A small-population based parallel differential evolution algorithm for short-term hydrothermal scheduling problem considering power flow constraints

Accepted Manuscript A Small-Population based Parallel Differential Evolution Algorithm for Short-term Hydrothermal Scheduling Problem Considering Power Flow Constraints

Jingrui Zhang, Shuang Lin, Houde Liu, Yalin Chen, Mingcheng Zhu, Yinliang Xu PII:

S0360-5442(17)30184-6

DOI:

10.1016/j.energy.2017.02.010

Reference:

EGY 10303

To appear in:

Energy

Received Date:

02 June 2016

Revised Date:

21 December 2016

Accepted Date:

02 February 2017

Please cite this article as: Jingrui Zhang, Shuang Lin, Houde Liu, Yalin Chen, Mingcheng Zhu, Yinliang Xu, A Small-Population based Parallel Differential Evolution Algorithm for Short-term Hydrothermal Scheduling Problem Considering Power Flow Constraints, Energy (2017), doi: 10.1016/j.energy.2017.02.010

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ACCEPTED MANUSCRIPT 1. Power flow constraints are introduced into the short-term hydrothermal scheduling (STHS) problem 2. A small-population based parallel DE algorithm is proposed to solve the considered STHS problem 3. The operations of gather and scatter and aggregative DE are introduced into the parallel algorithm 4. Four constraint handling rules as well as a lead operation are proposed to enhance the feasibility 5. Comparisons show the parallel DE approach performs effectively and yields competitive solutions

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A Small-Population based Parallel Differential Evolution Algorithm for Short-term Hydrothermal Scheduling Problem Considering Power Flow Constraints Jingrui Zhang1*, Shuang Lin2, 1, Houde Liu3, Yalin Chen1, Mingcheng Zhu1 and Yinliang Xu4 1. Department of Instrumental & Electrical Engineering, School of Aerospace Engineering, Xiamen University, Xiamen 361005, China 2. Electric Power Research Institute, Fujian Electric Power Co., LTD, State Grid Corporation of China, Fuzhou, 350007, China 3. Shenzhen Engineering Laboratory of Geometry Measurement Technology, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China 4. SYSU-CMU Joint Institute of Engineering, School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou 510275, China * Corresponding author: Jingrui Zhang ([email protected])

Abstract: Short-term optimal hydrothermal scheduling plays one of the most important roles in the modern power system plan and operation. The problem aims at minimizing the total fuel cost of the thermal units while satisfying various constraints such as power balance, water balance and other constraints on thermal units as well as hydro units. Except for the above various constraints, transmission network topology and valve point effects are also introduced into the mathematical optimizing model of the short-term hydrothermal scheduling (STHS) problem. Then a small-population based parallel differential evolution approach is proposed to solve the STHS problem considering power flow constraints. In the proposed approach, a large population is divided into several subpopulations each with a small population size and several parallel running processes of one or more CPUs are performed synchronously each evolving a certain subpopulation and searching for the optimal solution independently. Two different methods are employed in the proposed parallel DE approach in order to avoid low diversity of the small population in each process. One is implemented through the small population itself and the other through the communication mechanism among different running processes. Four constraint handling rules as well as a lead operation are proposed to enhance the feasibility of solutions. Numerical results for two well-known sample test systems are presented to demonstrate the capabilities of the proposed parallel DE algorithm to generate optimal solutions of STHS problem. Two other test systems with transmission networks of standard IEEE 9-bus and IEEE 39-bus are also employed to test the effectiveness of the proposed parallel DE algorithm. The results demonstrate the superiority of the proposed algorithm. Keywords: hydrothermal scheduling, differential evolution algorithm, parallel, small population, power flow I.

INTRODUCTION

Short-term hydrothermal scheduling (STHS) is one of the most important tasks in the operation of power systems [1]. The economic generation scheduling of a hydrothermal system is usually supposed to find an optimal generation scheme to minimize the operational cost of thermal units over a schedule time of a single day or a week while satisfying various constraints. The complicated constraints usually include the equality constraints such as power balance and water balance and the inequality constraints such as the discharge limits, reservoir volume limits, power output limits of generators and other system limits. The idea behind STHS is to allot the available water resources to hydro generators in each time interval and to dispatch the thermal generators in such a manner that the fuel cost of thermal units is minimized and operational constraints of both thermal and hydro units are fully satisfied [2]. The STHS problem is more complex than pure thermal economic dispatch or unit commitment problem for its limited storage capability of water reservoirs and the stochastic nature of water inflow. Moreover, the water transport time delay between the cascaded hydropower stations leads to the time coupling of water balance constraints in the hydro subsystem. The objective function and constraints of the STHS problem contain nonlinearity which gives rise to non-linear problem. All these make the optimal scheduling of hydrothermal power system be a complex programming problem involving nonlinear objective function and a mixture of linear and nonlinear constraints. The complex optimization problem of the STHS requires high efficiency computing technologies. The STHS problem is solved traditionally using mathematical optimization techniques such as dynamic programming [3-6], gradient search method [7], nonlinear programming [8-10] and decomposition and coordination techniques [11-13]. But these approaches may not perform satisfactorily due to the nonlinearity and composite constraints of the problem [2]. The dynamic programming method can overcome the difficulty of non-linearity and non-convexity of the STHS problem [1]. But it suffers from computational overburden for its large dimensionality when applied to a practical sized power system. The gradient search methods and nonlinear programming methods with network flow have difficulties in dealing with constraints of system. The decomposition and coordination techniques usually make some simplifying assumptions in order to make the original model easy to solve, which may lead to suboptimal solutions with great loss of revenue. 1 / 27

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In recent years, heuristic algorithms as alternative approaches to the STHS problem have attracted many attentions. These novel evolutionary algorithms such as generic algorithm (GA) [14-16], gravitational search algorithm [17-23], honey-bee mating optimization algorithm [24, 25], particle swarm optimization (PSO) [20, 22, 26-37], quantum-inspired evolutionary algorithm [38], teaching learning method [39], cuckoo search algorithm [40-43], simulated annealing algorithm [44] and differential evolution (DE) algorithm [45-48] have been employed to solve the STHS problem. In addition to the traditional mathematical programming methods and the heuristic methods, the hybrid methods such as the hybrid of real coded GA and artificial fish swarm algorithm [1], adaptive chaotic artificial bee colony algorithm [24] and modified chaotic differential evolution algorithm [49] have also been developed for different hydrothermal scheduling problems. Anyway, the above researches have shown a desirable ability of heuristic algorithms to search the global optimum of the STHS problem. Among these evolutionary methods, DE algorithm has many advantages for its features of simple principle, fast convergence, and few control parameters. Lakshminarasimman and Subramanian [50, 51] presented modified differential evolution and modified hybrid differential evolution algorithms for the solution of STHS problem in succession. Several equality and non-equality constraints on thermal units as well as hydro units and the effect of valve-point loading [52] were considered in the problem formulation of STHS problem. K.K. Mandal, et al. [53] compared the efficiency among different variants of DE algorithm, PSO algorithm and GA algorithm for STHS optimization. The numerical results reported in the above researches show the DE algorithm has the ability of dealing with the hydrothermal scheduling optimization problem. In addition, hybrid methods [49, 54-57] combining differential evolution algorithms and other heuristic algorithms have also been considered as the popular solution techniques for the STHS problem. Moreover, differential evolution algorithms [58, 59] have also been extended to solve multi-objective hydrothermal scheduling problems. As reviewed above, DE algorithms have many advantages and are very popular in solving complicated constrained optimization problems. This is also why the DE algorithm is employed as the solution technique in this work. However, many scholars usually employed a large population size (more than 50) to improve the algorithm’s ability to find the global optimal solution. The large population size will consume more CPU time and this will limit its application in practical power system. On the contrary, the small population size (less than 20, usually 3-10) will accelerate the searching process of the evolution algorithms and several scholars have employed micro-GA [60, 61], micro-PSO [62] and micro-DE [63-66] to solve optimization problems. But there are still some shortcomings for these small population based algorithms. On the one hand, small population techniques have to face more selection pressures which may produce premature convergence. On the other hand, most of the current approaches employed the small-population based approaches to solve the benchmark test functions and only few researches focused on practical engineering optimizations. According to the state of the art, small-population based PSO algorithms were successively proposed for the STHS problem [34] and the short-term hydrothermal unit commitment problem [67] in our last two works, respectively. However, the power transmission networks of the hydrothermal power systems considered in the two articles were neglected as well as other researches did. In fact, not only DE algorithms but also other evolutionary algorithms usually give the hypothesis that all generators provide power generation to the same load bus in order to simply the optimization of the STHS problem. Within this hypothesis, the security validation for the obtained plan of the STHS problem has to be performed in order to satisfy the demand of the schedule. Though the security validation considering security constraints has the ability of dealing with the effects of transmission networks, it usually requires solving the STHS problem repeatedly because the hypothesis will make the optimal solution deviate from the real operation point or even be unfeasible. Though modeling of transmission networks were considered in Refs. [15, 68], some simplifications were made in their solution techniques reported. For example, transmission networks are only considered for the obtained best result in Ref. [15] and the problem were decomposed into five separate easy-to-solve sub-problems in Ref. [68]. Considering power flow constraints in the modeling of the STHS problem is one of the best ways to represent the effect of transmission network, but it will increase the dimension of the optimization problem. The voltages of generator buses and the ratios of transformers have to be chosen as the extra control parameters in the power flow constrained STHS problem. Both of the large population size employed and the increment of control parameters will definitely cost more CPU time and this will limit the algorithm’s application in practical power systems. But the small-population based algorithms will face selection pressures and may have difficult when dealing with large-scale optimization problems. Fortunately, parallel techniques will provide an alternative and perfect approach to this dilemma. The main contribution of this paper aims at proposing a small-population based parallel differential evolution approach for the solution of short-term hydrothermal scheduling problem considering power flow constraints. In the proposed approach, the power flow equations at each bus of the system are employed to instead of the traditional power balance constraints of STHS problems. Other power flow constraints such as the bus voltage limits and the line transmission power limits are also introduced into the STHS problem. Then a message passing interface [69-71] (MPI) based parallel differential evolution algorithm is developed to solve the power flow constrained STHS problem by making full use of the calculation ability of a multi-process CPU. In the proposed parallel DE algorithm, every parallel process implements DE searching operation iteratively using a small population size. Two different approaches are employed in the proposed parallel DE algorithm in order to avoid low diversity and slow selection pressures of the small population in every process. One is implemented through the small population itself and the other through the communication mechanism among different running processes. The foregoing approach employs a regeneration operation which is performed when the population diversity goes below the preset tolerance level to enhance the diversity. The latter one is implemented through the gather and scatter operation and the aggregative DE operation. A lead operation is employed in the parallel DE algorithm to adjust the parameters of vectors so 2 / 27

