- Email: [email protected]
Optimization of hydro energy storage plants by using differential evolution algorithm
Energy xxx (2014) 1e11
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Optimization of hydro energy storage plants by using differential evolution algorithm Arnel Gloti c a, *, Adnan Gloti c b, Peter Kitak a, Jo ze Pihler a, Igor Ti car a a b
Faculty of Electrical Engineering and Computer Science, University of Maribor, SI-2000 Maribor, Slovenia Holding Slovenske elektrarne Group, SI-1000 Ljubljana, Slovenia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 November 2013 Received in revised form 22 April 2014 Accepted 4 May 2014 Available online xxx
The optimal dispatching of cascade Hydro Power Plants is known as a complex optimization problem. In order to solve this problem the authors have applied an adapted differential evolution algorithm by using a fixed and dynamic population size. According to the dynamic population size, the proposed algorithm uses novel random and minimum to maximum sort strategy in order to create new populations with decreased or increased sizes. This implementation enables global search with fast convergence. It also uses a multi-core processor, where all the necessary optimization data are sent to the individual core of a central processing unit. The main aim of the optimization process is to satisfy 24 h demand by minimizing the water quantity used per electrical energy produced. This optimization process also satisfies the desired reservoir levels at the end of the day. The models used in this paper were the real parameters' models of eight cascade Hydro Power Plants located in Slovenia (Europe). Also the standard model from the literature is used in order to compare the performance of the adapted optimization algorithm. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Cascade hydro power plants Energy storage Hydro generation scheduling Optimization algorithms
1. Introduction Energy storage is a physical storage for energy [1] and the different types and technologies [2] are available nowadays. The pumped [3] and conventional [2] energy storages are used with HPP (hydro power plants) where the storage source is water, and the electricity produced is classified as renewable electricity [4]. Different research and developments related to sustainable and renewable electricity production according to different energy sources has been already made [5e8]. The authors' research focused focuses on cascade HPPs, where each individual HPP has its own reservoir and energy storage, respectively. Various combinations of reservoirs' charging and discharging produce different amounts of electricity. The optimization process for HPPs electricity production could be used in at least in two different ways; to determine the optimal production for HPPs in regard to a day-ahead electricity market [9] and forecasted demands [10] or to produce energy in accordance with the system demand and all requirements of HPPs combined
* Corresponding author. E-mail addresses: [email protected], [email protected] (A. Gloti c), adnan. [email protected] (A. Gloti c), [email protected] (P. Kitak), [email protected] (J. Pihler), [email protected] (I. Ti car).
with thermal power plants in order to decrease thermal costs [11e16]. The optimization process used in this paper is focused on a 24h interval with 1-h time steps. Therefore it refers to the term “short-term” hydro scheduling optimization [13,17e21] problem. In the past, a wide-range of optimization techniques [17] has been used to for successfully solving this complex non-linear constrained optimization problem. The complexities of the problem arise from a large number of co-dependent variables and various HPPs' constrains [22], such as reservoir levels, turbines' flows, denivelation speed, etc. Due to the generators' characteristics of HPPs, the outputs of which are generally non-linear functions of water discharge and net head, this optimization problem can also be marked as a non-linear problem [23]. The modern evolutionary algorithms [24] such as genetic algorithms [19], particle swarm optimization algorithm [20,21,25], evolutionary programming [26,27] and the differential evolution algorithm [13,28e33] are attracting more attention because of their efficiencies in solving hydro scheduling optimization problems. The differential evolution (DE) algorithm is known as a powerful evolutionary algorithm for global optimization, as first mentioned by Storn and Price [32]. Because of its simple implementation, efficiency, and robustness it has been selected as an appropriate optimization algorithm.
http://dx.doi.org/10.1016/j.energy.2014.05.004 0360-5442/© 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
2
Nomenclature
Parameters natural inflow to hydro plant i in hour k ai,k bi,k biological minimum flow of hydro plant i in hour k vi maximal value of the storage of reservoir i vi minimal value of the storage of reservoir i dvi allowed denivelation of reservoir i pi maximal output power of hydro plant i pi minimal output power of hydro plant i qi maximal water discharge of hydro plant i qi minimal water discharge of hydro plant i K number of hours of the scheduling period I total number of hydro power plants D number of parameters CR crossover control parameter
The authors in Ref. [28] used suitable penalty factors during the fitness evaluation of the DE algorithm using modified initialization and mutation steps which helped to satisfy the final reservoirs state. The classic DE algorithm was used in Ref. [33] where the parameter settings were found for each test system through trial and error, and consequently provide the optimal result. Properly controlling the parameter set of a DE algorithm is a challenging task, therefore in Ref. [34] the authors highlighted the importance of an algorithm's control parameter set at such as differential factor F, crossover constant CR, and population size NP. In Ref. [28] the chaos theory [29] is proposed to determine the adequate parameters' settings of the DE algorithm. Specifically, they determine the sets of F and CR and the applied chaos sequences which consequently have an impact on population diversity and premature convergence towards a locally optimal solution. In Ref. [30] the authors combined an adaptive dynamic control mechanism for the CR parameter with a chaotic local search operation for the DE algorithm, which avoids premature convergence. The self-adaptive F and CR parameters for each individual in the population are proposed in Ref. [35]. The changing of the initial values for both control parameters did not have any significant impact on the final result. When the classic DE algorithm with fixed control parameters falls into locally optimal solution, usually it is hard to escape from it. Therefore, an adaptive Cauchy mutation is proposed in Ref. [13] which renews the diversity of the population and avoids premature convergence. The important control parameter of the DE algorithm is also the population size NP, which is chosen by the user and fixed throughout the generations in classic DE. In order to provide higher population diversity at the beginning of the evolutionary process and decreased total number of function evaluation, a gradual population size reduction is proposed throughout the generations in Refs. [36], where the population size is reduced by half in each block of predefined generation numbers. In order to avoid the efforts needed to determine a proper set of control parameter, the proposed algorithm uses the self-adaptive control parameters F and CR. It also uses a novel method of varying the population size along the generation number. Population size is therefore referred to as dynamic population size. In this process the population can be decreased or even increased and this has a significant impact on the total number of function evaluations. When the algorithm increases the population size, new additional members are needed to fill up the new population size. In order to provide the additional members, the proposed algorithm uses a multi-core processor, where the population is
NP F w m dp
population size difference factor weight for objectives core number digit precision of float variable
Variables qi,k si,k vi,k pi,k G xi,G ui,G f dwk owi,k
discharge of hydro power plant i in hour k water spillage by reservoir i in hour k reservoir i storage in hour k power generated by hydro plant i in hour k denotes generation through algorithm steps value of i-th vector trial vector of i-th vector in generation G objective function demand energy in hour k sum of optimal energy production in hour k
separately and simultaneous processed by all cores. By the multiprocessing of the DE algorithm, the appropriate diversity of population is provided and the algorithm is more efficient in avoiding locally optimal solutions. According to the algorithm's increase population size, the missing members for the next generation are provided by the RSMM (random or the minimum to maximum sort strategy) described in Section 4.3. However, this paper presents a modified DE algorithm which was applied to hydro generation scheduling problem in order to satisfy the system demand and other objectives with fast convergence and decreased number of function evaluations, respectively. The hydro scheduling problem formulation is described in Section 2, the classic DE algorithm in Section 3, the modification of DE algorithm in Section 4, the experimental results in Section 5, and the conclusion in Section 6.
