]. The total real power demand of this system is 600 MW and the total heat demand is 150 MWth. The cost and TU emission are both minimized in this example, simultaneously using the ε-constraint technique. It is worth mentioning that only SO2 and NOx emission generation is considered in this case and CO2 emission is neglected. Moreover, it is noted that total heat is considered as a constraint. The Pareto solutions of the MCHPEED problem have been found using lexicographic optimization and the augmented the ε-constraint method. In the ε-constraint method [26–28], the main objective function is F 1 (cost minimization), while 24 grid points (q2 = 24) are considered for F 2 (emission minimization) to attain Pareto optimal solutions. Therefore, the problem must be solved 25 times (q2 + 1 = 25) to obtain the Pareto set, while all solutions are feasible [26–28]. Table 1 ( 1 ) represents the resulting payoff table. If the emission generation is discarded, according to Table 1, the total cost can decrease to $10,045.23, while in this case, the emission is 29.36 kg. On the other hand, if cost is discarded, the emission can decrease to 7.18 kg, but the total cost in this case is $17,422.70. This means that taking into consideration the emission would lead to an increase in the total cost by $7377.47 (17,422.70 – 10,045.23 = 7377.47) in spite of a remarkable decrease in emission generation by 22.18 kg (29.36 – 7.18 = 22.18 kg). This state well shows the contradictory nature of these two objective functions considered in this case. Figure 3 depicts the Pareto optimal front obtained for the MCHPEED problem including only the first case. A fuzzy decision maker [27, 28] has been employed to select the most preferred solution of Pareto optimal solutions obtained for the MCHPEED problem. The weighting factor considered for the total cost in the fuzzy decision maker is the
10, 045.23 29.36 1 = 17, 422.70 7.18 TABLE 1. Payoff table
FIGURE 3. Pareto optimal solutions; emission versus cost for Case 1.
same as that used for the emission generation. Table 2 indicates the results obtained taking into consideration equal weighting factors. The membership value in Table 2 induces the degree of optimality of the solution. Considering equal weighting factors for two objective functions would result in the highest value of total membership [27, 28] that is equal to 0.639 for 25 Pareto solutions, and the most desired solution is the 17th Pareto solution. The best solution selected by incorporating lexicographic optimization and the augmented ε-constraint method is represented in Table 3 in detail along with the results reported by NSGA-II [22] and SPEA [22] to verify the effectiveness of the proposed approach. As it can be observed from Table 3, the proposed method results in better solutions compared to those reported in [22] in the case of quantity; e.g., the cost derived from the suggested method is $524.29 (13,433.19 – 12,908.900 = 524.29) and $540.05, i.e., much less than the value reported in [22]. The amount of emission generated using the presented method is 14.572 kg, while the amount of emission generated using NSGA-II [22] and SPEA-2 [22] methods are 25.8262 and 25.7810 kg, respectively. Therefore, the amount of emission generated using the presented method is 11.2542 kg (25.8262 Objective function Cost ($) Emission (kg)
Weighting factor
Objective function value
Membership value
1 1
12,908.900 14.572
0.612 0.667
TABLE 2. Optimum solution of MCHPEED problem with equal weighting factors for Case 1
Ahmadi et al.: Short-term Environmental/Economic Combined Heat and Power Scheduling Method/details
NSGA-II [22]
SPEA-2 [22]
Proposed
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) H 5 (MWth) H 6 (MWth) H 7 (MWth) Cost ($) Emission (kg) CPU time (sec)
73.5896 106.8761 119.0311 163.5563 188.4166 58.4850 26.8054 73.9970 49.1976 13433.19 25.8262 9.7188
73.3149 117.7996 117.7996 151.6436 195.1355 54.0988 25.8784 75.5331 48.5884 13448.95 25.7810 53.4688
61.913 72.971 89.726 121.321 220.270 40.000 0.000 75.000 75.000 12, 908.900 14.572 3.764
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
