Short-term environmental/economic hydrothermal ... - Semantic Scholar

Electric Power Systems Research 116 (2014) 117–127

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Short-term environmental/economic hydrothermal scheduling Mohammad Reza Norouzi a , Abdollah Ahmadi b , Adel M. Sharaf c,∗ , Ali Esmaeel Nezhad d a

Department of Electrical Engineering, Islamic Azad University, Lamerd Branch, Lamerd, Iran Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Fars, Iran c Life Senior Member IEEE, Sharaf Energy Systems, Inc, Fredericton, NB, Canada d Department of Electrical Engineering, Islamic Azad University, Najafabad Branch, Najafabad, Iran b

a r t i c l e

i n f o

Article history: Received 27 January 2014 Received in revised form 27 April 2014 Accepted 31 May 2014 Available online 18 June 2014 Keywords: Multi-objective optimization ε-Constraint technique Environmental/economic hydrothermal scheduling Fuzzy satisfying method

a b s t r a c t This paper proposes a new technique based on lexicographic optimization and ε-constraint method to solve the combined economic emission scheduling problem of hydrothermal systems comprising several equality constraints as well as non-equality ones. The hydrothermal scheduling problem is modeled as a multi-objective problem with two objective functions as fuel cost minimization and also pollutant emission minimization. After deriving the Pareto set, the most preferred solution is determined using a fuzzy satisfying method. The thermal plants are considered with valve point effect and emission level function. Besides, this paper has taken into consideration the multi-reservoir cascaded configuration of hydro units, while the relationship considered between water discharge rate and power generated through these hydro units is nonlinear. Also, there is a water transport delay between hydro units. The presented model is implemented on a sample test system comprising four cascaded hydro units and three thermal units to verify the efficiency of the proposed method. Furthermore, the proposed method is implemented on IEEE 118 bus test case. The results obtained from the simulation show the effectiveness of the presented technique in the case of fuel cost and emission output compared to other approaches recently used. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Short-term hydro-thermal scheduling (SHTS) problem is defined as a problem, in which the optimal hourly amount of water released by hydro units and the output power of thermal units over the scheduling period are determined, so that the total operation cost is minimized while subjected to different constraints [1,2]. So far, there are too many optimization techniques proposed to solve the SHTS problem. The problem of short-term hydrothermal optimal scheduling with economic emission issues is solved in Ref. [3] using three chaotic sequences based multi-objective differential evolution method wherein the non-dominated solutions are obtained applying Elitist archive mechanism. Ref. [4] employs an adaptive chaotic artificial bee colony algorithm to deal with the problem of SHTS while local optimum is skipped using chaotic search added to the artificial bee colony algorithm. The civilized swarm optimization based solution method is proposed in Ref. [5] for the problem of

∗ Corresponding author. Tel.: +1 506 453 4561. E-mail addresses: [email protected], [email protected] (A.M. Sharaf). http://dx.doi.org/10.1016/j.epsr.2014.05.020 0378-7796/© 2014 Elsevier B.V. All rights reserved.

multi-objective SHTS. Ref. [6] has taken into consideration the nonlinearities, such as valve loading effects pertaining to the thermal units and the prohibited discharge zone of the water reservoir in the SHTS problem wherein a teaching learning based optimization is proposed to solve the problem. The clonal real-coded quantum-inspired evolutionary algorithm with cauchy mutation is applied to the SHTS problem in Ref. [7], in which the realcoded rule is used to handle the continuous variables. Ref. [8] proposes a computational model that is in development to solve the problem of long-term hydrothermal scheduling of the Brazilian hydrothermal power system and mentioned several modeling issues probably leading to some problems in obtaining the optimal policy and currently entailed in the official long-term optimization model of the Brazilian regulatory framework. Ref. [9] presents a multi-objective framework for the problem of short-term economic/environmental hydrothermal scheduling wherein energy cost and pollutant emissions are considered as the two objective functions. The hydrothermal scheduling problem is solved in Ref. [10] utilizing a genetic-based algorithm. In the proposed problem, the GA is used to solve the hydro sub-problem while lambda iteration technique is employed to solve the thermal sub-problem. The economic/environmental dispatch problem of hydrothermal power system is solved in Ref. [11] as a true multi-objective

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M.R. Norouzi et al. / Electric Power Systems Research 116 (2014) 117–127

asi , bsi , csi , esi and fsi coefficients of thermal generating plant i ˛si , ˇsi , si , si and ısi emission coefficients of thermal plant i T total time intervals over scheduling horizon number of thermal and hydro plants, respectively Ns , Nh Psit and Phjt power generation of thermal and hydro plants at the tth scheduling period, respectively power demand at the tth scheduling period PDt C1j , C2j , C3j , C4j , C5j and C6j power generation coefficients pertaining to the hydro plant j Qhjt and Vhjt water discharge rate and reservoir storage volume of the jth hydro plant in the tth scheduling period PsiMin and PsiMax lower and upper bounds of power generation by ith thermal generating unit, respectively Min and P Max lower and upper bounds of power generation Phj hj by jth hydro generating unit, respectively Min and V Max lower and upper bounds of storage volumes Vhj hj of jth reservoir, respectively Isit , Ihjt is 1 if unit ith thermal generating unit/jth hydro plant is on-line at hour t, respectively inflow of jth hydro reservoir in the tth scheduling Inhjt period Shjt spillage of jth hydro plant in the tth scheduling period SUCsit , SDCsit start-up and shut down cost of ith thermal generating unit in the tth scheduling period, respectively water transport delay from reservoir m to j mj number of upstream hydro plants located directly Ruj above the jth reservoir length of a single scheduling period t p number of competing objective functions vector of decision variables which optimizes the x¯ i∗ objective function fi ˚ payoff table fU , fN , and fSN Utopia point, Nadir point and Pseudo Nadir point, respectively feasible region rn individual membership function (the degree of optimality) pertaining to the nth objective function in the rth Pareto optimal solution wn the weighting factor of the nth objective function in the multi-objective problem r total membership function of the rth Pareto optimal solution