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that the vectors can locate at the feasible domain, meanwhile the vectors are supposed to satisfy the power flow equations. Two traditional hydrothermal test systems and two other hydrothermal test systems employing the standard IEEE 9-bus and IEEE 39-bus as the transmission networks are employed to show the feasibility and efficiency of the proposed approach. The numerical results show a quite encouraging convergence and robustness of the proposed small-population based parallel DE algorithm. The remaining part of this paper is organized as follows. Section II gives the formulation of the STHS problem considering power flow constraints. Section III briefly describes the DE algorithm. Section IV describes the small-population based parallel DE algorithm. Section V proposes some improvement measures to the parallel DE algorithm. Section VI gives the hydrothermal scheduling optimization steps based on the proposed parallel DE algorithm. The numerical results and analysis of the results are presented in section VII. Section VIII draws a conclusion of the whole work.

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II.

PROBLEM FORMULATION

The optimal schedule for power flow constrained hydrothermal power system is modeled as a complex optimization problem with a nonlinear objective function and a set of linear, nonlinear, and dynamic constraints. In order to formulate the problem mathematically, the notations used in this paper are introduced first. Then the objective function and constraints employed for formulating the STHS problem are presented. A. Notations 1) Parameters F sum fuel cost of thermal units

19

fi ( Psi ,t ) fuel cost of thermal unit i at time t

20

Phj ,t power generation of hydro unit j at time t

21

Vhj ,t storage volume of reservoir j at time t

22

Vhjbegin initial storage volume of reservoir j

23

Vhjend final storage volume of reservoir j

24

Shj ,t spillage of reservoir j at time t

25

I hj ,t water inflow of reservoir j at time t

26

 l water transport delay from reservoir l to its directly downstream reservoir

27

Ruj directly upstream hydro unit set of the hydro unit j

28 29 30 31 32 33

T total time intervals over scheduling horizon

34

() min ,() max lower and upper limits of variables

35

N s , N h number of thermal and hydro units respectively

36

Pf mt , Qf mt injected active and reactive power of bus m at time t

37

Pf mn , Qf mn active and reactive power transmission between bus m and bus n

38

Gmn , Bmn conductance and susceptance of the line between bus m and bus n

39

 mnt phase between bus m and bus n at time t

40

Ll ,t transmission power of line l at time t

41

ai , bi , ci , di , ei coefficients of thermal unit i

42

c1 j , c2 j , c3 j , c4 j , c5 j , c6 j power generation coefficients of hydro unit j

43

2) Decision Variables

44

Qhj ,t water discharge rate of reservoir j at time t

45

Psi ,t power generation of thermal unit i at time t

busN number of the buses in the power system numK number of transformers in the power system numL number of transmission line N population size of the algorithm D dimension of problem considered

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K r ,t ratio of a transformer lie in line r at time t

2

U nt voltage of bus n at time t

3 4 5 6 7 8

B. Objective Function and Constraints The following objective function and constraints must be taken into account for the formulation of STHS problem with power flow constraints. 1) Objective Function The major purpose in hydrothermal scheduling is to minimize the total fuel cost of the thermal units. The objective function can be formulated by T

Ns

9

min F   fi ( Psi ,t )

10

where f i ( Psi ,t ) can be described as follow and the absolute term which represents the valve point effects of thermal units.

11 12 13 14 15

16

17 18 19 20 21 22 23 24 25

f i ( Psi ,t )  ai  bi Psi ,t  ci Psi2,t  | di sin(ei ( Psimin  Psi ,t )) |

busN  Pf   U mtU nt (Gmn cos  mnt  Bmn sin  mnt ) mt   nm  busN  Qf  U mtU nt (Gmn sin  mnt  Bmn cos  mnt )  mt n m t  1, 2,T , m, n  1, 2, busN

Phj ,t  c1 jVhj ,t 2  c2 j Qhj ,t 2  c3 jVhj ,t Qhj ,t  c4 jVhj ,t  c5 j Qhj ,t  c6 j

Vhj ,1  Vhj begin , Vhj ,T 1  Vhj end j  1, 2, N h

34

(3)

(4)

d) Dynamic Balance of the Reservoir storage Vhj ,t 1  Vhj ,t  Ihj ,t  Qhj ,t  Shj ,t   (Qhl ,t l  Shl ,t l ) lRuj

(5)

j  1, 2, Nh , t  1, 2,T

3) Inequality Constraints a) Voltage Constraint of the Buses

U mmin  U mt  U mmax m  1, 2, busN

(6)

b) Power Generation Limits of the Generators Phj min  Phj ,t  Phj max

j  1, 2 , N h

Psi min  Psi ,t  Psi max

i  1, 2 , N s

(7)

c) Reservoir Storage Volume Constraint and Water Discharge Rate Constraint

Vhj min  Vhj ,t  Vhj max Qhj min  Qhj ,t  Qhj max

j  1, 2 , N h

(8)

r  1, 2, numK

(9)

d) Maximum and Minimun of transformer ratios

K rmin  K r ,t  K rmax

32 33

j  1, 2, N h

c) Initial and Final Reservoir Storage Constraint

30 31

(2)

where n  m are the buses that connect to the bus m . b) Hydro Unit Generation

28 29

i = 1,2, N s

2) Equality Constraints a) Power Balance Constraint The power balance constraint is implicitly considered in (2) when power flow constraints at each bus are introduced into the STHS problem.

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(1)

t 1 i 1

e) Line Transmission Power Limits Ll ,t  Lmax l

l  1, 2, numL 4 / 27

(10)

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III.

Differential evolution relies on four simple arithmetic operations which include population initialization, mutation, crossover and selection. The latter three operations are performed iteratively until the termination criterion is reached. A. Initialization The vectors of initial population are randomly initialized based on uniform distribution according to (11)

X id0  R( X dmin , X dmax ) i  1, , N , d  1, , D

6 7

DIFFERENTIAL EVOLUTION ALGORITHM

where

0 id

X is the initial value of the dth decision variable of vector i , X R( X dmin , X dmax )

min d

and

X

(11)

max d

[X

are the lower and upper bounds of the dth

min d

, X max ]

8 9 10

d decision variable and generates a random value uniformly within . B. Mutation The mutation operation is an important step in the DE algorithm to generate variant vectors. The dth variant variable of

11

X di , is given by the vector i at kth generation, described as A

k

k k k k A X di  X dra  F *( X drb  X drc )

12 k

k

(12)

k

13 14

where X dra , X drb and X drc are randomly selected vectors and i  ra  rb  rc ; F is a scaling factor and it controls the

15 16

C. Crossover The crossover operation is a discrete recombination to create trial vectors shown as (13).

amplification of the difference between two vectors of ra and rb .

k  A X di if R(0,1)  CR or d =rd A X di   k otherwise  X di k

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(13)

k A R(0,1) where X di is the dth trial decision variable of vector i at kth iteration, is a uniform distributed random number within (0, 1) , CR  [0,1] is the crossover rate which is adopted to determine variables for trial vectors and rd is a random constant integer preset in [1, D] which ensures that at least one element of the trial vector comes from the variant vector.

D. Selection Afterwards, perform selection for each target vector by comparing its fitness with that of the trial vector, and the better one would survive for the next generation (14). k k  A k A X if fitness( X i i )
IV.