2. Hydro scheduling problem formulation In this paper, the hydro scheduling problem is formulated as a problem, where the main goal is to satisfy system demand by optimal scheduling across all HPPs. Cascade models of HPPs are used and therefore the production of downstream HPP is closely related with the upstream HPPs because of the upstream plant outflows' impact on downstream plant production. The system demand should be satisfied within the allowed deviation tolerance, usage of water quantity per electrical energy unit should be decreased and overflows should be also decreased and even eliminated, respectively. Since these goals should be achieved within acceptable CPU time, the hydro-scheduling problem can be solved by using the modified DE algorithm [32] with proposed dynamic population size and multi-processing. The final goal of the optimization problem is therefore conducted using three objectives, which are satisfied through single objective function by using the weighted sum method [37]. The optimal scheduling for connected HPPs can be obtained by searching the optimal schedule p, which is the function of flow through the turbine and the reservoir's volume,
p ¼ f ðv; qÞ;
(1)
where p is power, v volume of reservoir and q water discharge. The discharge q can be used as the optimization parameter, but within the allowed interval (2).
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
qi ; qi ¼ qi;k 2< ; qi qi;k qi ;
i ¼ 1; 2; 3; …; I;
(2)
k ¼ 1; 2; …; K: Changing the parameter q affects the variable v and consequently a different amount of electrical power (1) is obtained. The interval (2) consists of float variables, and the one digit precision (dp ¼ 10) has been selected in this paper. Therefore the interval for i-th HPP in hour k is conducted from zq variables:
qi qi $dp;
zqi;k ¼
i ¼ 1; 2; 3; …; I; k ¼ 1; 2; …; K:
(3)
The right mathematical approach should be used in order to evaluate the number of possible schedule combinations for test models I and II, as shown in Fig. 1a and b, respectively. The variation V with repetition [38] has been selected as an appropriate approach. It is used when the ordering of object matters and an object can be chosen from interval more than once,
The number of scheduled combinations according to Eq. (5) with the proposed one digit precision is 1.54 10696 for test model I and 1.06 10200 for test model II. Therefore, the HPPs' scheduling, where downstream plants operation depend on the upper plants' operation, in addition to a large solution space, is known as a complex optimization problem. It is impossible to obtain optimal schedule by calculating all combinations within reasonable CPU time. Therefore the modified DE optimization algorithm has been applied. 2.1. Objective function The optimization of HPPs' scheduling can be translated into simultaneous minimization of three different objectives:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 !!2 1 u X I X u@ K A$1=K : f1 ¼ t owi;k dwk
(4)
where z is the set of a size from which each r object can be chosen. The number of scheduled combinations for test models I and II with time interval K, can be obtained with:
f3 ¼
Vzr
¼
!K I Y qi qi $dp : i¼1
(5)
I X K X
qi;k $
i¼1 k¼1 I X K X
(6)
i¼1
k¼1
f2 ¼
Vzr ¼ zr ;
3
I X K X
!1 owi;k
:
(7)
i¼1 k¼1
si;k :
(8)
i¼1 k¼1
The first objective (6) evaluates the disagreement of system demand and energy obtained by the proposed algorithm within each individual hour k. The second objective (7) evaluates the used water quantity per electrical energy unit produced by summing up the discharge of all HPPs and by dividing this sum with the energy obtained. The last objective (8) evaluates water spillage. These three objective functions are merged by using the weighted sum method [37],
Minimize f ¼ w1 $f1 þ w2 $f2 þ w3 $f3 ;
(9)
where w1, w2, w3 are the weights which define priorities of each individual objective function. The weights selected by the authors were 0.65, 0.25 and 0.10. 2.2. Mathematical models and its constrains Two different mathematical models of HPP have been used in this paper. The first one represents the real parameters' models of eight cascade HPPs located in Slovenia (Dravske elektrarne Maribor) and the second one represents a test system used in several scientific papers [11,13e15,28,31] and consists of four HPPs. Both models are presented in Fig. 1. The model shown in Fig. 1 a consists of eight cascade HPPs where a1,k is the natural inflow of the first reservoir with volume v1,k in hour k. Each reservoir volume must be within the permitted limits
vi vi;k vi ; i ¼ 1; 2; 3; …; I; k ¼ 1; 2; …; K:
(10)
When the reservoir reaches the upper limit, the inflow ai,k of i-th HPP in hour k must be used as spillage si,k or can be used through turbines in order to produce energy. Water discharge of i-th HPP is also limited within minimal and maximal values.
qi qi;k qi ; i ¼ 1; 2; 3; …; I; k ¼ 1; 2; …; K:
Fig. 1. (a) Test model I and (b) test model II.