optimization problem using multi-objective differential evolution algorithm, while competing and non-commensurable objective functions are considered. The progressive hedging method is utilized in Ref. [12] to handle the medium term operation planning problem of hydrothermal power system. In this regard, the two critical aspects of the progressive hedging refer to: The use of warm start and the choice of a suitable penalty parameter. A method is presented in Ref. [13] based on the Particle Swarm Optimization (PSO) and employed a multi-reservoir cascaded hydroelectric system considering the valve point effect for thermal units that have Prohibited Operating Zones (POZs). Ref. [14] uses Benders decomposition method wherein the AC modeling of the network is proposed to consider the losses in the frame of hydrothermal scheduling optimization. The scheduling problem of a large-scale hydrothermal power system is presented in Ref. [14] using Branch

and Bond (B&B) on the weekly basis. Ref. [15] employs a dual approach to solve the SHTS problem for a system considering poolbilateral markets. A model is proposed in Ref. [16] taking into consideration the transmission network in DC and AC power flow for SHTS problem while the problem is solved utilizing Benders decomposition. An augmented Lagrangian method is used in Ref. [17] for scheduling the combined generation of thermal and hydro generating units. Ref. [18] used an adaptive chaotic differential evolution algorithm to solve the short-term generation scheduling problem of hydrothermal systems. The objective function considered in Refs. [19–21] is profit maximization. The problem of a hydro unit and thermal generating company self-scheduling in pool market is investigated in Refs. [19,20], respectively. The implementation of Mixed-Integer Programming (MIP) method to solve SHTS problem for generating units is addressed in Refs. [21,22]. The conventional pure economic scheduling does not meet the necessities of SHTS problem anymore after the Clean Air Amendments was established in 1990 [23], and inevitably, the emission issues should be included too. In this regard, the lexicographic optimization and ε-constraint method is employed in Ref. [24] to solve the hydro-thermal self-scheduling (HTSS) problem in a day-ahead joint energy and reserve market with the objectives of maximizing the Generation Company’s (GENCO) as well as minimizing the emissions caused by thermal units. Ref. [25] applies an interactive fuzzy satisfying method on the basis of the evolutionary programming approach to short-term multi-objective hydrothermal scheduling problem taking into consideration cost and emission as two objectives. The SHTS problem is solved in Refs. [26–28] using differential evolution, quantum-behaved PSO method, and cultural optimization, respectively while making attempts to simultaneously minimize fuel cost and pollutant emission. Refs. [29,30] propose a detailed literature review of optimization methods used to solve the SHTS problem. The main contributions of this paper can be summarized as: (a) Proposing a multi-objective framework for SHTS problem with cost and emissions as objectives. (b) Utilizing lexicographic optimization and ε-constraint to solve the SHTS problem. (c) Reduced fuel cost and emission compared to recently published papers using the proposed method. (d) The proposed fuzzy approach allows the decision maker to simply change the operation strategies. The remainder of this paper is organized as follows. Section 2 presents the mathematical modeling of the problem. The principles of the ε-constraint technique is described in Section 3. The simulation results are included in Section 4 and finally some relevant conclusions are drawn in Section 5. 2. Problem formulation The two objective functions of the proposed multi-objective SHTS problem can be stated as below:

Multiobjective functions =

f1 Cost minimization f2 Emmision minimization

(1)

where f1 and f2 denote the SHTS objective functions presented in details in the next section. 2.1. Economic scheduling The main objective function of SHTS considered in this paper is cost minimization. The non-smooth fuel cost function pertaining to


the thermal units along with valve-point effects are used as follows to model the fuel cost function more precisely and practically: f1 : Min F Cost =

Ns T

and lower and upper bounds of power generated by the jth hydro Min and P Max , respectively. unit are indicated by Phj hj 2.5. Startup cost and shut down cost

2 [Isit (asi + bsi Psit + csi Psit )

t=1 i=1

+ Isit |esi sin{fsi (PsiMin − PPsit )| + SUCsit + SDCsit ] (2) where the cost curve coefficients pertaining to the ith thermal unit are denoted by asi , bsi , csi , esi and fsi . Psit indicates the output power of the ith thermal generating unit at time t and the PsiMin denotes the lower bound of power generation by the thermal unit i. Ns and T indicate the number of thermal generating units and intervals of the scheduling period. Isit is 1 if unit ith thermal generating unit is on-line at hour t.

SUCsit ≥ SUsi [Isit − Isi(t−1) ]

2.6. Reservoir storage volume limits Min Max Vhj ≤ Vhjt ≤ Vhj

The emission caused by thermal units is the second objective function considered in the presented SHTS and can be stated as below:

Min Max Qhj Ihjt ≤ Qhjt ≤ Qhj Ihjt

2 Isit (˛si + ˇsi Psit + si Psit )

2.3. power balance limit The main constraint of the problem is the power balance limit, ensuring that the total power generated by thermal and hydro units at each time of scheduling period is equal to the total demand.

i=1

Psit +

Nh

(10)

Max and Qhj

are the lower and upper bounds of the water discharge rate of jth reservoir, respectively.

Eq. (11) represents the water balance equation:

(3)

where the emission coefficients pertaining to the ith thermal unit are denoted by ˛si , ˇsi , si , si and ısi . The equality and inequality constraints of the proposed SHTS problem are addressed in the next section.