MAINFRAME OF PARALLEL DE ALGORITHM

In this section, the procedures of the traditional serial DE algorithm and the proposed parallel DE algorithm, as well as the extra operations for parallel algorithm, would be shortly introduced. A. Procedure of Serial DE Algorithm The procedure of serial DE algorithm is introduced step by step in Fig.1 and more details of every operation are explained below. S1: Initial the parameter vectors of the initial population which is a large population (LP). S2: Evaluate the fitness of every vector. S3: Implement the mutation operation to create variant vectors. S4: Implement the crossover operation to create trial vectors. S5: Evaluate the fitness of every trial vector. S6: Implement the selection operation according to the fitness of every target vector and its trial vector. S7: If the termination criterion is not satisfied, implement S3 to S7 again, otherwise, end the DE iterations and print the detailed results. B. Procedure of Parallel DE Algorithm Increasing the population size is a straight way to enhance the diversity of the population. However, it will cost much CPU time in a serial way of DE. In the proposed parallel DE algorithm, a large population (LP) of individuals which is divided into several small populations is employed to search for the global optimum of an optimization problem. The parallel DE algorithm runs a number of processes to solve the optimization problem simultaneously. Every process implements the serial 5 / 27

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DE algorithm independently and evolves the small population (SP) until the criteria of gather and scatter operation expressed in the subsequent section is satisfied. Then the aggregative DE operation in the root process is performed on the gathered population through the traditional mutation and crossover. After that the new population generated by the aggregative DE operation will be divided into several small populations again and scattered to every running process of CPU. The parallel DE algorithm can take advantage of the multi-process CPU as much as possible and the detailed procedure of parallel DE algorithm is described in Fig.2. Rank index in this figure stands for the index of running process of CPU. Here Rank=0 identifies the root process of the parallel DE algorithm. The additional procedure in the parallel algorithm compared with the serial DE algorithm is exhibited below. S*1: Determine whether the criterion of gather operation is satisfied. S*2: Gather the information including decision vectors, scaling factor and crossover factor from all running processes. This procedure is implemented through the sending of every running process and the receiving of the root process. S*3: The root process implements the mutation to create variant parameter vectors. S*4: The root process implements the crossover to create trial parameter vectors. S*5: Scatter the new generated decision vectors to each process. This procedure is implemented through the sending of the root process and the receiving of other running processes. C. Gather and Scatter Operation The gather and scatter operation which includes two actions, gathering and scattering, plays one of the most important roles in the parallel DE algorithm. Both of the two actions need the continuity of the memory allocation for variables. The gathering action gathers information that the aggregative DE operation, which will be briefly introduced in the next subsection, needs from all running processes. The scattering action scatters information from the root process to all running processes. The gathered information includes decision vectors, scaling factors and crossover factors of all processes. Here, only the decision vectors consisting of a small population of individuals are considered as an example to introduce the gather and scatter operation, described as Fig.3. Assuming rank0 as the root process, SP0, SP1, mean different small populations of every running process. The frequency of the gather and scatter operation should be controlled at a suitable level to balance the CPU time consumed. If the frequency of the gather and scatter operation stays at a high level, it will take more CPU time to send and receive information among processes. The convergence will be slowed down for the frequent information communication. If the frequency stays at a low level, the diversity of small population of every process may be damaged. D. Aggregative DE operation The aggregative DE operation is another extra operation in the parallel DE algorithm. It is supposed to improve the population diversity for every running process. The operation is implemented by the root process with the gathered information from the gather and scatter operation. Here, the traditional mutation and crossover operations are performed on the gathered vectors of decision variables. After the aggregative DE operation of the root process, the new generated trial population is divided into several small populations again and scattered to every running process by the root process. This operation is supposed to improve the population diversity for every process, and it is a simple way to connect and share the information among different running processes. V.

SOME IMPROVEMENT MEASURES FOR THE DE ALGORITHM WITH SMALL POPULATION

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The vectors for the mutation operation are selected randomly, and F and CR are preset by the user in the traditional DE algorithm. It is not suitable for the searching of the small population based parallel DE algorithm. In this section, several improvement measures have been present to overcome the shortcomings such as low convergence and damaged diversity of small population. The sorting for mutation operation and self-adaptive selection for F and CR are introduced to accelerate convergence of the algorithm. In addition, a migration operation is also adopted to keep the population diversity above a desired level. A. Sorting for Mutation Operation The improvement for the mutation operation is to sort the three randomly selected vectors by their fitness. The one with

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the DE algorithm with small population employs this self-adaptive method for its F and CR.

k

k

k

best fitness is chosen as X dra , the one with the second best fitness is chosen as X drb and the last one is chosen as X drc . Then the mutation operation of equation (12) is performed within the sorted index. This modified mutation will help the algorithm to search along the direction towards the best individual selected. B. Self-Adaptive Selection for F and CR A self-adaptive method has been put forward to catch the suitable parameters F and CR for DE algorithm in [72] and here C. Migration Operation The convergence of small population based DE can be improved by the above two improvement measures. However, faster converging speed may increase the selection pressure which will lead to premature convergence. Aiming at balancing the convergence speed and the selection pressure, a migration method has been proposed for the small-population based particle swarm optimization in our last article [34]. Here this migration operation is also used to avoid the DE algorithm sinking into the local optima. This introduced operation regenerates a new population around the current best individual, and offers an opportunity to jump out local optima zones. Important parameters of the migration operation proposed in [34] 6 / 27

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f t include mig which represents the closeness degree on a variable between two vectors, and mig which represents the tolerated level of the swarm diversity. This migration operation is also another try to enhance the diversity of small population.

3

VI.

PARALLEL DE ALGORITHM FOR STHS PROBLEM CONSIDERING POWER FLOW CONSTRAINTS

4 5 6 7 8 9 10 11

Here in this section the structure of the decision vector for STHS problem considering power flow constraints, the constraint handling rules, and the detailed steps of the parallel DE algorithm for the problem are described. A. Structure of Decision Vector The hydrothermal power system consists of three subsystems, i.e., hydro subsystem, thermal subsystem and network topology subsystem. Discharge rates and thermal power generations are usually chosen as the control variables of hydro subsystem and thermal subsystem respectively [16]. Moreover, the ratios of transformers and voltages of the generator buses are also need to be optimized while considering power flow constraints. With above considerations, the structure of a decision vector i can be designed as following.

12

X i   Qhi Ps i U i K i 

13

T

where

14

 Qhi 1,1  Qhi 1,T  Qhi      i  Qi  Q hN h ,T  hNh ,1

15

 Psi1,1  Psi1,T  Ps      i i P  sN s ,1  PsN s ,T

16

i  U1,1  U1,i T  Ui      i U i  sumU ,1  U sumU ,T

    

(17)

17

i  K1,1  K1,i T  Ki      i  Ki  numK ,1  K numK ,T

    

(18)

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

i

    

(15)

    

(16)

where sumU denotes the number of generator buses. B. Rules Dealling with Constraints Four constraint handling rules are proposed to ensure the feasibility of the obtained solution of the STHS problem considering power flow constraints. 1) Rule 1:Boundary limited rule The voltages of generator buses, power generation outputs of thermal generators, water discharge rates of hydro generators and ratios of transformers are restricted to their boundaries when they violate their constraints. 2) Rule 2:Disposal strategy of the initial and final reservoir volumes The error between the expected final reservoir volume and the practical final reservoir volume with the obtained discharges for any reservoir j is shown as (19) in which zero spillages are supposed for simplification. Here two loops are designed to deal with this error. The inner loop is a coarse error adjustment and the error value is used to modify the water discharges at all time intervals. The outer loop is an intense error adjustment and the error value is used to modify the water discharge at only a time interval which is chosen randomly. The disposal strategy is operated as Fig.4 shown. The parameter ae is smaller than the parameter be in the figure and the decision of employing the inner loop or the outer loop depends on errorj the value of . T

T

t 1

t 1

errorj Vhj ,T 1 Vhjend Vhjbegin   Qhj ,t   I hj ,t  

T

Q

lRuj t 1 l

hl ,t  l

Vhjend

(19)

3) Rule 3:Disposal strategy of power flow constraints The typical power flow constraints of the STHS problem are power balance equality constraints. Here in this rule, NewtonRaphson method and a lead operation are employed to solve this power flow equations. The Newton-Raphson method is used to perform the famous power flow calculation. After the calculation, the power flow solution for a certain given decision vector will be obtained. The net injection of active power and positive power of the slack bus can be obtained through equation (20). The net injection of positive power of PV buses can be deduced by (21). Due to the stochastic of the generated decision vector, the obtained power flow solution especially the power generation of the slack bus may violate the constraints. 7 / 27

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Although penalty function method is serviceable to deal with this issue, it will waste a lot of CPU time to eliminate the illegal vectors through the iteration process and the penalty factors are quite difficult to confirm. Here a lead operation is proposed in order to avoid the above scene effectively and handle the violation of power generation of the slack bus. According to the power flow equations, the load demand is usually considered as known parameter, so the power outputs of PV buses play an important role in deciding the power output of slack bus. The lead operation offers a heuristic way to adjust the power generation of PV bus so that the power generation of slack bus can situate in its feasible domain. The power generations of hydro units are decided by the water discharge rate and reservoir volume according to (3). As a result, the lead operation only adjusts the power generation of thermal units. Moreover, the lead operation is performed only if the deltaP , which denotes



the violation value for the active power of slack bus, is larger than the preset constant . The process of lead operation for each time interval t  1, 2, , T can be described as following. busN   Pf slack ,t   U slack ,tU n ,t Gslack , n  nslack t  1, 2, , T  busN Qf  U slack ,tU n ,t Bslack , n  slack ,t n slack

11 12 13 14 15 16 17 18

Qf pvt 

21

pvt

U nt (G pvn sin  pvnt  B pvn cos  pvnt ) t  1, 2, , T

pv=1, 2 ,...,numPV (21)

Step 1: Calculate the injected power of slack bus through power flow calculation, described as Spower 0 , according to the corresponding power outputs and voltages of PV buses, the voltage and the phase of slack bus, the ratios of transformers and the known load demand. Step 2: If Spower 0 locates at the feasible domain of slack bus, then go to Step 6, otherwise, go to Step 3. Step 3: Calculate the violation value of active power injection denoted as deltaP by max  Psmax ,slack  Spower 0 if Ps ,slack  Spower 0  min deltaP   Psmin ,slack  Spower 0 if Ps ,slack  Spower 0  0 otherwise 

max

(22)

min

where Ps ,slack denotes the maximum power of slack bus, and Ps ,slack denotes the minimum power of slack bus. Step 4: Adjust the power generation of PV buses according to (23) if deltaP is larger than the preset constant  .