(11)
Due to the reservoir ground, the test model I also has the denivelation speed limit which is expressed as the permissible discharge per hour,
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
4
vi;k vi;k1 dvi ; i ¼ 1; 2; 3; …; I; k ¼ 2; 3; …; K;
(12)
where dvi represents the water discharge allowed within 1 h. The 7th and 8th HPP are canal based and therefore two old river beds are considered throughout the biological minimum flows b7,k and b8,k. Model II shown as Fig. 1b consists of four HPPs with the input data and coefficients described in Refs. [31,39]. This model is simpler in comparison to model I and without considering the denivelation speed limits. Time delay ti between the first and the third reservoir is 2 h, between the second and the third it is 3 h and between the third and the fourth reservoirs it is 4 h. The volume of reservoir i in the hour k is,
vi;kþ1 ¼ vi;k þ ai;k qi;k þ
up X
qi;kti ; i ¼ 1; 2; 3; …; I;
m¼1
(13)
k ¼ 2; 3; …; K; where up means the number of upstream reservoirs for the observed reservoir. The other mathematical equations presented above for model I are also used for model II. The hydro generator power output in both models is expressed as,
2 2 pi;k ¼ ci;1 $ vi;k þ ci;2 $ qi;k þ ci;3 $vi;k $qi;k þ ci;4 $vi;k
(14)
þ ci;5 $qi;k þ ci;6 ;
3.1. Initialization The initialization step is defined as a randomly-chosen population. Each individual xj,G of the initial population xi,G is composed of j variables:
xi;G ¼ ½x1 ; x2 ; …xD ; i ¼ 1; …; NP;
(16)
xj;G ¼ xj;upp randð0; 1Þ$ xj;upp xj;low ; j ¼ 1; 2; …; D;
(17)
where xj,upp and xj,low are the upper and lower bounds defined for each variable. The population therefore consists of NP D-dimensional vectors. 3.2. Mutation The mutant vector vi,Gþ1 is created for each target individual xi,G according to the selected strategy, which have an important impact on creating new vectors. The strategy applied and used in this paper is DE/rand/1/bin, which is formulated as,
vi;Gþ1 ¼ xr3 ;G þ F$ xr1 ;G xr2 ;G ; i ¼ 1; 2; …; NP;
(18)
where indexes r1, r2 and r3 are randomly-chosen from the interval [1, NP]. The randomly-chosen indexes are also chosen to be different from the running indexes i.
where ci,1, ci,2, … ci,6 represent the hydro power generation coefficients, for test model I obtained from the Dravske elektrarne Maribor Company and for model II from Refs. [31,39]. The hydro generator power output for i-th HPP is limited to the minimal and maximal output powers.
According to the strategy used in this paper, the DE algorithm produces the new trial vector ui,Gþ1 for each mutant vector vi,Gþ1 via binomial operation as,
pi qi;k pi :
ui;Gþ1 ¼ u1i;Gþ1 ; u2i;Gþ1 ; …uDi;Gþ1 ; i ¼ 1; …NP;
(15)
3.3. Crossover
(19)
where each variable in trial vector is formed as in 3. Differential evolution algorithm The DE algorithm was proposed by R. Storn and K. Price [32]. This is one among evolutionary algorithms which mimic the natural evolution. The main steps of the DE algorithm are initialization, mutation, crossover, evaluation, and selection. The pseudo-code of standard DE is shown in Fig. 2.
( uji;Gþ1 ¼
vji;Gþ1 if randð0; 1Þ CR or j ¼ rnbrðiÞ xji;G if randð0; 1Þ > CR or jsrnbrðiÞ
) ;
(20)
i ¼ 1; 2; …; NP; j ¼ 1; 2; …; D: The rnbr(i) is a randomly-chosen index on interval [1, … D]. 3.4. Selection In the last step the DE evaluates trial vector ui,Gþ1 and target vector xi,G, commonly referred as the parent (target) vector,
xi;Gþ1 ¼
ui;Gþ1 if f ui;Gþ1
< f xi;G
; i ¼ 1; 2; …; NP; xi;G if f ui;Gþ1 f xi;G
(21)
where those with lower objective values occupy positions in the next generation (G þ 1). The DE algorithm steps restarts once again in the following order; mutation, crossover, evaluation, and selection. The abovementioned steps are repeated until the stopping criterions are reached and the optimal solution is obtained, respectively. 4. Modification of differential evolution algorithm
Fig. 2. Pseudo-code for classic DE algorithm.
The control parameters (F, CR and NP) of the classic DE algorithm have a significant impact on global or local search and convergence times. In order to avoid the efforts needed to determine a proper set
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
of control parameters, the self-adaptive parameters F and CR and dynamic changes in population size throughout the generations are proposed in this paper. The algorithm uses a multi-core processor where the modified DE algorithm is processed in parallel in such a way that the DE steps (mutation, crossover, and evaluation) simultaneously run on all cores, as shown in Fig. 3. After the steps of DE algorithm (except the selection) have been processed by each CPU core, the new trial vector n (29) is created and used for the selection step. According to the dynamic population size, the new population is created by the RSMM strategy described in Section 4.3. 4.1. Self-adaptive parameters F and CR Solving the optimization problems considered in this paper by using classic DE algorithm may not always lead towards a global solution, regardless of the effort made in order to choose the adequate control parameters. In Ref. [40] the authors conducted a parameter study for the DE algorithm by testing different parameters on two and five-dimensional spheres, Rosenbrock's and Rastrigin's functions. The conclusion from Ref. [40] was that the best choice for the population size NP is between 3 and 5-times the problem's dimension, the crossover constant CR between 0.9 and 1, the differential factor F to 0.6, and the reasonable choices for strategy were DE/best/2/bin and DE/rand/1/bin. If the control parameter F is chosen as a small value, the probability of escaping from a locally optimal solution decreases. The convergence can be speeded up by choosing a higher value of CR. The suggested selection of control parameters by Storn and Price [32] are: NP between 5D and 10D, F ¼ 0.5, CR between 0.9 and 1.
Fi;Gþ1 ¼
Frand if randð0; 1Þ < a1 ; i ¼ 1; 2; …; NP; Fi;G otherwise
(22)
CRrand if randð0; 1Þ < a2 CRi;G otherwise
; i ¼ 1; 2; …; NP;
(23)
where Frand and CRrand are randomly generated numbers at interval [0, 1] and the values for a1, a2 are fixed and set to 0.1. The important modification of the DE algorithm in order to decrease the number of function evaluations followed by the decreased CPU time is presented in Sections 4.2 and 4.3.