Ns

Min where Qhj

2.8. Water dynamic balance

t=1 i=1

+|si exp(ısi Psit )|

(9)

where, the lower and upper bounds of storage volumes pertaining Min and V Max , respectively. to the jth reservoir are denoted by Vhj hj 2.7. Water discharge rate limits

Ns T

(8)

SDCsit ≥ SDsi [Isi(t−1) − Isit ]

2.2. Emission scheduling

f2 : Min F Emission =

119

Vhjt = Vhj,t−1 + [Inhjt − Qhjt − Shjt +

Ruj

(Qhm,t−mj + Shm,t−mj )]t

m=1

(11)

where, the inflow of jth hydro reservoir at time t is denoted by Ihjt . Shjt indicates the spillage of jth hydro unit at time t. The water transport delay from reservoir m to j and number of upstream reservoirs directly located above the jth reservoir are denoted by mj and Ruj , respectively. t is the length of a single schedule period. 2.9. Initial and final volume

Phjt = PDt

(4)

j=1

Eq. (12) represents the initial and final volume equation. Vhj0 = VhjB

where the load demand at time t, and output power of jth hydro unit at time t are indicated by PDt and Phjt , respectively. The number of hydro units is denoted by Nh . The hydropower generation can be stated in terms of water discharge rate and reservoir storage volume as Eq. (5):

where VhjB and VhjE denote the initial and final reservoir storage limits pertaining to the jth hydro unit, respectively.

2 2 Phjt = Ihjt (C1j Vhjt + C2j Qhjt + C3j Vhjt Qhjt + C4j Vhjt + C5j Qhjt + C6j )

3. Multi-objective mathematical programming

VhjT = VhjE

(12)

(5) where Qhjt and Vhjt denote the water discharge rate and reservoir storage volume pertaining to the jth hydro unit at the tth hour of scheduling period, respectively. The power generation coefficients of the jth hydro unit are denoted by C1j , C2j , C3j , C4j , C5j and C6j . Ihjt is 1 if jth hydro plant is on-line at hour t. 2.4. Hydrothermal plant power generation limits

PsiMin × Isit ≤ Psit ≤ PsiMax × Isit Max × I Min × I Phj hjt ≤ Phjt ≤ Phj hjt

Psit − Psi(t−1) ≤ RUsi × Isi(t−1) Psi(t−1) − Psit ≤ RDsi × Isit

(6)

(7)

where the lower and upper bounds of power generation pertaining to the ith thermal unit are denoted by PsiMin and PsiMax , respectively

The concept of multi-objective mathematical programming (MMP) involves with more than one objective function and as a result, the single optimal solution that concurrently optimizes all the objective functions no longer exists. In this regard, ε-constraint approach is an efficient technique for solving MMP problems. This method deals with one main objective function that is selected among the objective functions of the MMP problem. Thus, this paper employs ε-constraint technique to solve the presented multiobjective SHTS problem. The main principle of ε-constraint method [24,31], is to take and optimize the main objective function f1 while other objective functions are taken into consideration as constraints: Min f1 (¯x) subject to f2 (¯x) ≤ e2 f3 (¯x) ≤ e3 . . .fp (¯x) ≤ ep

(13)

where the number of competing objective functions of the MMP problem is denoted by subscript p and x¯ indicates the vector of

120


decision variables. It is assumed that all p objective functions are intended to be minimized in Eq. (13). In ε-constraint approach, the ranges of at least p − 1 objective functions that will be considered as extra objective function constraints should be available to be effectively applied to the MMP problem. These ranges are commonly calculated using payoff table. The first step to form the payoff table for an MMP problem comprising p competing objective functions is to determine the individual optima of the objective functions fi . The calculated optimum value corresponding to fi is denoted by fi∗ (¯xi∗ ) while the vector of decision variables optimizing the objective function fi is indicated by x¯ i∗ . Afterwards, the solution optimizing the objective function fi is used to calculate the corresponding values of other objective functions f1 , f2 ,. . ., fi−1 , fi+1 ,. . .,fp , which are denoted by f1 (¯xi∗ ), f2 (¯xi∗ ),. . ., fi−1 (¯xi∗ ), fi+1 (¯xi∗ ),. . ., fp (¯xi∗ ). The ith row of the payoff table represents f1 (¯xi∗ ), f2 (¯xi∗ ),. . ., fi∗ (¯xi∗ ),. . ., fp (¯xi∗ ). Therefore, all rows of the payoff table are calculated as below:

⎛

f1∗ (¯x1∗ )

···

fi (¯x1∗ )

···

fp (¯x1∗ )

···

fi (¯xp∗ )

···

f U = [f1U , . . ., fiU , . . ., fpU ] = [f1∗ (¯x1∗ ), . . ., fi∗ (¯xi∗ ), . . ., fp∗ (¯xp∗ )]

(14)

(15)

The Nadir point is a point located in the objective space at which all objective functions have their worst values, simultaneously. This point can be stated as below: (16)

where x¯

subject to x¯ ∈ ˝

(17)

And the feasible region is indicated by ˝. Another concept close to the Nadir point is Pseudo Nadir point defined as follows: f SN = [f1SN , . . ., fiSN , . . ., fpSN ]

(18)

fiSN = Max{fi (¯x1∗ ), . . ., fi∗ (¯xi∗ ), . . ., fi (¯xp∗ )}

(19)

Note that these three points, i.e. Utopia, Nadir and Pseudo Nadir points are defined in the objective space that is a vector space while its dimensions are objective functions. The Utopia and Pseudo Nadir points are used in the ε-constraint method to specify the range of each objective function as: fiU ≤ fi (¯x) ≤ fiSN

sub-problem (n2, . . .,np) are in the following form: Min f1 (¯x) subject to f2 (¯x) ≤ e2,n2 , . . ., fp (¯x) ≤ ep,np

e2,n2 = f2SN −

(20)