Pst, pvi  Pst, pvi    deltaP / N s , deltaP  

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U

n pv

(20)

where numPV denotes the number of PV bus.

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busN

t s , pvi

where P

pvi  1, 2 , ,N s

(23)

denotes the power injection of PV bus i at time interval t , and  is the scale factor of the deltaP for the

consideration of transmission losses. Step 5: Calculate the power flow equations using Newton-Raphson method again and the power generation of slack bus according to the adjusted power generation of PV bus and other parameters. Step 6: Implement the lead operation for next time interval. 4) Rule 4: Penalty function approch Penalty function approach has to be adopted if the above rules don’t work. The penalty terms are added to the fitness value of a vector through the penalty function described as (24). The choice of the penalty factor is significant in the penalty function approach: a small factor is beneficial to the global exploring for its weak constraint restriction but a large one will offer a hard restriction and would be suitable for local searching. Here in this approach penalty factors are set as (25) in order to balance the global exploring and the local searching. numS

fitness ( X ik )  F ( X ik )   penaltys ( X ik ) punishs (k ) i 1,, N

34

(24)

s 1

35

k k k k k k where fitness ( X i ) denotes the fitness of X i , F ( X i ) denotes the fuel cost of X i and penaltys ( X i ) is the penalty of X i for the

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s th penalty.

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where punishs (k ) denotes the penalty factor at the k th iteration for the s th penalty,  s denotes the minimum penalty factor

punishs (k )   s  k kmax   s

(25)

for the s th penalty,  s denotes the maximum increment of the penalty factor for the s th penalty and kmax denotes the number of iterations. 8 / 27

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C. Procedure of Parallel DE Algorithm for STHS problem

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* Each running process of the parallel DE algorithm is supposed to find a vector X that satisfies the constraints (2)-(10) * * independently, meanwhile the fitness value of the vector X is the best one in the corresponding process. All vectors X and their corresponding fitness values obtained by all processes are gathered to the root process if the criterion of gather and scatter operation is satisfied. The best fitness value and the corresponding decision vector among all the gathered processes are regarded as the current best fitness value and the current best solution, respectively. Fig.5 shows the simplified flowchart of the parallel DE algorithm for the solution of STHS problem with consideration of power flow constraints. The detailed procedure is shown as the following. Step 1: Set the data of the considering power system including the network topology data, the generators data, the power demand and the water inflow data and the parameters of the proposed parallel DE algorithm. Step 2: Allocate memory for the scaling factor, crossover factor, parameter vectors and fitness of the vectors. The memory allocation for the root process is equal to the total memory allocation for all processes. Step 3: Initialize the random decision vectors and set the iteration number K  1 . Step 4: Adjust the water discharges of hydro plants according to rule 2 and calculate the power generation of the hydro units for each vector. Step 5: Perform power flow calculation for the vectors. Before implementing Newton-Raphson to solve the power flow equations for the current time interval, the parameters of PQ and PV buses are needed to be updated according to the power demand and power outputs of the hydro and thermal units at current time interval. Reckon the unknown parameters of the slack bus, PV buses and PQ buses after the power flow calculation. Implement lead operation if the power generation of the slack bus goes beyond the power limits. Step 6: Count the penalty of the vectors including the errors of the power flow equations, the violations of power generation outputs, the errors of voltage amplitude of buses caused by exceeding the boundary condition and the errors of reservoir volumes at any time intervals caused by exceeding the boundary condition. Step 7: Calculate the fitness of each vector according to the rule 4. The one with the best fitness value is chosen as the best solution at the current iteration. Step 8: Determine whether the number of iteration satisfies with the GSF, which means the gather and scatter frequency. If the result is true, then go to Step 9, otherwise, go to Step10. Step 9: Every running process of CPU sends its information including decision vectors, scaling factor and crossover factor to the root process. The root process gathers information from all processes. Then the root process implements the aggregative DE operation, i.e., the mutation operation to create variant vectors and the crossover operation to produce the trial vectors, according to the gathered decision vectors. After that, the root process divides the population into several small populations again and scatters the trial vectors to every running process. Step 10: Every running process performs the mutation operation and the crossover operation independently to produce the trial vectors. Step 11: Repeat Step 4 to Step 7 and perform the selection operation after calculating the fitness values of the new generated vectors. Step 12: Update the scaling factor and crossover factor for every running process according to the self-adaptive selection mechanism mentioned in the section V. Step 13: Perform the migration operation if the population diversity fails to satisfy its tolerated value. Perform Step 4 to Step 7 and select the new vectors and their fitness values as the evolved population for next iteration. Step 14: Update the number of iteration as K  K  1 , and update the penalty factor as (25). If the termination criterion is achieved, go to Step 15. Otherwise, go to Step 8.

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VII. NUMERICAL RESULTS

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In this section, several test examples are used to verify the feasibility and effectiveness of the proposed parallel DE algorithm for solving the STHS problem. A classical test hydrothermal system [16] which contains four hydro units and an equivalent thermal unit as well as its extension [52] which considers more thermal units are adopted to evaluate the performance of the proposed algorithm with comparisons to other algorithms such as GA [16] Micro GA [61], MDE [50], SPPSO [34], MCDE [49], MAPSO [73] and TLBO [39]. This classical hydrothermal test system and its extension are denoted as Test System I and Test System II in the remaining work. After that, other two test hydrothermal systems considering IEEE 9-bus and IEEE 39-bus as their transmission networks are also selected to demonstrate the feasibility and effectiveness of the parallel DE algorithm. While in the remaining of this section, these two test systems are denoted as Test System III and Test System IV, respectively. A. Test System I, II The classical test system [16] which contains four multi-chain hydro units and an equivalent thermal unit is usually

Step 15: Gather vectors X * and their fitness values from all processes. The process which gets the best fitness is recorded, and the corresponding vector X * is regarded as best solution for the STHS problem.