4.2. Dynamic population size throughout the generations The important parameter of the DE algorithm which affects the number of function evaluation is the population size NP. In the classic DE algorithm this parameter is fixed and selected by the user. The population size can be also variable throughout the generations and for instance it can be decreased by the gradual reduction as by Brest et al. [36] proposed. The population size here is reduced by half within each block of the predefined generation number. The downside of this idea is that the expected optimization duration should be known in advance. To avoid this problem and to allow the algorithm not just to decrease but also to increase the population size, this paper proposes approaches which dynamically varies the population size (26) along with the generation number. The idea behind the proposed solution is to calculate factor fpm for the observed generation as shown in Eq. (24). The factor fpm1 from the previous generation is also needed in order to calculate the difference dfpm between these two factors.
fpm
If the population converges prematurely, then F and/or NP should be increased [40]. In order to overcome these problems, Brest et al. [35] proposed the self-adaptive F and CR. For the initial generation both control parameters are selected by the user and vary along with the iteration number,
CRi;Gþ1 ¼
5
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!, u u f xbest;Gþ1
¼t 1 0:05; f xbest;G
dfpm ¼ fpm fpm1 :
(24)
(25)
The new population size is afterwards calculated as
9 8 NPG f pm $ðNPG NPmin Þ if dpfm < 0 > > > > > > > > > > > > NP þ f p $ðNP NP Þ if dpf > 0 > > m max m G = < G NPGþ1 ¼ ; G > $NPG if dpfm ¼ 0 > NP $0:9 þ 0:1$ 1 > > > > > > G 100 > > > > > > ; : NPG $0:9 if dpfm ¼ 0 and G 100
(26)
where NPmin is the minimal and NPmax is the maximal allowed population size. The authors selection of NPmin and NPmax was fixed to 1D, 5D and 1D, 10D. When the population size is increased according to Eq. (26), the problem of missing members is solved by parallel processing of the DE algorithm, as described in Section 4.3.
4.3. Parallel processing of DE with RSMM strategy
Fig. 3. The parallel processing of DE with RSMM strategy.
The proposed algorithm simultaneously runs the mutation, crossover and evaluation steps on the multi-core processor. All of the processor's cores run the DE mutation and the crossover step by using the data obtained for the previous generation, as shown in Fig. 3. When the separated and simultaneous DE steps are processed by each core, the populations from all cores are collected:
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
6
Fig. 4. RSMM strategy.
0
x1;1 N¼@ « xm;1
… 1 /
1 x1;NP « A; xm;NP
(27) Fig. 5. Objective function values throughout the generations for test model I.
where xm,1, … xm,NP are the D-dimensional vectors obtained from the m-th core, and combined as
xs;i ¼ xs;1 ; xs;2 ; … xs;D ; i ¼ 1; 2; …; NP; s ¼ 1; 2; …; m;
(28)
changed, the vectors with lower (better) objective values occupy the positions in the G þ 1 population. Each i-th column in matrix N consists of m rows and therefore of m vectors. The vectors in each i-th column are sorted by their objective function values,
where s is the core number and the m is the total number of cores. The objective function values of the vectors (members) are compared at i-th position across all of the m cores. Therefore the vector which has the lowest objective function value occupies i-th place in vector n,
n ¼ ½z1 ; z2 ; …zNP ;
sort f x1;i ; f x2;i …; f xm;i ¼ f xpð1Þ;i ; f xpð2Þ;i ; …; f xpðmÞ;i ; i ¼ 1; 2; …; NP;
(29)
where zi 2 {x1,i, x2,i, … xm,i}. The vector n consists of NP Ddimensional vectors (31) where li 2 {1, 2, …, m}.
f xli ;i ¼ min f xs;i ; s ¼ 1; …; m ; i ¼ 1; 2; …; NP;
(30)
zi ¼ xli ;i ; i ¼ 1; 2; …; NP:
(31)
(32) where p : ℕm /ℕm is the permutation function which is subject to:
xpð1Þ;i xpð2Þ;i / xpðmÞ;i :
The vector n is used during the selection step for comparison purposes against the target vector x. If the population size is not
(33)
The sorted vectors from matrix N are collected into matrix S,
Table 1 Optimization results for test model I obtained with the proposed algorithm in comparison with the system demand from SCADA. Hour Total hydro generation in MW h
1 2 3 4 5 6 7 8 9 10 11 12 a b
MDEb
SCADA
542.57 543.16 537.87 532.66 542.35 557.64 517.70 557.47 529.88 526.09 524.13 542.69
542.80 543.40 538.10 532.90 542.60 557.90 517.70 557.70 530.10 526.10 524.30 542.90
Hour Total hydro generation in MW h
13 14 15 16 17 18 19 20 21 22 23 24
MDEb
SCADA
533.27 559.76 549.32 550.65 557.44 559.94 561.74 563.44 556.84 525.19 526.31 509.89
533.50 560.00 549.50 550.90 557.70 560.20 562.00 563.70 557.10 525.20 526.30 509.90
Hour Deviationa in
Hour Deviationa in
MW h % 1 2 3 4 5 6 7 8 9 10 11 12
0.23 0.24 0.23 0.24 0.25 0.26 0.00 0.23 0.22 0.01 0.17 0.21
0.04 0.04 0.04 0.05 0.05 0.05 0.00 0.04 0.04 0.00 0.03 0.04
13 14 15 16 17 18 19 20 21 22 23 24
Hour Total hydro discharge 104 m3
MW h %
MDEb
SCADA
0.23 0.24 0.18 0.25 0.26 0.26 0.26 0.26 0.26 0.01 0.01 0.01
1352.25 1368.03 1357.09 1334.97 1347.01 1392.87 1290.81 1387.03 1303.55 1300.84 1273.99 1343.30
1337.97 1344.07 1321.41 1341.65 1333.79 1394.67 1252.73 1413.30 1325.01 1306.45 1283.04 1341.07
0.04 1 0.04 2 0.03 3 0.05 4 0.05 5 0.05 6 0.05 7 0.05 8 0.05 9 0.00 10 0.00 11 0.00 12
Hour Total hydro discharge 104 m3
13 14 15 16 17 18 19 20 21 22 23 24
MDEb
SCADA
1311.33 1406.87 1372.50 1387.31 1410.48 1428.17 1431.84 1434.79 1415.20 1306.52 1315.14 1255.14
1319.27 1395.34 1355.27 1361.95 1413.72 1427.13 1431.79 1435.08 1405.74 1303.70 1298.23 1252.86
Deviation between total hydro generation obtained by MDE and SCADA. Modified Differential Evolution Algorithm.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
7
For the RSMM strategy purposes the vectors from each i-th column of matrix N are chosen in random sequence as
rand f x1;i ; f x2;i ; …; f xm;i ¼ f xpð1Þ;i ; f xpð2Þ;i ; …; f xpðmÞ;i ; i ¼ 1; 2; …; NP; (36) where p : ℕm /ℕm is the random generation of permutation. The vectors are then collected in the vector r.
i h r ¼ xpð1Þ;1 ; xpð1Þ;2 ; …; xpð1Þ;NP ; …; xpðmÞ;NP :
The vectors n, s, r are therefore used for RSMM strategy as shown in Fig. 4. When the population in the generation G þ 1 is increased, the new additional members are provided from vector s. When the population is decreased and the randomly-chosen number from interval [0, 1] is lower than the constant b1, the members for the new target vectors are obtained from vector r. The constant b1 is defined by the user and the authors' selection was 0.2. After the trial vectors are combined according to RSMM strategy and the selection step has already been processed, the population and other data are provided to m cores where the DE algorithm steps are repeated until the stopping criterion is met.