TABLE 3. Detailed results of different methods for Case 1
– 14.572 = 11.2542) and 11.209 kg (25.7810 – 14.572 = 11.209), less than that obtained in [22] at the same time. In addition, the time required by the proposed method to solve the problem is noticeably less than other methods. The total time to find the Pareto set using the proposed method is 3.764 sec to find 25 Pareto solutions, while the times required by the method employed in [22] are 9.7188 and 53.4688 sec to find only 1 solution. However, the decision maker is able to simply find the desired solution by changing weighting factors. For instance, if cost is more preferred by the decision maker rather than the emission generation, the weighting factor assigned for cost would be higher than that considered for the emission generation, e.g., two for cost and one for the emission generation. In such conditions, a Pareto solution having a high value of cost membership and a low value of emission membership is selected, as indicated in Table 4. The best solution in this state is the fifth Pareto solution, while the cost has considerably improved from 0.612 in case of assigning equal weighting factors to 0.979 in the case of different weights. The cost value in Table 4 is $10,198.45, which is indeed very close to its ideal value ($10,045.23). On the contrary, the emission has been exposed to increase, since its membership value is low (0.167). This test case verifies the potential and capability of the proposed method compared to others reported previously. 4.2.
Case 2
This case study is the same as that in [23]. The difference between this case and Case 1 is the modified SO2 and NOx emission coefficients and also that the CO2 emission coefficients are taken into consideration. To make a fair comparison, all data and formulation are the same as in [23]. Table 5 represents the Pareto optimal solutions obtained for the MCHPEED problem in the second case. For this case, 19 grid points (q2
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) H 5 (MWth) H 6 (MWth) H 7 (MWth) Cost ($) Emission (kg)
953
75.000 99.891 122.135 166.246 103.169 40.000 4.577 75.000 70.423 10, 198.453 25.661
TABLE 4. Detailed results of optimum Pareto solution with different weighting factors (w1 = 2 for cost and w2 = 1 for emission) for Case 1
= 19) are considered for F 2 (emission minimization) to attain Pareto optimal solutions. Therefore, the problem must be solved 20 times (q2 + 1 = 20) to obtain the Pareto set, while all solutions are feasible [26–28]. Choosing the number of Pareto solutions is arbitrary, but 20 Pareto solutions are considered to compare the results with those reported in [23]. The contradictory nature of the two objective functions considered can be well observed from Table 5. The best compromise solution obtained using the proposed method and that reported using MLCA [23] is depicted in Table 6, with details verifying the efficiency of the proposed method. Table 6 shows that the results obtained using the proposed method are Pareto solution number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Cost ($)
Emission (kg)
10, 045.23 10, 045.95 10, 048.92 10, 057.56 10, 185.98 10, 388.19 10, 605.06 10, 838.22 11, 089.64 11, 361.71 11, 657.17 11, 977.48 12, 325.76 12, 706.39 13, 124.98 13, 588.96 14, 172.16 14, 866.70 15, 689.23 16, 728.45
22.69 21.72 20.76 19.79 18.82 17.85 16.89 15.92 14.95 13.99 13.02 12.05 11.09 10.12 9.15 8.19 7.22 6.25 5.29 4.32
TABLE 5. Pareto solutions for Case 2
954
Electric Power Components and Systems, Vol. 42 (2014), No. 9
Method/details
MLCA [23]
Proposed
P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) H 5 (MWth) H 6 (MWth) H 7 (MWth) Cost ($) Emission (t) CPU time (sec)
62.9074 98.519682 100.028509 105.850926 193.003287 40.419851 4.7380965 75.301553 69.96035 12,451.38 11.1 —
69.366 84.624 96.514 117.880 197.775 40.000 2.907 75.000 72.093 12,325.760 11.087 3.370
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
TABLE 6. Detailed results of different methods for Case 2 FIGURE 4. Pareto optimal solutions; emission versus cost for Case 3.