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applied to evaluate the performance of an algorithm. Here this system is also adopted to verify the effectiveness of the proposed parallel algorithm with comparisons to other algorithms. The proposed parallel DE algorithm is implemented and performed on a PC (Intel core i3-3200, 8G-RAM, 3.3GHz). The population size of each process, number of total processes and maximum iterations have been selected as 10, 4, and 2000 for this system, respectively. The other parameters which are set through trial and errors are shown in Table 1. The parameter limits of the considered multi-chain hydro stations are shown in Table 2 and other data of this system come from [16]. The best optimal cost obtained in 10 independent runs is $917276.5 and the other statistical performances as well as their comparisons with other algorithms in literature are shown in Table 3. In order to compare the detailed performances of these algorithms for the STHS problem, GA and microGA algorithms are reimplemented in the same environments of the proposed parallel DE algorithm. Their statistic results (denoted as RE-GA and RE-microGA, respectively) in 10 independent runs are also attached at the end of this table. While the optimal hydro discharges as well as the corresponding hydro power generations and the optimal thermal generations are shown in Table 4. The optimal convergences of the proposed parallel DE and the re-implemented GA and microGA for this test system are shown in Fig. 6. It is seen from Table 3 that the best, worst and average cost obtained by the proposed parallel DE algorithm in 10 independent runs outperform the corresponding values obtained by other algorithms considered. It is also found from the table that the parallel DE algorithm consumes the least CPU time which benefits from the small population and parallel process of the proposed algorithm. By the way, the optimal cost obtained by other algorithms as well as the CPU time consumed is only listed in this table according to the original references [16, 39, 50, 73-75]. The average CPU time of the all three algorithms in 10 runs can be obtained in the algorithm and are also listed in this table. The feasibility of the obtained solution can be verified using the optimal hydro discharges and thermal generations given in Table 4. It is also found from the table that the obtained decision vectors satisfy all constraints. From Fig. 6, it can be clearly seen that the parallel DE converges to the better results in shorter generations compared with other two algorithms. In order to show the superiority of the parallel DE algorithms, the box plots of different methods in 30 independent runs with 2000 iterations and 20000 iterations are shown in Fig.7 and Fig. 8, respectively. The corresponding non-parametric tests of the proposed parallel DE (PDE) with other methods are reported in Table 5. It is shown from Fig. 7 and Fig. 8 that the PDE obtains the best results among the considered algorithms with 2000 iterations and that the PDE nearly obtains the same statistic results compared with microGA with 20000 iterations. The non-parametric tests also show the same implications. An extension of the Test system I denoted as Test system II is considered to evaluate the performance of the parallel DE algorithm for larger hydrothermal power systems. The extended system consists of the last multi-chain cascaded hydro units and thermal equivalent units. The detailed data of the system come from [52]. The GA and microGA are also implemented on this test system. The obtained results by the three algorithms each with 2000 iterations in 10 independent runs are shown in Table 6. To show the effects of the generation number, the statistic results of the three algorithms with 20000 iterations are also attached at the end of the table. It is quite sure from this table that the parallel DE algorithm obtains better performance not only on the total fuel cost but also the consumed CPU time in the considered algorithms. In addition, the obtained results also show that the larger iterations play well on the GA and the microGA but not on the parallel DE. It also means that the parallel DE algorithm in 2000 iterations obtains nearly the same statistic results with the algorithm in 20000 iterations. It should be noted that the proposed parallel DE algorithm needs multi-core processor and more RAM to perform all operations simultaneously. However, the prices of the CPU and the RAM are becoming cheaper and cheaper with the development of the hardware manufacturing technologies. Hence, it can get a conclusion that the proposed parallel DE algorithm shows quite competitive performances on both the efficiency and convergence compared with other algorithms in literature and we have enough reasons to recommend this parallel algorithm to solve the STHS problem considering power flow constraints. B. Test System III This test system considers a multi-chain cascade of two reservoir hydro plants and an equivalent thermal plant [76]. The entire scheduling period is 1 day and divided into 24 intervals. What a more important is this test system considers the transmission network employing IEEE 9-bus grid. The detail system data and topology information for this test system come from [72]. The parameters employing in the parallel DE algorithm are given in Table 7. In order to compare the statistical performance of the proposed algorithm in detail, a serial DE algorithm is also implemented for this test system. Both of the algorithms are performed independently for 20 times, and Table 8 gives the obtained results by the parallel and serial DE algorithm for this test system. It is found from this table that the obtained best, worst and average cost by parallel DE algorithm are $18188.81, $18471.19 and $181320.50, while the corresponding cost values obtained by serial DE algorithm are $18295.53, $18519.45 and $18405.07 respectively. It is also noted that not only the statistical performance of the fuel cost obtained by parallel DE is better than that by serial DE, but also the average CPU time consumed. This also shows the effect of the parallel algorithm. The optimal hourly discharge and hydrothermal generation obtained by the proposed parallel DE algorithm are shown in Table 9. Table 10 shows the optimal voltages of generator buses, i.e., bus 1, 2 and 3 corresponding to the best fuel cost $18188.81, while Fig. 9 depicts the convergence curve of the fuel cost for parallel DE algorithm and serial DE algorithm. It is clearly seen from the figure that both the parallel DE algorithm and the serial DE algorithm converge at about 250-300 iterations. But the parallel converges to the smaller cost than the serial DE and this comparison shows the parallel DE algorithm is better in convergence of the solution than the serial DE algorithm. Within these decision variables from Table 9 and Table 10, the feasibility of the obtained solution will be verified. After verifying, the obtained solution is found to be feasible for this test system. 10 / 27

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As described above, all the parameters for this parallel DE algorithm are preset through trials and errors. So what solution will get if one of the parameters adapted changes? Aiming at this, several important parameters are disturbed independently around their original values. Here the population size of every running process (PS), the number of parallel processes (PN), the gather and scatter operation frequency (GSF) are selected as the disturbed parameters. It is noted that the rest parameters are not changed in these runs. The best, average and worst values and the average CPU time consumed in 20 independent runs for each disturbed parameter have been shown in Table 11, Table 12 and Table 13, respectively. It is seen from table 11 that the proposed parallel DE algorithm yields better result on the average and worst values except for the best value when the PS is increasing. In fact, the employed parameters are not the best ones. They are only set through several trial and errors. Moreover, as seen in Table 11, the results obtained by the proposed parallel DE algorithm with 6 as the PS are better than the results obtained by the serial DE algorithm with 60 as its population size. It proves that the proposed parallel DE algorithm works efficiently while searching for optimal results with a small population size, and the efficiency of the proposed DE algorithm behaves better for the higher population diversity level when the population size of each process increases. It is no doubt that the CPU time consumed will increase with increasing population size. The PN and GSF are two important parameters of the proposed parallel DE algorithm. It is shown from Table 12 that more process number turns out to be more efficient to search the optimum values. A larger PN means that the population size of the gather and scatter operation will be larger while the population size of each running process stays the same. As a consequence, the aggregate operation would share more information, and the population diversity of each process is also increased. However, it would take more CPU time to send information among processes. Furthermore, table 13 compares the results obtained by parallel DE with GSF equals to 10, 20 and 30, respectively. The efficiency turns out to be improved when GSF is small. A larger GSF means the reductive number of aggregative DE operation among the processes and decreases the population diversity of each running process, while it could accelerate the solution speed for saving the time to send the information among processes. It is also seen from Table 13 that the GSF has very low sensitivity to CPU consumed. C. Test System IV This test system contains four cascade hydro units of last hydro subsystem and six equivalent thermal units. Moreover IEEE 39-bus grid shown in Fig. 10 is employed as the transmission net work of the system. The entire scheduling period is 1 day and divided into 24 time intervals. The parameters adopted in parallel DE algorithm for this system have been given in Table 14. Table 15 gives the reference load demand of 19 load buses of this system, and Table 16 gives load curve factors along 24 hours horizon. The cascaded hydraulic network, hydro power generation coefficients and hourly inflow involved in this system come from [16].The fuel cost coefficients of considered thermal units are shown in Table 17. The connected buses of the considered hydro and thermal units are given in Table 18. The PV buses and slack bus are labeled in Fig. 10, and the rest buses are select as PQ buses. The voltage base value is 345KV, and the power base value is 100MVA. The voltages of the buses are limited in the range of [0.94, 1.06] and other system data can be found in the software of MATPOWE [77]. The best, worst and median values of fuel cost obtained by the proposed parallel DE algorithm in 10 independent runs are $365193.438, $376510.60 and $369931.90, respectively. The obtained optimal hourly water discharges and power generation of hydro and thermal units are shown in Table 19. The obtained optimal voltages of the PV buses and slack bus are listed in Table 20. The optimal ratios of transformers are displayed in Table 21. It is clearly seen from these tables that the obtained optimal result satisfies with all constraints considered. In order to test the effects of lead operation, the parallel algorithm without lead operation is also performed independently for this test system 10 times. The best, worst and median values of fuel cost obtained by the algorithm without lead operation are $372315.094, $388102.188 and $383053.25, respectively. The optimal convergence rate comparisons for this system are shown in Fig. 11. It is found from the comparisons that lead operation plays one of the most roles in the parallel DE algorithm. VIII.

CONCLUSION

The optimal short-term hydrothermal scheduling plays one of the most important roles in the modern power systems plan and operation. In recent years more and more researches paid attention to the solution of this optimization problem, but most of them neglected the power flow constraints. This neglect will make the obtained schedule far away from the real operation point. This work focuses on making the optimal problem of short-term hydrothermal scheduling more in line with the actual situation. The power flow constraints are introduced into the mathematical model of the STHS problem in this paper. In addition, a small-population based parallel DE algorithm which employs self-adaptive mechanism, migration operation, gather and scatter operation, aggregative DE operation and lead operation is proposed to solve the power flow constrained short-term hydrothermal scheduling problem. Four constraint handling rules are also adopted to deal with the complex constraints of the STHS problem. The optimal results obtained by the proposed parallel DE algorithm on the famous hydrothermal test systems have been compared with those obtained by other algorithms. The comparisons have proved that the proposed parallel DE algorithm performs better than other algorithms in terms of both optimal value and average CPU time. After verifications on the traditional STHS problems, the parallel DE algorithm is employed to solve two test systems considering transmission networks. The results also show the competiveness of the proposed parallel DE algorithm. In order to show the effects of different parameters on the final solutions, the disturbed parameters are also employed for the solution of test system III. The numerical results obtained from these test systems show that the population size (PS), the process number (PN) and the gather and scatter operation frequency (GSF) play important roles in the proposed parallel DE algorithm. The numerical results also show that the lead operation would greatly increase the feasible of solutions and 11 / 27

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improve the convergence of the proposed parallel DE algorithm.

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ACKNOWLEDGMENT The authors gratefully acknowledge the financial support by National Natural Science Foundation of China (No.61403321 and 51507193), Natural Science Foundation of Guangdong Province, China (No. 2014A030310003 and 2014A030310318) and Knowledge Innovation Program of Shenzhen City (No. JCYJ20130327150859765).