Fig. 6. Population size variations throughout the generations for test model I.
0
xpð1Þ;1 « S¼@ xpðmÞ;1
… 1 /
1 xpð1Þ;NP A; « xpðmÞ;NP
(37)
(34) 5. Experimental results
where the vectors in each i-th column are sorted from minimal to maximal objective function value and afterwards the vectors are arranged in vector s.
i h s ¼ xpð1Þ;1 ; xpð1Þ;2 ; …; xpð1Þ;NP ; …; xpðmÞ;NP :
The performance of the proposed self-adaptive DE algorithm was implemented within the MATLAB and evaluated using a Intel Core i7, 3.50 GHz computer and two HPPs test models. The input data for test model I was obtained from the Dravske elektrarne Maribor Company and the HSE Group, respectively. The input data for test model II were obtained from Refs. [31,39].
(35)
Table 2 Experimental results for test model I averaged over 10 independent runs of the modified and classic DE by different sets and self-adaptive F and CR. Algorithm
Proposed modified DE
Population size
Fixed
Type F, CR Best objective value [p.u.] nfevala CPU time [s]
1D 3D Self-adaptive 0.24608 0.24485
5D
[1D, 5D]
0.24483
1,381,440 159
6,907,200 615
a
Classic DE Dynamic
4,144,320 366
Reduction by Brest et al. [36]
Fixed
Reduction by Brest et al. [36]
[1D, 10D]
5D/2.5D/1.25D
1D
3D
5D
5D/2.5D/1.25D
0.24443
0.24416
0.24472
0.24761
0.24739
0.24767
0.24674
2,181,450 325
3,993,870 391
3,955,200 440
1,381,440 241
4,144,320 691
6,907,200 1002
6,907,200 582
Number of function evaluations.
Table 3 Experimental results for test model I averaged over 10 independent runs of the classic DE by fixed population size and different F and CR values. Algorithm
Classic DE
Population size
Fixed
Type F,CR Best objective value [p.u.] nfevala CPU time [s]
1D 0.1, 0.1 0.25184
3D
5D
0.25109
1,381,440 96
4,144,320 630
a
3D
5D
0.25066
1D 0.5, 0.5 0.25319
3D
5D
0.25306
1D 0.5, 0.9 0.24805
0.25317
6,907,200 823
1,381,440 129
4,144,320 395
3D
5D
0.25244
1D 0.9, 0.9 0.28255
0.25072
0.29142
0.29592
6,907,200 1003
1,381,440 118
4,144,320 461
6,907,200 1148
1,381,440 357
4,144,320 1149
6,907,200 2313
Number of function evaluations.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
0.005 0.010 0.016 0.002 0.002 0.014 0.011 0.010 0.008 0.014 0.000 0.013 13 14 15 16 17 18 19 20 21 22 23 24
436.37 406.35 447.19 424.23 484.20 494.16 446.35 486.33 483.31 511.12 499.91 461.62
436.39 406.39 447.26 424.22 484.21 494.23 446.40 486.38 483.35 511.19 499.91 461.68
System demand [31] in MW h Total hydro generation in MW h Hour
0.017 0.012 0.013 0.013 0.014 0.009 0.003 0.010 0.007 0.013 0.007 0.007
Deviationa in % 4 4
Deviation between total hydro generation obtained by modified Differential Evolution Algorithm and system demand [31]. a
117.92 116.73 118.63 116.90 117.32 118.99 121.60 128.01 136.12 142.71 150.34 169.98
3 2
89.22 92.21 93.48 94.31 93.07 89.54 89.15 87.63 86.68 83.67 81.21 70.00 148.13 126.41 111.86 101.02 104.75 100.38 106.22 106.89 107.92 108.79 109.74 112.94
3 2
80.39 79.86 82.09 82.07 83.09 83.50 83.50 84.50 85.66 85.45 86.77 88.65 98.56 98.08 97.72 94.73 92.09 92.26 93.04 94.87 96.82 98.98 102.04 106.28 1 2 3 4 5 6 7 8 9 10 11 12
1
109.63 99.00 87.60 74.60 91.57 108.38 125.38 142.31 145.47 156.25 157.81 159.95
13 14 15 16 17 18 19 20 21 22 23 24
109.54 115.26 118.70 123.06 123.65 122.80 123.22 121.95 121.84 120.15 120.21 120.00
1
159.66 159.91 159.89 160.00 159.70 159.98 159.97 159.99 158.03 153.72 149.04 140.00
1 2 3 4 5 6 7 8 9 10 11 12
352.49 335.55 305.55 309.47 346.42 318.60 373.81 391.06 409.39 452.41 436.36 405.48
352.55 335.59 305.59 309.51 346.47 318.63 373.82 391.10 409.42 452.47 436.39 405.51
System demand [31] in MW h Total hydro generation in MW h Hour Hydro storage volume 104 m3 of reservoir no. Hour
The system demand from Ref. [31] was successfully satisfied using the adapted DE algorithm and the dynamic population size. The hydro storage volumes and the comparison between the total hydro generations against the system demand [31] are shown in Table 4. The final states of the reservoirs were also satisfied by using the adapted DE. The optimal hydro generation by each HPP and the associated hydro discharges obtained by the adapted DE algorithm and dynamic population size at intervals [1D, 10D] are shown in Table 5. The comparison between the objective function values' convergence obtained with the adapted DE algorithm and the different types of populations, are shown in Fig. 7. The best objective function value was obtained with adapted algorithm and dynamic changes of population sizes at intervals [1D, 10D]. The variations in the population sizes throughout the generations with different population types are shown in Fig. 8. The gradual reduction sizes of the population were set to 5D, 2.5D and 1.25D in the 1st, 300th, and 600th generation blocks. The numerical comparisons between final results, the objective function evaluations and the CPU times obtained by the adapted and classic DE algorithms by different sets and self-adaptive F and CR, are shown in Table 6. Experimental tests with the classic DE algorithm were also made using fixed population size and different fixed F and CR values. The associated results are shown in Table 7. When the classic DE algorithm with dynamic and fixed population sizes is compared against the adapted algorithm by using
Hydro storage volume 104 m3 of reservoir no.
5.2. Test model II
Table 4 Optimization results for test model II obtained with the proposed algorithm in comparison with the system demand from Ref. [31].