Solving the MMP problems leads to the set of Pareto optimal solutions. A point x¯ ∗ ∈ ˝ is the Pareto optimal or efficient solution in a general multi-objective optimization problem provided that there is no x¯ ∈ ˝ such that fi (¯x) ≤ fi (¯x∗ ) for all i = 1, 2, . . .,p with at least one strict inequality.

f2SN − f2U fpSN − fpU

(21)

q2

The payoff table includes p rows and columns. The obtained values for the objective function fj are located in the jth column of the table wherein the range of the objective function fj for the ε-constraint approach is indicated by the minimum and maximum values. However, several concepts must be defined to improve the performance of the ε-constraint method applied to the proposed MMP problem. In general, it is again assumed that all objective functions are intended to be minimized. The first concept corresponds to Utopia point which is a specific point usually located outside the feasible region at which all objective functions have their best values, simultaneously. This point can be stated as follows:

fiN = Max fi (¯x),

(qi + 1) optimization sub-problems must be solved while the

i=2

e2,np = fpSN −

fp∗ (¯xp∗ )

f N = [f1N , . . ., fiN , . . ., fpN ]

p

⎞

⎜ ⎟ .. ⎟ ⎜ .. .. ⎜ . . . ⎟ ⎜ ⎟ ∗ ∗ ∗ ∗ ⎟ ˚=⎜ ⎜ f1 (¯xi ) · · · fi (¯xi ) · · · fp (¯xi ) ⎟ ⎜ ⎟ ⎜ .. .. ⎟ .. ⎝ . . . ⎠ f1 (¯xp∗ )

The next step after the range of all objective functions are determined is to divide the range of p − 1 objective functions f2 , . . ., fp to q2 , . . ., qp equal intervals exploiting (q2 − 1), . . ., (qp − 1) intermediate equidistant grid points, respectively. There are totally (q2 + 1), . . ., (qp + 1) grid points for f2 , . . ., fp , respectively taking into consideration the minimum and maximum values of the range. Hence,

× n2 n2 = 0, 1, . . ., q2

(22)

× np np = 0, 1, . . ., qp

(23)

qp

where the value of the objective function in the Utopia and Pseudo Nadir points is indicated by the subscripts U and SN as expressed in (15) and (19), respectively. Furthermore, each optimization subproblems must include the constraints of the MMP problem as well as objective function constraints indicated in (21). Solving each optimization sub-problem leads to one Pareto optimal solution in the ε-constraint method. The optimization sub-problems with infeasible solution space will be discarded in this technique. Then, the most desired solution among Pareto optimal solutions is selected by the decision maker (DM). The advantages of the presented approach over weighting method can be found in Ref. [31]. In spite of the advantages of ε-constraint approach have, there is a key point that should be noted. There is no guarantee that the range of the objective functions over the efficient set is optimal while the lexicographic optimization method is employed to overcome this deficit. In general, lexicographic optimization of a series of objective functions is interpreted as optimization of the first objective function, then among the possible alternative optima, optimize for the second objective function and so on. At the beginning of lexicographic optimization the first objective function is optimized and min f1 = z1 * is attained. The next step is to optimize the second objective function taking into account the f1 = z1 * to maintain the optimal solution of the first optimization. Therefore, the optimal solution would be min f2 = z2 * subject to f1 = z1 *. After that, the third objective function is going to be minimized while constrained to f1 = z1 * and f2 = z2 * to keep the optimal solutions obtained previously, etc. This process continues to complete the first row of the payoff table, i.e. ˚ in (13). The procedure for the second row of the payoff table would be the same as the first row, e.g. the second objective function is going to be optimized first, which is f2 = z2 *. Then, taking into consideration the constraint f2 = z2 *, the first (or third) objective function will be optimized and so forth. It is noted that the results derived from the lexicographic optimization are all non-dominated or efficient solutions [32]. 3.1. Fuzzy decision maker After deriving the Pareto solutions by ε-constraint, the most preferred solution should be selected by the decision maker according to the application of the model and preferences of the DM. In this regard, this paper proposes a fuzzy satisfying method that is able to choose the most desired solution among Pareto solutions [24,32]. In this method, a linear membership function is defined for each objective function in each Pareto optimal solution. This membership function measures the relative distance between the value of the objective function in the Pareto optimal solution from


121

Fig. 1. Flowchart of the ε-constraint optimization method for the MMP problem.

its values in the corresponding Utopia and Pseudo Nadir points. This linear function can be stated in the mathematical form for the MMP problem as below:

rn n=1

=

⎧ ⎪ ⎪ ⎨

0 fnr − fnSN

U SN ⎪ ⎪ ⎩ fn − fn

1

rn =

n=2

⎧ ⎪ ⎪ ⎨

1 fnSN − fnr

SN U ⎪ ⎪ ⎩ fn − fn

0

fnr ≤ fnSN fnSN ≤ fnr ≤ fnU

(24)

p w · rn n=1 n = p

fnr ≥ fnU

r

n=1

fnr ≤ fnU fnU ≤ fnr ≤ fnSN fnr ≥ fnSN

The fuzzification process described in (24) and (25) corresponds to the objectives that should be maximized and minimized, respectively. In addition, the total membership function is defined as (26) for each Pareto optimal solution taking into consideration the individual membership functions as well as the relative significance of objective functions (wn values). This total membership function determines the degree of optimality corresponding to each Pareto solution.

(25)

wn

(26)

The most desired solution is defined as the one with the highest value of total membership [32,33]. The procedure of the proposed framework for ε-constraint approach is depicted in Fig. 1 as a flowchart.

122


Table 1 Payoff table for case 1.

˚1 =

40766.834 41145.141

18278.567 15666.605

Table 2 Optimum solution of SHTS with equal weighting factors for case 1.