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[57] S. Sivasubramani and K. Shanti Swarup, "Hybrid DE–SQP algorithm for non-convex short term hydrothermal scheduling problem," Energy Conversion and Management, vol. 52, pp. 757-761, 2011. [58] H. Zhang, J. Zhou, Y. Zhang, N. Fang, and R. Zhang, "Short term hydrothermal scheduling using multi-objective differential evolution with three chaotic sequences," International Journal of Electrical Power and Energy Systems, vol. 47, pp. 85-99, 2013-01-01 2013. [59] H. Zhang, J. Zhou, Y. Zhang, Y. Lu, and Y. Wang, "Culture belief based multi-objective hybrid differential evolutionary algorithm in short term hydrothermal scheduling," Energy Conversion and Management, vol. 65, pp. 173-184, 2013-01-01 2013. [60] C. A. Coello Coello and G. T. Pulido, "A micro-genetic algorithm for multiobjective optimization,", Zurich, Switzerland, 2001, pp. 126-140. [61] M. Burhan, K. J. E. Chua and K. C. Ng, "Sunlight to hydrogen conversion: Design optimization and energy management of concentrated photovoltaic (CPV-Hydrogen) system using micro genetic algorithm," Energy, vol. 99, pp. 115-128, 2016-0101 2016. [62] J. C. F. Cabrera and C. A. C. Coello, "Micro-MOPSO: A multi-objective particle swarm optimizer that uses a very small population size,": Springer, 2010, pp. 83-104. [63] C. Brown, Y. Jin, M. Leach, and M. Hodgson, "μJADE: adaptive differential evolution with a small population," Soft Computing, pp. 1-10, 2015-01-01 2015. [64] Y. Li, Z. Zhan, Y. Gong, J. Zhang, Y. Li, and Q. Li, "Fast micro-differential evolution for topological active net optimization," IEEE transactions on cybernetics, vol. 46, pp. 1411-1423, 2016-01-01 2016. [65] K. Rajesh, A. Bhuvanesh, S. Kannan, and C. Thangaraj, "Least cost generation expansion planning with solar power plant using Differential Evolution algorithm," Renewable Energy, vol. 85, pp. 677-686, 2016-01-01 2016. [66] T. W. Lau, C. Y. Chung, K. P. Wong, T. S. Chung, and S. L. Ho, "Quantum-inspired evolutionary algorithm approach for unit commitment," IEEE Transactions on Power Systems, vol. 24, pp. 1503-1512, 2009-01-01 2009. [67] J. Zhang, Q. Tang, Y. Chen, and S. Lin, "A hybrid particle swarm optimization with small population size to solve the optimal short-term hydro-thermal unit commitment problem," Energy, vol. 109, pp. 765-780, 2016-01-01 2016. [68] F. Y. K. Takigawa, E. L. Da Silva, E. C. Finardi, and R. N. Rodrigues, "Solving the hydrothermal scheduling problem considering network constraints," Electric Power Systems Research, vol. 88, pp. 89-97, 2012-01-01 2012. [69] W. Gropp, E. Lusk, N. Doss, and A. Skjellum, "A high-performance, portable implementation of the MPI message passing interface standard," Parallel computing, vol. 22, pp. 789-828, 1996-01-01 1996. [70] W. Gropp, E. Lusk and R. Thakur, Using MPI-2: Advanced features of the message-passing interface: MIT press, 1999. [71] E. Gabriel, G. E. Fagg, G. Bosilca, T. Angskun, J. J. Dongarra, J. M. Squyres, V. Sahay, P. Kambadur, B. Barrett, and A. Lumsdaine, "Open MPI: Goals, concept, and design of a next generation MPI implementation,", 2004, pp. 97-104. [72] J. Zhang, S. Lin, X. Zeng, and Q. Tang, "Short-term optimal hydrothermal scheduling problem considering power flow constraint," in 2015 IEEE Congress on Evolutionary Computation (CEC), 2015, pp. 3235-3242. [73] N. Amjady and H. R. Soleymanpour, "Daily Hydrothermal Generation Scheduling by a new Modified Adaptive Particle Swarm Optimization technique," Electric Power Systems Research, vol. 80, pp. 723-732, 2010. [74] P. K. Hota, A. K. Barisal and R. Chakrabarti, "An improved PSO technique for short-term optimal hydrothermal scheduling,". vol. 79, 2009, pp. 1047-1053. [75] N. Sinha, R. Chakrabarti and P. K. Chattopadhyay, "Fast evolutionary programming techniques for short-term hydrothermal scheduling," IEEE Transactions on Power Systems, vol. 18, pp. 214-220, 2003. [76] M. Basu, "Quasi-oppositional group search optimization for hydrothermal power system," International Journal of Electrical Power & Energy Systems, vol. 81, pp. 324-335, 2016. [77] R. D. Zimmerman, C. E. Murillo-Sánchez and R. J. Thomas, "MATPOWER: Steady-state operations, planning, and analysis tools for power systems research and education," IEEE Transactions on power systems, vol. 26, pp. 12-19, 2011-0101 2011. 15 / 27

ACCEPTED MANUSCRIPT 1

2 3 4

S1

S2

S3

S4

S5

S6

I

F

M

C

F

S

S7

Fig.1 Procedure of serial DE algorithm Rank index S1 Rank=0

I

S2

S 1

N

F

S3

S4

S5

S6

M

C

F

S

S 2

Y S1 Rank=1

I

S2

S 1

N

F

S3

S 4

M

C

Rank=2

5 6 7

I

S2

S5

S6

M

C

F

S

S5

S3

S4

S5

S6

M

C

F

S

N

.. .. ..

S 2

S5

Y

S6

Fig.2 Procedure of parallel DE algorithm

SP0*D

Rank 0 Gather

SP0

Root

8 9

SP2

.. .. ..

SP1

SP2

.. .. ..

Rank 0 SP0

LP*D

SP1*D SP2*D

Rank2

Scatter

SP0

Rank1 SP1

S7

S6

S 1

F

S6

S4

Y S1

S5

S3

S 2

S7

Rank1 SP1 Rank2 SP2

Fig.3 Gather and scatter of decision vectors

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S7

ACCEPTED MANUSCRIPT Start Calculate errorj Y N

errorj≤ae N be
avg_error=errorj/T Qhj,t=Qhj,t+avg_error Qhjmin
1 2

End

Fig.4 Disposal strategy of the initial and final reservoir volumes

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ACCEPTED MANUSCRIPT Start Perform step1 Perform step2 Preform step3, here the iteration number is set as K=1 Preform step4 to step7 K%GSF=0?

Y

N Preform step10

Preform step9

Preform step11 Satisfy the migration condition? Y Preform migration operation and repeat step4 to step7 N

Y

The new vectors and its fitness after the migration operation are selected for next iteration. K=K+1 Update the penalty factor K
1 2

End

Fig.5 Simplified flowchart of the parallel DE algorithm for STHS problem considering power flow constraints 944

RE-GA Parallel DE RE-microGA

942 940

3

Thermal Cost (10 $)

938 936 934 932 930 928 926 924 922 920 918 916 0

500

1000

1500

2000

Generation

3 4 5

Fig. 6 Optimal convergence of fuel cost for test system I

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ACCEPTED MANUSCRIPT RE-GA RE-microGA Parallel DE

Thermal Cost (103$)

926

924

922

920

918

916

1 2 3

Fig. 7 Box plot of different methods for test system I with 2000 iterations

Thermal Cost (103$)

922

RE-GA RE-microGA Parallel DE

921

920

919

918

917

4 5

Fig. 8 Box plot of different methods for test system I with 20000 maximum iterations

6 7

Fig. 9 Optimal convergence of fuel cost for proposed parallel DE and serial DE

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ACCEPTED MANUSCRIPT G PV 30

G PV 37 25

26

28

2

29

27 38

1 G

3

18

PV G

17

PV 39

21

16 15

PV G 14

4

36

13

5

9

24

23

6

12

19

11

7

20

22

10 8

1 2 3

4 5 6 7

Slack G

31 G

32 PV

PV

34 G

33

35

G PV

G PV

Fig. 10 IEEE 39-bus electric network structure

Fig. 11 Comparisons of optimal convergence curve of fuel cost for test system IV Table 1 Parameters of the parallel DE algorithm for test system I Parameter

Value

min

0.1

F max

0.9

CR min

0

CR max

0.9

f mig

0.15

tmig

0.85

F

8 9 10 20 / 27

ACCEPTED MANUSCRIPT 1

Table 2 Parameters limits of the considered multi-chain hydro stations

2 3

4 5

Plant

Qhmin

Qhmax

Vhmin

Vhmax

Vhbegin

Vhend

Phmin

Phmax

Ru



1

5

15

80

150

100

120

0

500

0

2h

2

6

15

60

120

80

70

0

500

0

3h

3

10

30

100

240

170

170

0

500

2

4h

4

6

20

70

160

120

140

0

500

1

1h

Table 3 Comparisons of statistical performances among different algorithms for test system I Techniques

Best cost ($)

Worst cost ($)

Average cost ($)

Average CPU time (s)

Parallel DE

917276.5

917622.3

917446.3

2

GA [16]

926707.00

-

-

-

TLBO [39]

922373.49

922462.24

922873.81

-

MDE [50]

922555.44

-

-

45

IPSO [74]

922553.49

-

-

38.46

IFEP [75]

930129.82

930881.92

930290.13

1033.20

MAPSO [73]

922421.00

923508.00

922544.00

-

RE-GA

924159.6

925613.6

924880.4

60.40

RE-microGA

918806.1

920010.8

919522.5

6.20

Table 4 Optimal hourly hydro discharges and power generation of the hydro and thermal units for test system I Hour

Hydro discharge (104m3)

Hydro power output (MW)

Thermal power output (MW)