The test model I is combined of eight HPPs with real parameters and constraints. The main aim of optimization was to satisfy the real system demand obtained from SCADA (Supervisory control and data acquisition). The deviation between the system demand and the schedule obtained by the proposed algorithm was within the allowed tolerance and therefore the system demand was successfully satisfied. The optimal generation scheduling obtained by the proposed algorithm and the comparison with SCADA is shown in Table 1. The adapted algorithm with dynamic population size was compared with the gradual reduction [36] and the fixed population sizes throughout the generations. The comparison is shown in Fig. 5. The gradual reducing population method must include predefined blocks of generations as each population size will be reduced. The authors prepared for gradual reductions being made in such a way that the population sizes 5D, 2.5D and 1.25D were applied to the 1st, 300th and 500th generations. The best result was obtained by the adapted algorithm with the dynamic population size. The population sizes' variations for the tested algorithms during the optimization process are shown in Fig. 6. The impact of varying the population sizes throughout generation number was evident from the total number of function evaluations. The comparison of the results obtained with the adapted and classic DE algorithms by the different sets and self-adaptive control parameters F and CR are shown in Table 2. The experimental results obtained by the classic DE algorithm, but with the fixed population sizes and different fixed control parameters F and CR are shown in Table 3. When the results in Tables 2 and 3 for the adapted and classic DE algorithms are compared, it can be seen that the adapted algorithm provided better final results with at least a 3-times lower total number of function evaluations.
Deviationa in %
5.1. Test model I
Hour
8
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
9
Table 5 Optimal hydro generations and hydro discharges for test model II obtained with the proposed algorithm. Hour
1 2 3 4 5 6 7 8 9 10 11 12
Hydro generation in MW h for HPP no.
Hour
1
2
3
4
91.17 82.81 76.59 83.93 76.19 64.88 67.83 68.13 74.41 79.78 81.33 60.17
59.88 64.98 55.50 69.02 57.48 55.17 51.00 51.57 57.99 72.07 64.08 54.66
0.00 0.00 0.00 0.00 34.29 0.00 38.01 37.67 40.36 41.23 40.66 41.36
201.44 187.76 173.46 156.52 178.46 198.55 216.97 233.69 236.63 259.33 250.29 249.29
13 14 15 16 17 18 19 20 21 22 23 24
Hydro generation in MW h for HPP no.
Hour
1
2
3
4
75.93 65.84 76.14 61.07 82.68 85.36 69.11 74.35 73.17 89.91 85.54 92.61
63.84 55.66 67.99 64.77 70.84 76.11 63.56 74.91 76.52 83.29 75.48 80.95
36.92 34.96 42.70 35.66 41.87 44.63 45.29 48.43 49.28 50.98 52.57 0.00
259.68 249.89 260.36 262.73 288.81 288.06 268.39 288.64 284.34 286.94 286.32 288.06
1 2 3 4 5 6 7 8 9 10 11 12
Hydro discharge 104 m3 of reservoir no. 1
2
3
4
11.44 9.48 8.36 9.99 8.64 6.83 7.22 7.17 8.05 8.84 8.95 5.75
7.61 8.53 6.77 9.02 6.98 6.60 6.00 6.00 6.83 9.21 7.68 6.12
29.97 29.82 30.00 29.93 16.16 25.12 14.83 15.14 13.78 13.30 14.09 14.47
13.17 13.03 13.00 13.00 13.00 13.01 13.00 13.00 13.00 14.35 13.27 13.00
Hour
13 14 15 16 17 18 19 20 21 22 23 24
Hydro discharge 104 m3 of reservoir no. 1
2
3
4
7.75 6.27 7.56 5.65 8.41 8.84 6.59 7.26 7.11 9.69 8.94 10.21
7.43 6.01 7.74 7.17 8.24 9.53 7.39 9.51 9.95 12.01 10.46 15.00
17.18 17.62 14.97 17.44 15.14 13.72 13.97 11.67 10.00 10.07 10.00 0.00
14.07 13.06 14.12 14.36 17.48 17.34 14.97 17.42 17.10 18.03 18.65 20.71
An adapted DE algorithm for dynamic population sizes is proposed for optimal dispatching of cascade HPPs by considering all constraints and the final states of the reservoirs. The classic DE algorithm has control parameters which are selected by the user. Choosing the proper values for all control parameters can be a challenging task because different control parameters have different impacts on the functions being optimized. Therefore the
control parameters F and CR are used as self-adaptive throughout the generation number. In order to decrease the number of function evaluations and consequently to decrease the convergence times, this paper presented a novel approach which dynamic population size. In order to ensure the missing population's members when expending it's the proposed algorithm uses a novel RSMM strategy. This strategy ensures new members by using parallel processing of DE algorithm's steps on multi-core processors. This implementation enables global search with fast convergence. The results show that the proposed adapted DE algorithm by parallel processing and self-adaptive F and CR with RSMM strategy achieves satisfactory system demand by decreasing the water quantity used per electrical energy unit produced and also the number of function evaluations in comparison with the other tested algorithm. The proposed algorithm is appropriate for use in the real-time applications. This is evident from the results section where the algorithm's results are shown for the application on real parameter's model as well as on standard test model used in many scientific publications.
Fig. 7. Objective function values throughout the generations for test model II.
Fig. 8. Population size variations throughout the generations for test model II.
results from Tables 6 and 7, it is obvious that the adapted algorithm achieved better objective function values at a lower number of function evaluations. Consequently, in comparison to other algorithms this means that the optimal solution is obtained at lower CPU times. The adapted algorithm with the dynamic population size on interval [1D, 10D] obtained better objective function value comparing to interval [1D, 5D]. However, these values are relatively close to each other, while the numbers of function evaluations are notably lower for narrower interval.
6. Conclusion
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
10
Table 6 Experimental results for test model II averaged over 10 independent runs of the modified and classic DE by different sets and self-adaptive F and CR. Algorithm
Proposed modified DE
Population size
Fixed
Type F, CR Best objective value [p.u.] nfevala CPU time [s]
1D 3D Self-adaptive 0.20929 0.20884
5D
[1D, 5D]
0.20851
460,320 37
2,301,600 104
a
Classic DE Dynamic
1,380,960 63
Reduction by Brest et al. [36]
Fixed
Reduction by Brest et al. [36]
[1D, 10D]
5D/2.5D/1.25D
1D
3D
5D
5D/2.5D/1.25D
0.20818
0.20810
0.20821
0.21629
0.21077
0.21103
0.20846
1,021,674 69
1,728,666 100
1,437,600 86
460,320 10
1,380,960 27
2,301,600 56
2,301,600 65
Number of function evaluations.