Objective function

Weighting factor

Cost ($) Emission (lb)

1 40,854.280 1 16,180.200 Total membership of all objective functions

The system used to implement the SHTS problem is a test system comprising four hydro generating units with multi-reservoir cascaded configuration as well as three thermal units [25–28]. The data of hydro and thermal units for the first case study can be found in [25]. Furthermore, the presented framework is implemented on IEEE 118-bus test system for the second case study. The scheduling period is considered to be 24 h on the hourly basis. In order to verify the effectiveness of the proposed approach, two case studies have been simulated. 4.1. Case study 1

Emission(lbs)

Membership value 0.772 0.800 0.786

180 160 140 120 100 80 60

5

0

10

15

20

Hour Reservoir 1

The computer system employed to solve the proposed SHTS problem is a Pentium IV PC with 2.6 GHz CPU and 2 GB RAM using solver CONOPT implemented in the general algebraic modeling system (GAMS) [34]. The Pareto solutions of SHTS are found using lexicographic optimization and ε-constraint method. The main objective function in the ␧-constraint technique in this paper is the first objective function f1 , i.e. cost minimization. The Pareto optimal solutions are obtained considering 10 grid points (q2 = 10) for f2 , i.e. emission minimization. Thus, the problem should be solved totally for 11 times (q2 + 1 = 11) to derive the Pareto solutions, that all the obtained solutions are feasible [24,33]. The obtained payoff table (˚1 ) is illustrated in Table 1. As it can be observed from Table 1, the total cost can be reduced to 40,766.834 ($) provided that the emission generation is discarded, that in this case the emission is 18,278.576 (lb). On the other hand, the emission can be reduced to 15,666.605 (lb) provided that the cost is discarded while in this case, the total cost has been exposed to a remarkable increase such that it reaches 41,145.141 ($). In other words, considering the emission minimization in the multi-objective function of SHTS causes the total cost to be faced by 378.307 ($) increase, i.e. (41,145.141–40,766.834 = 378.307($)) while emission generation decreases by 2611.962 (lb), i.e. (18,278.567–15,666.605 = 2611.962 (lb)). This state demonstrates the conflicting nature of these two objective functions. The obtained Pareto set for the proposed SHTS problem is illustrated in Fig. 2. As previously mentioned, the most preferred solution among all Pareto solutions can be chosen using a fuzzy satisfying method [24,33]. In the proposed fuzzy approach, at first, the weighting factors corresponding to two objective functions are the same. Table 2 represents the results obtained for these two objective functions 41200 41150 41100 41050 41000 40950 40900 40850 40800 40750 40700 15500

Reservoir storage volume (cu.m)

4. Simulation results

Objective function value

Reseroir 2

Reservoir 3

Reservoir 4

Fig. 3. Reservoir storage volumes with equal weighting factor for case 1.

wherein the membership value implies the degree of optimality. With the same weighting factors, the highest total membership value is obtained as 0.786 for the 9th Pareto solution. Hence, this solution is considered as the best one. Table 3 indicates the details of the most desired solution determined by lexicographic optimization and ε-constraint method for equal weighting factors. Fig. 3 shows the optimal values of reservoir storage volumes derived using the presented approach for equal weighting factors. However, the DM can change the weighting factors to attain the most preferred solution. For instance, the DM may intend to obtain a solution with less cost and higher emission. Therefore, by assigning the weighting factors to 3 and 1 for cost and emission, respectively, the proposed method is solved again. In this state, it is obvious that the Pareto solution with high value of membership for cost and low value of membership for emission is searched as depicted in Fig. 4. As it can be observed from Fig. 4 and Table 4, the membership value of cost that was 0.772 in the case of equal weighting factors enhanced to 0.938 in the case of different weighting factors. In this state, Pareto solution 6 is the most desired solution. The corresponding cost of this solution is represented in Table 4 as 40,790.190 ($) which is very close to its ideal value, i.e. 40,766.834 ($). On the other hand, the emission increases due to its low membership value, i.e. 0.500. The economic load scheduling (ELS) in Refs. [26–28] represents the scheduling strategy, in which the economic aspect is merely taken into consideration. Besides, the goal of implementing economic emission scheduling (EES) strategy is to minimize the 1 0.8 0.6 0.4 0.2 0

1

2

3

4

5

6

7

8

9

10

11

Pareto soluon number

16000

16500

17000

17500

18000

18500

Total membership

Cost membership

Emission membership

Cost ($) Fig. 2. Pareto optimal solutions the emission versus cost for case 1.

Fig. 4. Variation of total membership, cost and emission functions Vs. Paretooptimal solutions for case 1.


123

Table 3 The details of the best compromise solution with equal weighting factors for case 1. Hour

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Water discharge rates (× 104 m3 )

Hydro generations in MW

Qh1t

Qh2t

Qh3t

Qh4t

Ph1t

Ph2t

8.82 8.84 8.44 8.10 7.92 8.21 8.51 8.60 8.87 8.77 8.82 9.01 8.79 8.38 8.20 8.26 8.04 8.22 7.77 7.55 6.84 6.59 6.73 6.74 Cost ($)

6.00 6.00 6.00 6.00 6.00 6.00 6.62 7.03 7.67 7.69 7.96 8.53 8.47 8.18 8.32 8.78 8.99 9.70 9.94 10.46 10.62 11.37 12.35 13.31

22.47 21.22 20.69 19.99 19.01 17.62 16.59 17.07 17.03 17.03 17.25 17.62 18.02 18.79 18.95 17.97 16.98 15.88 15.07 14.09 11.63 12.15 12.63 13.05

6.00 6.00 6.00 6.00 6.00 6.85 9.76 11.84 14.07 15.28 16.59 17.07 17.03 17.03 17.25 17.62 18.02 18.79 18.95 20.00 20.00 20.00 20.00 20.00