Hydro1

Hydro2

Hydro3

Hydro4

Hydro1

Hydro2

Hydro3

Hydro4

1

8.993

6.004

29.988

6.077

80.944

49.029

0.000

132.925

1107.102

2

8.782

6.056

29.988

6.008

80.096

50.557

0.000

129.068

1130.279

3

8.480

6.014

17.395

6.025

78.416

51.363

39.606

125.998

1064.617

4

8.294

6.014

17.091

6.036

77.159

52.993

38.815

122.001

999.031

5

8.304

6.100

16.401

6.009

76.819

55.177

40.656

115.794

1001.554

6

8.280

6.118

16.342

7.808

75.931

56.263

41.252

161.353

1075.200

7

8.759

6.368

16.328

10.763

78.236

58.486

42.040

215.867

1255.372

8

8.557

6.515

17.782

13.715

76.838

59.340

38.090

252.607

1573.125

9

8.403

6.344

16.582

15.249

76.110

58.378

41.214

269.823

1794.474

10

8.460

7.760

16.840

16.395

76.989

68.646

40.220

280.607

1853.537

11

8.616

7.296

17.248

16.470

78.732

66.286

38.635

281.163

1765.184

12

8.231

8.433

16.847

17.634

77.448

74.194

39.317

290.042

1828.998

13

8.809

8.432

17.313

16.479

81.395

73.979

37.878

281.242

1755.506

14

7.922

8.327

17.295

16.883

76.516

73.142

39.159

284.559

1726.624

15

8.073

8.774

17.872

17.322

78.415

76.075

37.742

287.911

1649.857

16

7.955

9.531

17.867

16.753

78.202

80.330

38.694

283.429

1589.344

17

8.137

8.774

17.412

17.331

79.778

75.427

40.379

287.999

1646.417

18

7.758

9.380

15.970

17.324

77.332

77.833

44.652

287.928

1652.255

19

7.904

9.821

14.630

18.092

78.380

78.252

48.082

293.532

1741.754

20

7.376

11.296

13.900

19.589

74.504

83.250

50.236

303.249

1768.762

21

7.946

11.358

10.890

19.977

78.295

81.455

51.272

303.833

1725.145

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ACCEPTED MANUSCRIPT

1 2

22

7.122

11.759

11.624

19.999

72.285

81.462

53.901

301.294

1611.058

23

6.893

12.459

12.929

19.988

70.688

82.017

56.026

296.964

1344.304

24

6.945

13.066

11.642

19.965

71.368

80.659

57.108

291.008

1089.857

Table 5 Non-parametric test of the proposed parallel DE with other methods for test system I in 30 independent runs Methods

Compared Methods

PDE

Decisions of the considered nonparametric tests Mann-Whitney U

Wald-Wolfowitz

Kolmogorov-Simirnov

Jonckheere-Terpstra

GA

Reject

Reject

Reject

Reject

PDE

MicroGA

Reject

Reject

Reject

Reject

PDE

PDE(20000)

Accept

Accept

Accept

Accept

Zero hypotheses H 0 : Parallel DE obtains the same distributional solutions with other methods. Significance level   0.05

3 4

Table 6 Comparisons of statistical performances among different algorithms for larger hydrothermal test system Techniques

Iterations

Best cost ($)

Worst cost ($)

Average cost ($)

Average CPU time (s)

Parallel DE

2000

168769.2

177777.7

173034.6

5

RE-microGA

2000

188834.6

190598.7

189608.4

9.08

RE-GA

2000

201644.17

206310.94

203225.02

90.03

MDE [52]

-

170964.15

-

-

96.40

Parallel DE

20000

168750.3

176453.1

173635.2

19.73

RE-microGA

20000

174864.2

176841.4

176095.9

99.98

RE-GA

20000

190585.3

192715.3

191491.7

923.77

5 6

Table 7 Parameters of the parallel DE algorithm Parameters

7 8

9 10

value

F

min

0.1

F

max

0.9 0

CR min CR max

0.9

f mig

0.15

tmig

0.85

Number of running process

10

Population size for each process

10

GSF

10

Iteration number

600

Table 8 Results of serial and parallel DE for test system III Method

Best ($)

Worst ($)

Average ($)

Average CPU Time (s)

Serial DE Parallel DE

18295.53 18188.81

18519.45 18471.19

18405.07 18320.50

80 34

Table 9 Optimal hydro discharge and power generation of hydro and thermal units for test system III Hour 1

Hydro Discharge (104m3) Qh1 Qh 2 8.694

8.535

Hydro Generation (p.u.)

Thermal Generation (p.u.)

Ph1

Ph 2

Ps

0.793

0.651

1.753

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1 2

2

8.444

11.097

0.782

0.770

1.753

3

5.471

8.657

0.570

0.634

1.753

4

11.305

13.608

0.930

0.893

0.926

5

12.383

14.920

0.946

0.948

0.943

6

8.269

12.925

0.747

0.889

1.753

7

5.015

14.128

0.508

0.956

2.578

8

7.946

14.088

0.733

0.980

2.580

9

5.009

8.422

0.518

0.747

3.400

10

12.961

14.993

0.966

1.033

2.593

11

11.096

14.997

0.903

1.043

2.734

12

5.001

14.510

0.526

1.027

3.400

13

5.023

14.866

0.537

1.068

3.145

14

7.243

14.638

0.727

1.082

2.580

15

5.903

14.997

0.628

1.087

2.580

16

8.800

14.954

0.849

1.082

2.580

17

8.029

14.989

0.799

1.084

2.580

18

5.004

14.989

0.552

1.074

3.153

19

9.642

14.818

0.906

1.069

2.577

20

8.152

14.979

0.809

1.074

2.580

21

5.007

14.562

0.551

1.055

2.261

22

8.156

14.946

0.809

1.083

1.753

23

7.445

14.977

0.758

1.094

1.753

24

5.002

14.956

0.551

1.084

1.753

Table 10 Optimal voltages (p.u.) of generator buses obtained by the proposed parallel DE algorithm for test system III Bus index

Hour

3 4

Bus index

Hour

1

1 0.965

2 0.900

3 1.098

13

1 1.038

2 1.055

3 1.100

2

1.100

1.006

1.024

14

1.100

0.946

1.100

3

1.069

1.094

1.093

15

1.098

1.035

1.099

4

1.096

1.090

1.053

16

1.070

1.100

1.100

5

1.081

1.074

1.081

17

1.100

1.100

1.100

6

1.100

1.027

1.100

18

1.100

1.099

1.096

7

1.084

1.027

1.099

19

1.074

1.067

1.100

8

1.100

1.076

1.100

20

1.100

1.100

1.100

9

1.099

1.022

1.094

21

1.052

1.100

1.095

10

1.100

1.100

1.099

22

1.100

1.100

1.043

11

1.099

1.087

1.100

23

1.027

1.092

1.089

12

0.982

1.100

1.070

24

1.100

1.100

1.081

Table 11 Best, worst and average fuel cost and average CPU time with disturbed PS PS 6 10 15 Serial DE

Best ($) 18182.48 18188.81 18149.72 18295.53

Worst ($) 18652.23 18471.19 18372.07 18519.45

Average ($) 18404.47 18320.50 18268.42 18405.07 23 / 27

Average CPU time (s) 18 34 44 80

ACCEPTED MANUSCRIPT 1 2

Table 12 Best, worst and average fuel cost and average CPU time with disturbed PN PN 5 10 20 Serial DE

3 4

Best ($) 18348.22 18188.81 18135.22 18295.53

Worst ($) 18872.19 18471.19 18342.51 18519.45

Average ($) 18583.227 18320.50 18194.14 18405.07

Table 13 Best, worst and average fuel cost and average CPU time with disturbed GSF GSF 10 20 30 Serial DE

5 6

Best ($) 18188.81 18208.36 18278.32 18295.53

Worst ($) 18461.19 18463.91 18615.51 18519.45

Average ($) 18320.50 18335.66 18382.32 18405.07

Average CPU time(s) 34 30 29 80

Table 14 Parameters of the parallel DE for test system IV Parameters

Value

min

0.1

F max

0.9

CR min

0

F

CR

7 8

Average CPU time (s) 25 34 43 80

0.9

max

f mig

0.15

tmig

0.85

PN

10

PS

10

GSF

15

Iteration number

2500

Table 15 Reference power demand value (p.u.) of different load buses

9 10

Bus index

PD

QD

Bus index

PD

QD

3

3.220

0.024

21

2.740

1.150

4

5.000

1.840

23

2.475

0.846

7

2.338

0.840

24

3.086

-0.922

8

5.220

1.766

25

2.240

0.472

9

0.065

-0.666

26

1.390

0.170

12

0.085

0.880

27

2.810

0.755

15

3.200

1.530

28

2.060

0.276

16

3.290

0.323

29

2.835

0.269

18

1.580

0.300

31

0.092

0.046

20

6.800

1.030

39

11.040

2.500

Table 16 Power demand factors of load buses in 24 hours Hours

factor

Hours

factor

Hours

Factor

1

0.438

9

0.5

17

0.505

2

0.422

10

0.494

18

0.622

3

0.394

11

0.484

19

0.688

24 / 27

ACCEPTED MANUSCRIPT 4

0.315

12

0.524

20

0.635

5

0.434

13

0.564

21

0.599

6

0.413

14

0.518

22

0.489

7

0.457

15

0.542

23

0.433

8

0.51

16

0.479

24

0.444

1 2

Table 17 Fuel cost coefficient of thermal units Thermal index

1

2

3

4

5

6

a

150

115

40

122

125

120

b

1.89

2

3.5

3.15

3.05

2.75

c

0.005

0.0055

0.006

0.005

0.005

0.007

d

300

200

200

150

150

150

e

0.035

0.042

0.042

0.063

0.063

0.063

P

min

0.5

0.5

0.5

0.5

0.5

0.5

P

max

10.4

6.48

7.25

6.52

5.08

6.87

3 4

Table 18 Location of the thermal and hydro units

5 6

Generation unit

Bus index

Generation unit

Bus index

Hydro1 Hydro2 Hydro3 Hydro4 Thermal1

36 37 38 39 30

Thermal2 Thermal3 Thermal4 Thermal5 Thermal6

31 32 33 34 35

Table 19 Optimal discharges (104m3) and power generation (p.u.) obtained by parallel DE algorithm Hour