Table 7 Experimental results for test model II averaged over 10 independent runs of the classic DE by fixed population size and different F and CR values. Algorithm
Classic DE
Population size
Fixed
Type F, CR Best objective value [p.u.] nfevala CPU time [s]
1D 0.1, 0.1 0.21711
3D
5D
0.21306
460,320 9
1,380,960 21
a
3D
5D
0.21629
1D 0.5, 0.5 0.27155
3D
5D
0.27378
1D 0.5, 0.9 0.21959
0.27108
2,301,600 41
460,320 14
1,380,960 37
3D
5D
0.23042
1D 0.9, 0.9 0.37816
0.22244
0.40844
0.39627
2,301,600 70
460,320 10
1,380,960 31
2,301,600 66
460,320 20
1,380,960 31
2,301,600 70
Number of function evaluations.
References [1] Olabi AG. State of the art on renewable and sustainable energy. Energy 2013;61:2e5. [2] Ferreira HL, Garde R, Fulli G, Kling W, Lopes JP. Characterisation of electrical energy storage technologies. Energy 2013;53:288e98. [3] Nazari ME, Ardehali MM, Jafari S. Pumped-storage unit commitment with considerations for energy demand, economics, and environmental constraints. Energy 2010;35:4092e101. [4] Ostergaard PA. Reviewing optimisation criteria for energy systems analyses of renewable energy integration. Energy 2009;34:1236e45. [5] Olabi AG. The 3rd international conference on sustainable energy and environmental protection SEEP 2009 e Guest Editor's Introduction. Energy 2010;35:4508e9. [6] Carton JG, Olabi AG. Wind/hydrogen hybrid systems: opportunity for Ireland's wind resource to provide consistent sustainable energy supply. Energy 2010;35:4536e44. [7] Lund H. Renewable energy strategies for sustainable development. Energy 2007;32:912e9. [8] Olabi AG. Developments in sustainable energy and environmental protection. Energy 2012;39:2e5. [9] Catalao JPS, Pousinho HMI, Contreras J. Optimal hydro scheduling and offering strategies considering price uncertainty and risk management. Energy 2012;37:293e8. [10] Dursun B, Alboyaci B, Gokcol C. Optimal wind-hydro solution for the Marmara region of Turkey to meet electricity demand. Energy 2011;36:864e72. [11] Yuan X, Wang L, Yuan Y. Application of enhanced PSO approach to optimal scheduling of hydro system. Energy Convers Manag 2008;49:2966e72. [12] Narang N, Dhillon JS, Kothari DP. Multiobjective fixed head hydrothermal scheduling using integrated predatoreprey optimization and Powell search method. Energy 2012;47:237e52. [13] Qin H, Zhou J, Lu Y, Wang Y, Zhang Y. Multi-objective differential evolution with adaptive Cauchy mutation for short-term multi-objective optimal hydrothermal scheduling. Energy Convers Manag 2010;51:788e94. [14] Wang Y, Zhou J, Mo L, Zhang R, Zhang Y. Short-term hydrothermal generation scheduling using differential real-coded quantum-inspired evolutionary algorithm. Energy 2012;44:657e71. [15] Zhang H, Zhou J, Fang N, Zhang R, Zhang Y. Daily hydrothermal scheduling with economic emission using simulated annealing technique based multiobjective cultural differential evolution approach. Energy 2013;50:24e37. [16] Basu M. Artificial immune system for fixed head hydrothermal power system. Energy 2011;36:606e12. [17] Farhat IA, El-Hawary ME. Optimization methods applied for solving the shortterm hydrothermal coordination problem. Electr Power Syst Res 2009;79: 1308e20.
[18] Catalao JPS, Pousinho HMI, Mendes VMF. Hydro energy systems management in Portugal: profit-based evaluation of a mixed-integer nonlinear approach. Energy 2011;36:500e7. [19] Wong SYW. Hybrid simulated annealing/genetic algorithm approach to shortterm hydro-thermal scheduling with multiple thermal plants. Electr power & Energy Syst 2001;23:565e75. [20] Yu B, Yuan X, Wang J. Short-term hydro-thermal scheduling using particle swarm optimization method. Energy Convers Manag 2007;48:1902e8. [21] Soares J, Silva M, Sousa T, Vale Z, Morais H. Distributed energy resource shortterm scheduling using signaled particle swarm optimization. Energy 2012;42: 466e76. [22] Wood AJ, Wollenberg BF. Power generation, operation and control. 2nd ed. New York: Wiley; 1996. [23] Martins LSA, Soares S, Azevedo AT. A nonlinear model for the long-term hydro-thermal generation scheduling problem over multiple areas with transmission constraints. Power Systems Conference and Exposition; 2009. pp. 1e7. [24] Michalewicz Z, Schoenauer M. Evolutionary algorithms for constrained parameter optimization problems. Evol Comput 1996;4:1e32. [25] Khanmohammadi S, Amiri M, Haque MT. A new three-stage method for solving unit commitment problem. Energy 2010;35:3072e80. [26] Christober C, Rajan A. Hydro-thermal unit commitment problem using simulated annealing embedded evolutionary programming approach. Electr Power Energy Syst 2011;33:939e46. [27] Hong YY, Lin JK. Interactive multi-objective active power scheduling considering uncertain renewable energies using adaptive chaos clonal evolutionary programming. Energy 2013;53:212e20. [28] Yuan X, Zhang Y, Wang L, Yuan Y. An enhanced differential evolution algorithm for daily optimal hydro generation scheduling. Comput Math Appl 2008;55:2458e68. [29] Caponetto R, Fortuna L. Chaotic sequences to improve the performance of evolutionary algorithms. IEEE Trans Evol Comput 2003;3:289e304. [30] Lu Y, Zhou J, Qin H, Wang Y, Zhang Y. An adaptive chaotic differential evolution for the short-term hydrothermal generation scheduling problem. Energy Convers Manag 2010;51:1481e90. [31] Lakshminarasimman L, Subramanian S. Short-term scheduling of hydrothermal power system with cascaded reservoirs by using modified differential evolution. Generation, Transm Distrib, IEE Proc 2006;153:693e700. [32] Storn R, Price K. Differential evolution e a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 1997;11:341e59. [33] Mandal KK, Chakraborty. Differential evolution techniques-based short-term economic generation scheduling of hydrothermal systems. Electr Power Syst Res 2008;78:1972e9. [34] Mandal KK, Chakraborty N. Parameter study of differential evolution based optimal scheduling of hydrothermal system. Hydro-Environ Res 2013;7:72e80.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11 [35] Brest J, Greiner S, Boskovi c B, Mernik M, Zumer V. Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans Evol Comput 2006;10:646e57. [36] Brest J, Maucec MS. Population size reduction for the differential evolution algorithm. Appl Intell 2008;29:228e47. [37] Vahidinasab V, Jadid S. Joint economic and emission dispatch in energy markets: a multiobjective mathematical programming approach. Energy 2010;35:1497e504.