80.34 50.16 80.54 51.30 78.08 52.93 75.68 54.50 73.97 55.50 75.34 55.99 76.92 60.11 77.59 62.88 79.48 67.20 79.68 67.94 80.98 70.17 82.32 73.33 81.67 72.70 80.14 71.44 79.60 72.59 80.31 74.85 79.01 74.98 80.18 76.67 76.99 76.14 75.17 77.05 70.05 76.71 68.25 78.24 69.65 78.70 70.06 77.48 40,854.28 ($)

Thermal generations in MW Ph3t

Ph4t

Ps1t

Ps2t

27.93 29.90 29.44 31.57 35.20 40.51 43.64 41.98 41.52 41.24 40.44 39.59 39.17 36.77 36.69 40.64 44.22 47.85 50.20 52.65 54.96 56.83 58.14 59.00

129.03 125.74 121.63 115.82 131.23 154.91 200.72 231.04 257.94 271.15 282.20 286.01 285.71 285.65 287.35 290.13 293.09 298.41 299.44 303.92 300.78 296.40 291.03 284.40

137.31 174.72 146.94 186.58 122.98 156.94 108.36 138.71 108.89 139.37 140.75 178.96 171.32 216.32 175.00 229.98 175.00 256.01 175.00 243.01 175.00 246.42 175.00 269.26 175.00 248.90 167.29 211.44 160.17 202.78 173.07 218.43 168.15 212.48 175.00 241.30 170.88 215.78 162.55 205.68 119.63 152.77 104.44 133.82 101.93 130.68 87.96 113.17 Emission (lb)

Ps3t 150.51 159.00 138.00 125.37 125.83 153.53 180.98 191.54 212.85 201.99 204.79 224.49 206.85 177.28 170.82 182.58 178.07 200.59 180.57 172.98 135.10 122.01 119.86 107.92

Total in MW

Ph +

Ps

750 780 700 650 670 800 950 1010 1090 1080 1100 1150 1110 1030 1010 1060 1050 1120 1070 1050 910 860 850 800 16,180.20

emission issues while combined emission scheduling (CEES) tries to optimize two objective functions comprehensively by considering a set of weights to make the original problem a single optimization problem. However, the reported results of CEES for same weighting factors for all cases are taken to make a fair comparison between the results obtained in Refs. [25–28] and the presented approach. One of the most significant factors making an optimization method applicable to real time and real size systems is the computational burden of that method. The averaged CPU time mainly depends on the computer system employed to implement other methods. Thus, it cannot be directly compared for different approaches. So, this paper introduces the scaled CPU time for each technique as below [35,36]:

for all cases in the case of quantity, e.g. the obtained cost from the proposed approach is 7051.72 (47,906 − 40,854.28 = 7051.72), 4059.72, 2652.72 and 3489.72($) less than reported by Refs. [25–28], respectively, while the obtained emission by the proposed method is 10,053.8, 3434.8, 2002.8 and 1227.8 (lb) less than the reported ones by Refs. [25–28], respectively, at the same time. Moreover, the solution time required by this method is less than other solution methods. The scaled solution time cannot be compared, since enough information is not available. The results presented in this paper verify the efficiency of incorporating lexicographic optimization and ε-constraint method to optimally solve the daily generation scheduling of hydrothermal systems on the hourly basis in a reasonable time.

Scaled CPU time =

4.2. Case study 2

Given CPU speed 2.6 GHz

× given average CPU time

(27) The CPU execution times of the referred methods are obtained using the scaled CPU time for different methods of the same problem considering the same base, i.e. 2.6 GHz. In this technique, the method with the least scaled CPU time is the fastest one and vice versa [35,36]. The optimal solution derived from ε-constraint technique along with the solutions reported by Refs. [25–28] is represented in Table 5. This paper tries to obtain ELS and EES by optimizing the economic load scheduling as well as economic emission scheduling, respectively. Note that the results of ELS and EES construct the payoff table (Table 1). As it can be seen in Table 5, the results obtained from the proposed problem is better than those reported in [25–28] Table 4 Optimum solution of SHTS with different weighting factors for case 1. Objective function

Weighting factor


Cost ($) Emission (lb)



In this case, IEEE 118-bus test system has been used to implement the model in order to show the efficiency of the presented method. This test system includes 54 thermal units (33 coal-fired units, 11 gas-fired units and 10 oil-fired units) and 8 hydro units. Ref. [19] includes the detailed data of the case 2 for hydro units, while other required information can be found in Ref. [37]. It is worth-mentioning that thermal units 5, 10, 11, 28, 36, 43, 44 and 45 have valve loading effects cost and thermal units 7, 10, 30, 34, 35 and 47 have prohibited operation zone (POZ) limitations. Ref. [37] presents the data on demand and thermal units. Table 6 represents the obtained payoff table. The results of the first column relate to the first objective function (f1 = FCost ) and the second column represents the results pertaining to the second objective function (f2 ), i.e. emission. The problem has been solved for 11 times (q2 + 1 = 11) to obtain the Pareto solutions, while all of the solutions are feasible. In order to select the most preferred solution among Pareto optimal solutions, the fuzzy decision maker has been utilized in this paper. Note that the weighting factors of two objective functions in the proposed decision making process are the same. Fig. 5 shows Pareto optimal solutions. Fig. 6 demonstrates the variation of objective functions and total membership value versus Pareto-optimal solutions while the best solution is Pareto

˚2 =

631976.59 779167.60

760000 740000 720000 700000 680000 660000

44,344 17,408 NR –

640000 620000

47,871 17,019 NR –

EES

40000

60000

80000

100000

120000

Cost ($)