better than those reported in [23] in the case of quantity; e.g., the value of cost obtained employing the proposed method is $12,451.38. Therefore, the cost obtained employing the proposed method is $125.62 (12,325.760 – 12,451.38 = 125.62) less than the value reported in [23]. Also, the emission obtained using the proposed method is 0.013 t (11.1 – 11.087 = 0.013) less than that reported in [23]. Furthermore, the time required by the proposed method to solve the problem to find 20 optimal solutions is 3.370 sec. Unfortunately, the solution time was not reported in [23]. This test case shows the capability and effectiveness of the proposed method to solve the MMP problem compared to other previously established methods.
4.3.
Case 3
This case includes 400 conventional TUs, 200 CHP units, and 100 heat-only units, obtained by multiplying the Case 1 by 100. The electric power demand is 60,000 MW, while the heat demand is 15,000 MWth. It is noted that the power loss is neglected in this case due to lack of information. Figure 4 depicts 20 Pareto optimal solutions for Case 3. A fuzzy decision maker has been employed to select the most preferred solution from the Pareto optimal solutions. The values of cost membership, emission membership, and total membership value for each Pareto solution considering equal weighting factors are shown in Figure 5. In such conditions, the 13th Pareto solution is chosen as the most desired solution, while the values of cost and emission are $1,229,639.45 and 1069.31 kg, respectively. The effectiveness of the proposed method is verified through this case in solving the problem of economic/emission dispatch for such a large-scale system.
4.4.
Case 4
This case study tries to show the effectiveness of proposed method for solving a multi-period multi-objective problem while investigating the contribution of CHP units. The test system is comprised of ten TUs obtained from [30] by adding a TU, two CHP units, and a heat-only unit obtained from [31] to demonstrate the computational efficiency of the proposed method. Lexicographic optimization and the augmented epsilon-constraint technique have been used to concurrently optimize competing objective functions, including minimizing cost and minimizing emissions of TUs. The data for TUs cost and emission coefficients, TU power generation limits, forecasted electric load demand for 24 hr, TU ramp rate limits, and valve-point loading coefficients are taken from [30], while the associated data of a TU, two CHP units, and a heat-only unit are taken from [31]. It is supposed that heat demand of each hour is equal to 1000 MWth. Table 7 shows Pareto optimal solutions for the fourth case. This table shows that using CHP units reduces cost and
FIGURE 5. Variation of total membership, cost, and emission functions versus Pareto-optimal solutions.
Ahmadi et al.: Short-term Environmental/Economic Combined Heat and Power Scheduling
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
Pareto solution number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
With CHP by proposed method
Without CHP by proposed method
955
With CHP by weighted sum method
Cost ($)
Emission (lb)
Cost ($)
Emission (lb)
Cost ($)
Emission (lb)
5, 861, 678.40 5, 861, 702.33 5, 861, 811.11 5, 862, 023.34 5, 862, 361.68 5, 862, 826.26 5, 863, 435.01 5, 864, 228.71 5, 865, 281.43 5, 866, 603.24 5, 868, 197.02 5, 870, 143.66 5, 872, 436.95 5, 875, 122.57 5, 878, 320.34 5, 882, 229.35 5, 887, 084.03 5, 893, 270.98 5, 902, 074.78 5, 917, 927.54 6, 018, 651.41 6, 182, 150.44 6, 381, 449.75 6, 643, 153.16 7, 173, 002.92