Qh1

Qh 2

Qh 3

Qh 4

Ph1

Ph 2

Ph 3

Ph4

Ps1

Ps 2

Ps 3

Ps 4

Ps 5

Ps 6

1

5.618

6.013

20.551

13.001

0.578

0.491

0.425

2.113

5.886

4.988

3.859

4.493

1.456

2.887

2

7.032

8.735

20.959

13.462

0.698

0.674

0.367

2.041

4.900

4.988

2.761

2.377

4.793

2.529

3

10.563

6.347

30.000

13.010

0.907

0.522

0.000

1.873

2.296

2.744

5.666

2.994

3.473

3.973

4

11.344

8.328

18.062

13.062

0.929

0.663

0.366

1.736

3.182

2.744

2.413

2.490

2.492

2.494

5

7.279

10.325

22.783

13.005

0.702

0.770

0.115

1.561

3.213

3.491

2.743

6.352

3.495

4.538

6

7.874

6.434

17.989

13.079

0.738

0.534

0.354

1.667

4.946

4.988

2.645

3.042

3.813

2.852

7

8.784

6.837

13.136

13.003

0.789

0.564

0.471

1.761

4.967

3.492

2.196

3.607

5.021

5.500

8

9.414

6.498

17.260

15.156

0.819

0.536

0.415

2.141

5.868

5.736

2.743

4.438

4.510

4.407

9

5.373

9.118

20.582

13.339

0.548

0.698

0.293

2.028

6.773

4.238

4.140

3.870

4.996

3.502

10

14.074

6.227

16.077

14.093

0.981

0.514

0.441

2.199

6.738

4.239

3.491

3.504

4.022

4.512

11

8.984

9.178

14.400

13.611

0.801

0.711

0.476

2.200

5.983

3.527

4.241

3.041

3.517

5.513

12

5.888

10.779

11.393

13.004

0.601

0.786

0.484

2.140

6.734

5.736

4.327

4.724

3.493

3.518

13

8.369

6.158

20.158

13.446

0.788

0.508

0.349

2.225

6.782

5.736

3.492

4.897

4.700

5.539

14

5.579

9.227

14.507

13.432

0.589

0.707

0.511

2.297

6.423

5.000

2.090

5.575

4.995

3.993

15

5.360

7.406

17.145

17.839

0.579

0.601

0.469

2.675

5.873

4.236

4.240

4.490

4.466

6.040

16

14.025

6.266

14.253

13.814

1.055

0.534

0.538

2.324

7.683

4.996

3.499

2.466

2.212

4.377

17

6.692

8.384

13.814

13.052

0.692

0.679

0.540

2.229

6.783

4.988

4.987

3.055

2.913

4.489

18

5.523

11.981

16.671

14.336

0.597

0.839

0.504

2.417

8.576

5.728

4.119

5.443

5.053

5.411

19

7.012

7.652

11.679

13.618

0.723

0.591

0.563

2.354

8.630

5.728

7.114

6.520

5.077

5.489

20

8.516

11.073

15.002

13.436

0.829

0.760

0.559

2.371

6.770

5.733

6.483

6.487

5.080

4.469

25 / 27

ACCEPTED MANUSCRIPT 21

6.250

11.404

15.989

13.795

0.659

0.752

0.543

2.413

6.744

4.983

7.169

5.964

3.413

4.588

22

7.294

7.770

17.408

14.482

0.742

0.559

0.525

2.475

4.094

4.412

5.315

5.488

4.620

2.114

23

6.539

9.416

10.500

14.862

0.684

0.657

0.564

2.530

4.987

4.235

2.730

2.994

3.470

3.991

24

11.615

10.445

23.200

16.283

0.992

0.694

0.301

2.614

4.988

2.744

4.210

5.486

1.497

4.034

1 2

Table 20 Optimal voltages (p.u.) of generator buses obtained by parallel DE algorithm Hour

Bus index 30

31

32

33

34

35

36

37

38

39

1

1.040

1.049

1.054

1.040

0.982

0.999

1.046

0.970

1.002

0.964

2

1.002

1.043

0.979

1.045

1.028

0.992

1.025

0.987

1.042

1.031

3

0.980

1.001

1.050

1.036

0.952

0.994

0.968

1.014

0.976

0.960

4

1.015

1.004

1.010

1.019

0.973

0.979

0.999

1.046

1.021

0.941

5

0.975

1.042

1.021

1.000

1.052

1.016

0.958

1.036

0.967

0.990

6

1.015

1.033

0.953

1.041

0.980

1.057

1.009

1.046

1.012

0.959

7

0.951

0.986

1.019

1.057

1.060

0.966

0.944

0.980

1.060

0.976

8

1.038

1.016

1.041

1.034

1.021

1.041

0.985

0.947

0.984

1.018

9

0.954

1.014

1.039

1.034

0.998

1.043

0.979

1.027

1.060

1.009

10

0.971

1.001

1.001

1.054

1.008

1.021

1.017

1.025

1.049

1.055

11

1.024

0.941

1.016

0.977

1.041

1.010

1.045

0.951

1.056

0.986

12

0.974

0.998

1.025

0.982

1.028

1.028

0.950

0.984

0.985

1.018

13

0.998

0.960

0.999

0.983

0.956

1.043

0.990

0.969

0.955

1.039

14

0.986

1.030

1.011

1.003

0.952

1.060

0.990

0.999

0.941

1.040

15

0.994

1.010

0.941

1.030

1.032

0.999

1.053

1.039

0.953

1.018

16

1.033

0.986

0.978

1.021

1.028

1.037

0.967

1.012

0.995

1.049

17

1.034

1.017

0.979

0.948

0.987

0.940

1.029

1.020

1.024

0.995

18

1.002

1.012

0.983

0.994

0.964

0.940

0.998

0.964

1.060

0.989

19

1.015

1.042

1.041

1.060

0.982

1.036

0.988

0.977

1.030

0.940

20

0.940

0.999

0.951

0.999

0.981

1.024

0.969

0.950

1.020

0.976

21

1.021

1.016

0.959

0.984

0.999

1.014

1.051

1.055

0.952

1.033

22

0.959

1.058

0.985

1.050

1.029

1.040

1.025

0.991

1.006

1.017

23

0.982

0.975

0.966

0.940

0.989

0.989

0.960

0.979

1.037

0.967

24

1.008

1.060

0.965

1.012

0.977

0.953

1.006

0.957

1.021

1.058

3 4

Table 21 Optimal ratios of transformer obtained by parallel DE algorithm Hour

Transformer between the head bus and tail bus 2, 30

6, 31

10, 32

12, 11

12, 13

19, 20

19, 33

20, 34

22, 35

23, 36

25, 37

29, 38

1

1.025

1.025

1.025

1.025

1.025

1.025

1.025

1.025

1.025

1.025

1.025

1.025

2

0.960

0.960

0.960

0.960

0.960

0.960

0.960

0.960

0.960

0.960

0.960

0.960

3

1.027

1.027

1.027

1.027

1.027

1.027

1.027

1.027

1.027

1.027

1.027

1.027

4

1.003

1.003

1.003

1.003

1.003

1.003

1.003

1.003

1.003

1.003

1.003

1.003

5

1.013

1.013

1.013

1.013

1.013

1.013

1.013

1.013

1.013

1.013

1.013

1.013

6

1.015

1.015

1.015

1.015

1.015

1.015

1.015

1.015

1.015

1.015

1.015

1.015

7

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

8

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

9

1.024

1.024

1.024

1.024

1.024

1.024

1.024

1.024

1.024

1.024

1.024

1.024

26 / 27

ACCEPTED MANUSCRIPT 10

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

11

0.990

0.990

0.990

0.990

0.990

0.990

0.990

0.990

0.990

0.990

0.990

0.990

12

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

0.950

13

1.039

1.039

1.039

1.039

1.039

1.039

1.039

1.039

1.039

1.039

1.039

1.039

14

1.009

1.009

1.009

1.009

1.009

1.009

1.009

1.009

1.009

1.009

1.009

1.009

15

0.998

0.998

0.998

0.998

0.998

0.998

0.998

0.998

0.998

0.998

0.998

0.998

16

1.035

1.035

1.035

1.035

1.035

1.035

1.035

1.035

1.035

1.035

1.035

1.035

17

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

0.975

18

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

19

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

0.984

20

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

1.002

21

1.008

1.008

1.008

1.008

1.008

1.008

1.008

1.008

1.008

1.008

1.008

1.008

22

1.019

1.019

1.019

1.019

1.019

1.019

1.019

1.019

1.019

1.019

1.019

1.019

23

1.016

1.016

1.016

1.016

1.016

1.016

1.016

1.016

1.016

1.016

1.016

1.016

24

1.041

1.041

1.041

1.041

1.041

1.041

1.041

1.041

1.041

1.041

1.041

1.041

1 2 3 4

27 / 27