43,278 17,984 NR –

Membership value

43,507 18,183 NR –

1.2

41,909 30,724 NR – 44,914 19,615 74.97 –

0.8 0.6 0.4 0.2 1

2

3

Total membership

4

5 6 7 8 Pareto solution number

Cost membership

9

10

11

Emission membership

Fig. 6. Variation of objective functions and total membership value versus Paretooptimal solutions for different weighting factor case for case 2.

solution 8, since it is associated with the highest total membership as 0.687. The results of the Pareto solution 8 with equal weighting factors are shown in Table 7. Fig. 7 shows the optimal values of reservoir storage volumes derived using the presented approach for equal weighting factors. Note that the total membership value indicates the degree of optimality. DM is able to easily find the most desired solution by changing the weighting factors, while DM tends toward minimizing cost rather than emission. Hence, by assigning the weighting factors as 2 and 1 for cost and emission, respectively, the obtained solutions are shown in Table 8. It is obvious that in such conditions, the solution with higher membership value for cost and lower membership value for emission is intended to be found. As it can be observed from Table 8 and Fig. 6, the cost considerably improves from 0.674 with equal weighting factors to 0.860 with different weighting factors. Table 8 shows the cost value as 65,2635.64 ($), that is very close to its ideal value, i.e. 63,1976.59 ($), but with higher value of emission, i.e. 62,132.02 (lb),

47,906 26,234 4582 – NR NR NR – NR NR NR –

Table 7 Optimum solution of SHTS with equal weighting factors for case 2. NR = not reported.

Cost ($) Emission (lb) Solution time (s) Scaled Solution time (s)

1

0

43,500 21,092 72.96 –

51,449 18,257 72.74 –

45,392 17,659 NR –

ELS

Ref. [28]

CEES EES ELS CEES ELS

EES Ref. [26]

EES

CEES ELS

Ref. [27]

20000

0

Fig. 5. Pareto optimal solutions the emission versus cost for case 2.

Ref.[25] Method

780000

600000

Table 5 Result of different methods for case 1.

100499.27 4581.16

800000

Emission (lbs)

40,854.28 16,180.20 15.68 15.68 41,145.14 15,666.61 1.12 1.12

Table 6 Payoff table for case 2.

40,766.83 18,278.57 1.78 1.78

CEES CEES

ELS

EES


Proposed method

124

Objective function

Weighting factor


Cost Emission



Reservoir storage volume (Hm3)


1000

with good approximation like the initial volume of hydro plants and reservoir inflow, etc. Since the scope of this paper is the short term scheduling, the uncertainty of drought periods has not been taken into consideration. In short term scheduling, independent system operator (ISO) can use the proposed approach for day-ahead scheduling of generating units.

100 10 1

125

5. Conclusion 5

0

10

15

20

our Ho Reservoir 1

Resservoir 2

Reservoir 3

Reservoirr 4

Reservoir 5

Resservoir 6

Reservoir 7

Reservoirr 8

Fig. 7. Reservoir storage volumes with equal weighting factors for case 2.

Table 8 Optimum solution of SHTS with different weighting factors for case 2. Objective function

Weighting factor

Cost Emission




as its membership value is low (0.400). Fig. 6 depicts the variation of objective functions and total membership value versus Pareto optimal solutions relating to the case with different weighting factors. It is worth-mentioning that in test case 2, the linearization technique has been used to accelerate the optimization process while taking into consideration the unit commitment process. The CPLEX solver under GAMS [34] has been employed to solve the Mixed Integer Programming (MIP) optimization problem. It is noted that in the first case, the authors tried to verify the effectiveness of the proposed method using the well-known case study, while the first case considers the economic dispatch for 24 h disregarding the unit commitment of generating units. In this paper, the default gap for CONOPT and CPLEX solvers have been used. The Pareto solutions found by CONOPT solver are locally optimal while the Pareto solutions found by the CPLEX solver are integer solution. The integer solution means that a feasible integer solution has been found that this solution satisfies the termination criteria. The optimization statistics for multi-objective SHTS problem for both cases has been represented in Table 9. It can be observed from this table that 15.68 (s) and 1092 (s) is required for the presented method to solve the problem and find the Pareto optimal solutions for the first case and the second case, respectively. Thus, for the second case, 92.273 (s) (1092/11 = 92.273) is needed on average to find one Pareto solution. This solution time for IEEE 118-bus test system is rational showing the effectiveness of the proposed method for solving STHS problem. It is worth to mention that in general, the hydrothermal scheduling can be divided to 3 planning horizons as long term, middle term and short term. In long-term and middle term scheduling, many parameters are uncertain like drought and flood periods, etc. But, in short term scheduling, most of the parameters are known

This paper successfully incorporates the lexicographic optimization and ε-constraint method to solve the short-term combined economic emission load dispatch of hydro and thermal units considering non-smooth fuel cost as well as the emission level function for thermal plants and multi-reservoir cascaded configuration of hydro units, as well. In addition, in comparison with the methods reviewed, i.e. Refs. [25–28], the fuel cost and emission can be reduced by this method. In addition, the simulation results of IEEE 118-bus test case are another evidence for the effectiveness of the proposed method. The proposed method makes the DM able to decide on different operation strategies by conveniently changing the weighting factors of the objective functions, thus leading to a more flexible operation of generating units. The ongoing research is to present a stochastic model to consider renewable energy resources with hydrothermal units and investigating their impacts on pollutant emissions. Appendix A. Appendix Table A1 shows the cost curve coefficients and operating limits of thermal generators for case 2 [37]. Table A2 shows the emission curve coefficients of thermal generators for case 2 [37,38]. In addition, Fig. A1 [25] and Fig. A2 [39–42] describe how the hydro plants hydraulically connected in cases 1 and 2, respectively.

Fig. A1. Hydraulic topology for case 1.