159, 352.07 157, 630.78 155, 909.49 154, 188.20 152, 466.91 150, 745.62 149, 024.33 147, 303.04 145, 581.75 143, 860.47 142, 139.18 140, 417.89 138, 696.60 136, 975.31 135, 254.02 133, 532.73 131, 811.44 130, 090.15 128, 368.86 126, 647.57 124, 926.28 123, 204.99 121, 483.70 119, 762.41 118, 041.12
7, 591, 323.42 7, 591, 384.75 7, 591, 652.92 7, 592, 121.40 7, 592, 745.35 7, 593, 548.08 7, 594, 542.66 7, 595, 751.71 7, 597, 218.29 7, 598, 950.80 7, 600, 978.30 7, 603, 293.36 7, 605, 887.60 7, 608, 777.37 7, 612, 043.43 7, 615, 737.39 7, 619, 900.48 7, 624, 631.79 7, 630, 045.12 7, 636, 314.03 7, 643, 683.16 7, 652, 574.08 7, 663, 848.11 7, 679, 648.12 7, 722, 709.09
248, 325.41 246, 522.57 244, 719.74 242, 916.90 241, 114.07 239, 311.23 237, 508.40 235, 705.57 233, 902.73 232, 099.90 230, 297.06 228, 494.23 226, 691.39 224, 888.56 223, 085.72 221, 282.89 219, 480.05 217, 677.22 215, 874.38 214, 071.55 212, 268.71 210, 465.88 208, 663.04 206, 860.21 205, 057.37
5, 861, 687.32 5, 861, 709.10 5, 861, 800.86 5, 861, 946.39 5, 862, 113.29 5, 862, 452.31 5, 862, 878.83 5, 863, 315.33 5, 863, 880.82 5, 864, 427.99 5, 866, 045.62 5, 866, 900.66 5, 868, 766.58 5, 869, 188.47 5, 873, 171.50 5, 875, 903.43 5, 879, 049.51 5, 882, 577.68 5, 887, 163.57 5, 892, 487.64 5, 898, 027.58 5, 904, 998.95 5, 913, 837.49 5, 924, 943.57 7, 520, 794.63
159, 355.34 157, 457.88 156, 024.89 154, 708.38 153, 672.377 152, 093.96 150, 576.81 149, 330.51 148, 002.44 146, 937.04 144, 544.29 143, 582.80 141, 595.40 141, 463.72 138, 196.14 136, 521.47 134, 903.67 133, 396.14 131, 786.95 130, 282.40 129, 068.45 127, 941.56 126, 962.71 126, 269.67 118, 041.12
TABLE 7. Pareto solutions for Case 4
emission. For instance, by comparing the first Pareto solution in Table 7, it can be observed that cost and emission of a system with two CHP units are $1,729,645.02 (7,591,323.42 – 5,861,678.40 = 1,729,645.02) and 88,973.34 lb (248,325.41 – 159,352.07), which are less than a system without CHP units. Table 7 also illustrates the information on cost and emission relating to each Pareto solution obtained by the weighted sum method for the case with CHP units. To compare the Pareto solutions obtained using the proposed method with those obtained by the weighted sum
Case number 1 2 3 4 (with CHP by proposed method) 4 (without CHP by proposed method) 4 (with CHP by weighted sum method)
method, a fuzzy decision-maker method has been proposed in Section 3.1. It is noted that in this case, weighting factors of the two objective functions in the fuzzy decision maker are considered to be two for cost and one for emission (w1 = 2, w2 = 1) to select the most preferred solution of Pareto optimal solutions. Figure 6 illustrates the results obtained by considering w1 = 2 and w2 = 1. As it can be observed from Figure 6, the total membership pertaining to all Pareto solutions obtained by lexicographic optimization and the augmented epsilon-constraint technique
Total solution time (sec)
Number of constraints for each Pareto
Number of variables for each Pareto
3.764 3.370 16.619 19.489 10.281 22.625
17 17 610 790 622 785
18 18 908 392 296 387
TABLE 8. Optimization statistics for different case studies
956
Electric Power Components and Systems, Vol. 42 (2014), No. 9
extension to include more objectives, such as security indices, would be quite straightforward. The ongoing research work is to incorporate security indices (overload index, voltage drop index, and voltage stability margin [VSM]) into the existing problem. Moreover, the uncertainty of outages of generating units and load forecast errors can be taken into consideration as another aspect in future research. REFERENCES
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
FIGURE 6. Total membership of Pareto solutions obtained by the proposed and weighted sum method for Case 4.
and the weighted sum method are demonstrated. As Figure 6 shows, by using the proposed method, the most preferred solution with total membership 0.901959 occurs at Pareto solution 20, while in the case of employing the weighted sum method, Pareto solution 23 is the most preferred Pareto solution with total membership 0.901495. Therefore, for this case, the obtained Pareto solution by the proposed method is better than that obtained by the weighted sum method. Table 3 represents the detailed information of the most desired Pareto solutions obtained by the proposed technique and weighted sum method. It is worth-mentioning that the total membership of 20 Pareto optimal solutions of the proposed method is equal to or more than the total membership of Pareto optimal solution obtained by the weighted sum method. Also, the solution time needed by lexicographic optimization and the augmented epsilon-constraint technique is less than that needed by the weighted sum method. This case study shows the effectiveness of the proposed method. The detailed statistics of the multi-objective optimization problem are indicated in Table 8. As it can be observed from this table, the solution time for all cases is reasonable, demonstrating the capability of the proposed method to solve the MCHPEED problem. 5.
CONCLUSION
In this article, the short-term CHP economic/emission dispatch problem is investigated. This problem has been modeled in a multi-objective framework considering energy production cost and emission as objective functions. Furthermore, a fuzzy decision-making technique has been employed to select the most preferred solution among Pareto optimal solutions, and the results obtained from the proposed method have been compared to those obtained from NSGA-II [22], SPEA-2 [22], and MLCA [23]. Comparison results show the capability of the proposed method to generate better solutions in the case of quality and execution time. As the proposed approach does not impose any limitation on the number of objective functions, its
[1] Hosseini, S. S. S., Jafarnejad, A., Behrooz, A. H., and Gandomi, A. H., “Combined heat and power economic dispatch by mesh adaptive direct search algorithms,” Expert Syst. Appl., Vol. 38, No. 6, pp. 6556–6564, June 2011. [2] Basu, M., “Bee colony optimization for combined heat and power economic dispatch,” Expert Syst. Appl., Vol. 38, No. 11, pp. 13527–13531, October 2011. [3] Abdolmohammadi, H. R., and Kazemi, A., “A Benders decomposition approach for a combined heat and power economic dispatch,” Energy Convers. Manag., Vol. 71, pp. 21–31, July 2013. [4] Subbaraj, P., Rengaraj, R., and Salivahanan, S., “Enhancement of combined heat and power economic dispatch using selfadaptive real-coded genetic algorithm,” Appl. Energy, Vol. 86, No. 6, pp. 915–921, June 2009. [5] Sashirekha, A., Pasupuleti, J., Moin, N. H., and Tan, C. S., “Combined heat and power (CHP) economic dispatch solved using Lagrangian relaxation with surrogate sub-gradient multiplier updates,” Int. J. Elect. Power Energy Syst., Vol. 44, No. 1, pp. 421–430, January 2013. [6] Basu, M., “Artificial immune system for combined heat and power economic dispatch,” Int. J. Elect. Power Energy Syst., Vol. 43, No. 1, pp. 1–5, December 2012. [7] Liu, X., “Optimization of a combined heat and power system with wind turbines,” Int. J. Elect. Power Energy Syst., Vol. 43, No. 1, pp. 1421–1426, December 2012. [8] Haghifam, M. R., and Manbachi, M., “Reliability and availability modeling of combined heat and power (CHP) systems,” Int. J. Elect. Power Energy Syst., Vol. 33, No. 3, pp. 385–393, March 2011. [9] Chen, C. L., Lee, T. Y., Jan, R. M., and Lu, C. L., “A novel direct search approach for combined heat and power dispatch,” Int. J. Elect. Power Energy Syst., Vol. 43, No. 1, pp. 766–773, December 2012. [10] Geem, Z. W., and Cho, Y. H., “Handling non-convex heat-power feasible region in combined heat and power economic dispatch,” Int. J. Elect. Power Energy Syst., Vol. 34, No. 1, pp. 171–173, January 2012. [11] Yildirim, U., and Gungor, A., “An application of exergoeconomic analysis for a CHP system,” Int. J. Elect. Power Energy Syst., Vol. 42, No. 1, pp. 250–256, November 2012. [12] Karamia, H., Sanjaria, M. J., Tavakolia A., and Gharehpetiana, G. B., “Optimal scheduling of residential energy system including combined heat and power system and storage device,” Elect. Power Compon. Syst., Vol. 41, No. 8, pp. 765–781, April 2013. [13] Basu, M. “Combined heat and power economic dispatch by using differential evolution,” Elect. Power Compon. Syst., Vol. 38, No. 8, pp. 996–1004, May 2010.