1

Reservoi r

2 Hydro po ower plant 3 4

Table 9 Optimization statistics for multi-objective SHTS problem. Case number 1 2

No. of single constraints 14,096 675,555

No. of single variables 5456 678,540

No. of discrete variables 1808 364,770

6

5 Solution time (s)

7

15.68 1092

8 Fig. A2. Hydraulic topology for case 2.

126


Table A1 Cost curve coefficients and operating limits of thermal generators for case 2. Unit

PsiMin (MW)

PsiMax (MW)

a ($/MW2 )

b ($/MW)

c ($)

e ($)

f (1 MW–1 )

SUi /SDi ($)

1 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

5 5 5 100 10 25 5 5 100 100 8 8 25 8 25 8 8 25 50 50 25 25 50 50 25 100 100 80 30 10 5 5 25 25 150 25 10 50 8 20 100 100 100 8 25 25 8 25 25 25 25 25

30 30 30 300 30 100 30 30 300 350 30 30 100 30 100 30 30 100 250 250 100 100 200 200 100 420 420 300 80 30 30 20 100 100 300 100 30 200 20 50 300 300 300 20 100 100 20 50 100 100 100 50

0.0697 0.0697 0.0697 0.0109 0.0697 0.0128 0.0697 0.0697 0.0109 0.0030 0.0697 0.0697 0.0128 0.0697 0.0128 0.0697 0.0697 0.0128 0.0024 0.0024 0.0128 0.0128 0.0044 0.0044 0.0128 0.106 0.0106 0.0109 0.0459 0.0697 0.0697 0.0283 0.0128 0.0128 0.0109 0.0128 0.0697 0.0109 0.0283 0.0098 0.0109 0.0109 0.0109 0.0283 0.0128 0.0128 0.0283 0.0098 0.0128 0.0128 0.0128 0.0098

26.2438 26.2438 26.2438 12.8875 26.2438 17.8200 26.2438 26.2438 12.8875 10.7600 26.2438 26.2438 17.8200 26.2438 17.8200 26.2438 26.2438 17.8200 12.3299 12.3299 17.8200 17.8200 13.2900 13.2900 17.8200 8.3391 8.3391 12.8875 15.4708 26.2438 26.2438 37.6968 17.8200 17.8200 12.8875 17.8200 26.2438 12.8875 37.6968 22.9423 12.8875 12.8875 12.8875 37.6968 17.8200 17.8200 37.6968 22.9423 17.8200 17.8200 17.8200 22.9423

31.67 31.67 31.67 6.78 31.67 10.15 31.67 31.67 6.78 32.96 31.67 31.67 10.15 31.67 10.15 31.67 31.67 10.15 28.00 28.00 10.15 10.15 39.00 39.00 10.15 64.16 64.16 6.78 74.33 31.67 31.67 17.95 10.15 10.15 6.78 10.15 31.67 6.78 17.95 58.81 6.78 6.78 6.78 17.95 10.15 10.15 17.95 58.81 10.15 10.15 10.15 58.81

0 0 0 100 0 0 0 0 100 120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 150 0 0 0 0 0 0 0 105 0 0 0 0 0 200 200 200 0 0 0 0 0 0 0 0 0

0 0 0 0.084 0 0 0 0 0.084 0.0077 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.063 0 0 0 0 0 0 0 0.081 0 0 0 0 0 0.042 0.042 0.042 0 0 0 0 0 0 0 0 0

40 40 40 440 110 40 50 40 100 100 40 40 50 40 50 40 40 59 100 100 50 50 100 100 50 250 250 100 45 40 40 30 50 50 440 50 40 440 400 30 45 100 110 30 50 50 30 45 50 50 50 45

Table A2 Emission curve coefficients of thermal generators for case 2. Unit

8 9 10 11 15 16 17 18 21 22 23 24 28 29 30 31

Coefficient of SO2 emission function

Coefficient of NOx emission function

˛si (lb/MW2 )

ˇsi (lb/MW)

si (lb)

˛si (lb/MW2 )

ˇsi (lb/MW)

si (lb)

0.024382 0.024382 0.003806 0.001050 0.020899 0.003840 0.020899 0.020899 0.000600 0.003200 0.003200 0.001100 0.002118 0.002175 0.009185 0.013933

9.185337 9.185337 4.510625 3.766000 7.873146 5.346000 7.873146 7.873146 3.082474 4.455000 4.455000 3.322500 1.667830 2.577500 3.094155 5.248764

11.0845 11.0845 2.37300 11.5360 9.50100 3.04500 9.50100 9.50100 7.00000 2.53750 2.53750 9.75000 12.8320 1.35600 14.8660 6.33400

0.009753 0.009753 0.001523 0.000420 0.008360 0.001536 0.008360 0.008360 0.000240 0.001280 0.001280 0.000440 0.000847 0.000870 0.003674 0.005573

3.674135 3.674135 1.804250 1.506400 3.149258 2.138400 3.149258 3.149258 1.232990 1.782000 1.782000 1.329000 0.667132 1.031000 1.237662 2.099506

4.4338 4.4338 0.9492 4.6144 3.8004 1.2180 3.8004 3.8004 2.8000 1.0150 1.0150 3.9000 5.1328 0.5424 5.9464 2.5336


127

Table A2 (Continued) Unit

Coefficient of SO2 emission function ˛si (lb/MW )

ˇsi (lb/MW)

0.010449 0.004245 0.001920 0.001920 0.001631

3.936573 5.654519 2.673000 2.673000 1.933125

2

32 33 34 35 36

Coefficient of NOx emission function si (lb) 4.75050 2.69250 1.52250 1.52250 1.01700

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1.574629 2.261807 1.069200 1.069200 0.773250

1.9002 1.0770 0.6090 0.6090 0.4068

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