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
Ahmadi et al.: Short-term Environmental/Economic Combined Heat and Power Scheduling [14] Jubrila, A. M., Adedijia, A. O., and Olaniyana, O. A., “Solving the combined heat and power dispatch problem: A semi-definite programming approach,” Elect. Power Compon. Syst., Vol. 40, No. 12, pp. 1362–1376, August 2012. [15] Wong, K. P., and Algie, C., “Evolutionary programming approach for combined heat and power dispatch,” Elect. Power Syst. Res., Vol. 61, No. 3, pp. 227–232, April 2002. [16] Song, Y. H., Chou, C. S., and Stonham, T. J., “Combined heat and power economic dispatch by improved ant colony search algorithms,” Elect. Power Syst. Res., Vol. 52, No. 2, pp. 115–121, November 1999. [17] Su, C. T., and Chiang, C. L. “An incorporated algorithm for combined heat and power economic dispatch,” Elect. Power Syst. Res., Vol. 69, No. 2–3, pp. 187–195, May 2004. [18] Khorram, E., and Jaberipour, M., “Harmony search algorithm for solving combined heat and power economic dispatch problems,” Energy Convers. Manag., Vol. 52, No. 2, pp. 1550–1554, February 2011. [19] Chatterjee, A., Ghoshal, S. P., and Mukherjee, V., “Solution of combined economic and emission dispatch problems of power systems by an opposition-based harmony search algorithms,” Int. J. Elect. Power Energy Syst., Vol. 39, No. 1, pp. 9–20, July 2012. [20] Abusoglu, A., and Kanoglu, M., “Exergoeconomic analysis and optimization of combined heat and power production: A reviews,” Renew. Sustain. Energy Rev., Vol. 13, No. 9, pp. 2295–2308, December 2009. [21] Le, K. D., Golden, J. L., Stansberry, C. J., Vice, R. L., Wood, J. T., Balance, J., Brown, G., Kamya, J. Y., Nielsen, E. K., Nakajima, H., Ookubo, M., Iyoda, I., and Cauley, G. W., “Potential impacts of clean air regulations on system operations,” IEEE Trans. Power Syst., Vol. 10, No. 2, pp. 647–656, May 1995. [22] Basu, M., “Combined heat and power economic emission dispatch using non-dominated sorting genetic algorithm-II,” Int. J. Elect. Power Energy Syst., Vol. 53, pp. 135–141, December 2013. [23] Shi, B., Yan, L. X., and Wu, W., “Multi-objective optimization for combined heat and power economic dispatch with power transmission loss and emission reduction,” Energy, Vol. 56, No. 1, pp. 135–143, July 2013. [24] Javadi, M. S., Esmaeel Nezhad, A., and Sabramooz, S., “Economic heat and power dispatch in modern power system harmony search algorithm versus analytical solution,” Scientia Iranica, Vol. 19, No. 6, pp. 1820–1828, December 2012. [25] Aghaei, J., Akbari, M. A., Roosta, A., and Baharvandi, A., “Multi-objective generation expansion planning considering power system adequacy,” Elect. Power Syst. Res., Vol. 102, pp. 8–19, September 2013. [26] Ahmadi, A., Aghaei, J., Shayanfar, H. A., and Rabiee, A., “Mixed integer programming of multi-objective hydro-thermal self-scheduling,” Appl. Soft Comput., Vol. 12, No. 8, pp. 2137–2146, August 2012. [27] Mavrotas, G., “Effective implementation of the ε-constraint method in multi-objective mathematical programming problems modified augmented,” Appl. Math. Computat., Vol. 213, No. 2, pp. 455–465, July 2009. [28] Norouzi M. R., Ahmadi A., Esmaeel Nezhad, A., and Ghaedi, A., “Mixed integer programming of multi-objective security-
957
constrained hydro/thermal unit commitment,” Renew. Sustain. Energy Rev., Vol. 29, pp. 911–923, January 2014. [29] “Generalized algebraic modeling systems (GAMS),” available at: http://www.gams.com [30] Basu, M., “Dynamic economic emission dispatch using nondominated sorting genetic algorithm-II,” Int. J. Elect. Power Energy Syst., Vol. 30, No. 2, pp. 140–149, February 2008. [31] Su, C. T., and Chiang, C. L., “An incorporated algorithm for combined heat and power economic dispatch,” Elect. Power Syst. Res., Vol. 69, No. 2–3, pp. 187–195, May 2004.
APPENDIX A A.2. Cost and Emission Functions of Each Unit of Test System 1 Cost and emission coefficients of power-only units of Test System 1 are given in Tables A1 and A2, respectively. Cost and emission coefficients of cogeneration units of Test System 1 are given in Tables A3 and A4, respectively. Cost and emission coefficients of heat-only units of Test System 1 are given in Tables A5 and A6, respectively. Network loss coefficients are given as ⎡ ⎢ ⎢ ⎢ ⎢ B=⎢ ⎢ ⎢ ⎣
49 14 15 15 20 25
14 45 16 20 18 19
15 16 39 10 12 15
15 20 10 40 14 11
20 18 12 14 35 17
25 19 15 11 17 39
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ 10−6 , ⎥ ⎥ ⎦
−0.3908 −0.1297 0.7074 0.0591 B0 = ∗ 10−3 , 0.2161−0.6635 B00 = 0.056.
A.2. Cost and Emission Functions of Each Unit of Test System 2 Emission coefficients of power-only units of test system 2 are given in Table A7, emission coefficients of cogeneration units of Test System 2 are given in Table A8, and emission coefficients of heat-only units of Test System 2 are given in Table A9. The other data are the same as Case 1. A.3. Cost and Emission Functions of Each Unit of Test System 3 The data for this case can be found by multiplying Case 1 by 100, but due to lack of information, power loss is discarded.
958 Unit 1 2 3 4
Electric Power Components and Systems, Vol. 42 (2014), No. 9 ai
bi
ci
di
ei
piT U
P¯iT U
Unit
25 60 100 120
2 1.8 2.1 2
8 ∗ 10−3 3 ∗ 10−3 1.2 ∗ 10−3 1 ∗ 10−3
100 140 160 180
4.2 ∗ 10−2 4 ∗ 10−2 3.8 ∗ 10−2 3.7 ∗ 10−2
10 20 30 40
75 125 175 250
5 6
αj
βj
γj
1.5 ∗ 10−6 1.5 ∗ 10−6
1.5 ∗ 10−5 1.5 ∗ 10−5
2 ∗ 10−3 3 ∗ 10−3
TABLE A8. Emission coefficients of cogeneration units of Test System 2
TABLE A1. Cost coefficients of power-only units of Test System 1 Unit
4.091 ∗ 2.543 ∗ 4.258 ∗ 5.326 ∗
1 2 3 4
Downloaded by [Universidad De Concepcion] at 03:18 30 May 2014
αi
βi 10−4 10−4 10−4 10−4
γi
−5.554 ∗ −6.047 ∗ −5.094 ∗ −3.550 ∗
10−4 10−4 10−4 10−4
6.490 ∗ 5.638 ∗ 4.586 ∗ 3.370 ∗
ξi 10−4 10−4 10−4 10−4
2∗ 5∗ 1∗ 2∗
λi
τi
10−4 2.857 ∗ 10−2 10−4 3.333 ∗ 10−2 10−6 8 ∗ 10−2 10−3 2 ∗ 10−2
0 0 0 0
TABLE A2. Emission coefficients of power-only units of Test System 1 Unit 5 6
αi
βi
γi
ξi
λi
τi
2650 14.5 3.45 ∗ 10−2 4.2 3 ∗ 10−2 3.1 ∗ 10−1 ∗ −2 ∗ −1 1250 36 4.35 10 6 10 2.7 ∗ 10−1 1.1 ∗ 10−1
TABLE A3. Cost coefficients of cogeneration units of Test System 1 Unit
αj
βj
γj
5 6
0 0
0 0
1.65 ∗ 10−3 1.65 ∗ 10−3
TABLE A4. Emission coefficients of cogeneration units of Test System 1
Unit 7
H
ah
bh
ch
H hH
Hh
950
2.0109
38 ∗ 10−2
0
2695.2
TABLE A5. Cost coefficients of heat-only unit of Test System 1 Unit
αh
βh
γh
7
0
0
18 ∗ 10−3
TABLE A6. Emissions coefficients of heat-only unit of Test System 1 Unit 1 2 3 4
αi 4.091 ∗ 2.543 ∗ 4.258 ∗ 5.326 ∗
10−2 10−2 10−2 10−2
βi
γi
−2.777 ∗ 10−2 3.0235 ∗ 10−2 −2.547 ∗ 10−2 −1.775 ∗ 10−2
6.490 ∗ 10−4 5.638 ∗ 10−4 4.586 ∗ 10−4 3.38 ∗ 10−4
ξi 2∗ 2∗ 1∗ 2∗
λi
10−4 2.857 ∗ 10−2 10−4 2.857 ∗ 10−2 10−6 8 ∗ 10−2 10−3 2 ∗ 10−2
τi 6.4 ∗ 6.4 ∗ 7.6 ∗ 8.8 ∗
10−3 10−3 10−3 10−3
TABLE A7. Emission coefficients of power-only units of Test System 2
αh
Unit 7
∗
βh −6
8 10
∗
γh −5
1 10
∗
8 10−3
TABLE A9. Emissions coefficients of heat-only unit of Test System 2
BIOGRAPHIES Abdollah Ahmadi was born in Janah, Iran, in 1984. He received his B.S. in electrical engineering from Shiraz University, Shiraz, Iran, in 2007 and his M.S. in electrical engineering from Iran University of Science and Technology, Tehran, Iran, in 2011. His research interests include power system operation and economics in a deregulated market environment, particularly the issues of distributed generation and its economic impacts on restructured power systems. Mohammad Reza Ahmadi was born in Janah, Iran, in 1987. He received his B.S. in civil engineering from Hormozgan University, Bandar Abbas, Iran, in 2011 and his M.S. in civil engineering from Payam Noor University, Tehran, Iran, in 2014. His research interest is energy management. Ali Esmaeel Nezhad was born in Shiraz, Iran, in 1989. He received his B.S. in electrical engineering from Science and Research Branch, Islamic Azad University, Fars, Iran, in 2011 and his M.S. at Islamic Azad University of Najafabad, Isfahan, Iran, in 2013. His current research interests are planning in restructured power systems, optimization and power system protection, plug-in electric vehicles, and renewable energy sources.