11
[38] Mazur DR. Combinatorics: a guided tour. United States of America; 2010. pp. 1e15. [39] Orero SO, Irving MR. A genetic algorithm modeling framework and solution technique for short term optimal hydrothermal scheduling. IEEE Trans Power Syst 1998;13:501e18. [40] Gamperle R, Muller SD, Koumoutsakos P. A parameter study for differential evolution. WSEAS International Conference on Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation; 2002. pp. 293e298.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Optimization of hydro energy storage plants by using differential evolution algorithm Arnel Gloti c a, *, Adnan Gloti c b, Peter Kitak a, Jo ze Pihler a, Igor Ti car a a b
Faculty of Electrical Engineering and Computer Science, University of Maribor, SI-2000 Maribor, Slovenia Holding Slovenske elektrarne Group, SI-1000 Ljubljana, Slovenia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 November 2013 Received in revised form 22 April 2014 Accepted 4 May 2014 Available online xxx
The optimal dispatching of cascade Hydro Power Plants is known as a complex optimization problem. In order to solve this problem the authors have applied an adapted differential evolution algorithm by using a fixed and dynamic population size. According to the dynamic population size, the proposed algorithm uses novel random and minimum to maximum sort strategy in order to create new populations with decreased or increased sizes. This implementation enables global search with fast convergence. It also uses a multi-core processor, where all the necessary optimization data are sent to the individual core of a central processing unit. The main aim of the optimization process is to satisfy 24 h demand by minimizing the water quantity used per electrical energy produced. This optimization process also satisfies the desired reservoir levels at the end of the day. The models used in this paper were the real parameters' models of eight cascade Hydro Power Plants located in Slovenia (Europe). Also the standard model from the literature is used in order to compare the performance of the adapted optimization algorithm. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Cascade hydro power plants Energy storage Hydro generation scheduling Optimization algorithms
1. Introduction Energy storage is a physical storage for energy [1] and the different types and technologies [2] are available nowadays. The pumped [3] and conventional [2] energy storages are used with HPP (hydro power plants) where the storage source is water, and the electricity produced is classified as renewable electricity [4]. Different research and developments related to sustainable and renewable electricity production according to different energy sources has been already made [5e8]. The authors' research focused focuses on cascade HPPs, where each individual HPP has its own reservoir and energy storage, respectively. Various combinations of reservoirs' charging and discharging produce different amounts of electricity. The optimization process for HPPs electricity production could be used in at least in two different ways; to determine the optimal production for HPPs in regard to a day-ahead electricity market [9] and forecasted demands [10] or to produce energy in accordance with the system demand and all requirements of HPPs combined
* Corresponding author. E-mail addresses: [email protected], [email protected] (A. Gloti c), adnan. [email protected] (A. Gloti c), [email protected] (P. Kitak), [email protected] (J. Pihler), [email protected] (I. Ti car).
with thermal power plants in order to decrease thermal costs [11e16]. The optimization process used in this paper is focused on a 24h interval with 1-h time steps. Therefore it refers to the term “short-term” hydro scheduling optimization [13,17e21] problem. In the past, a wide-range of optimization techniques [17] has been used to for successfully solving this complex non-linear constrained optimization problem. The complexities of the problem arise from a large number of co-dependent variables and various HPPs' constrains [22], such as reservoir levels, turbines' flows, denivelation speed, etc. Due to the generators' characteristics of HPPs, the outputs of which are generally non-linear functions of water discharge and net head, this optimization problem can also be marked as a non-linear problem [23]. The modern evolutionary algorithms [24] such as genetic algorithms [19], particle swarm optimization algorithm [20,21,25], evolutionary programming [26,27] and the differential evolution algorithm [13,28e33] are attracting more attention because of their efficiencies in solving hydro scheduling optimization problems. The differential evolution (DE) algorithm is known as a powerful evolutionary algorithm for global optimization, as first mentioned by Storn and Price [32]. Because of its simple implementation, efficiency, and robustness it has been selected as an appropriate optimization algorithm.
http://dx.doi.org/10.1016/j.energy.2014.05.004 0360-5442/© 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
2
Nomenclature
Parameters natural inflow to hydro plant i in hour k ai,k bi,k biological minimum flow of hydro plant i in hour k vi maximal value of the storage of reservoir i vi minimal value of the storage of reservoir i dvi allowed denivelation of reservoir i pi maximal output power of hydro plant i pi minimal output power of hydro plant i qi maximal water discharge of hydro plant i qi minimal water discharge of hydro plant i K number of hours of the scheduling period I total number of hydro power plants D number of parameters CR crossover control parameter
The authors in Ref. [28] used suitable penalty factors during the fitness evaluation of the DE algorithm using modified initialization and mutation steps which helped to satisfy the final reservoirs state. The classic DE algorithm was used in Ref. [33] where the parameter settings were found for each test system through trial and error, and consequently provide the optimal result. Properly controlling the parameter set of a DE algorithm is a challenging task, therefore in Ref. [34] the authors highlighted the importance of an algorithm's control parameter set at such as differential factor F, crossover constant CR, and population size NP. In Ref. [28] the chaos theory [29] is proposed to determine the adequate parameters' settings of the DE algorithm. Specifically, they determine the sets of F and CR and the applied chaos sequences which consequently have an impact on population diversity and premature convergence towards a locally optimal solution. In Ref. [30] the authors combined an adaptive dynamic control mechanism for the CR parameter with a chaotic local search operation for the DE algorithm, which avoids premature convergence. The self-adaptive F and CR parameters for each individual in the population are proposed in Ref. [35]. The changing of the initial values for both control parameters did not have any significant impact on the final result. When the classic DE algorithm with fixed control parameters falls into locally optimal solution, usually it is hard to escape from it. Therefore, an adaptive Cauchy mutation is proposed in Ref. [13] which renews the diversity of the population and avoids premature convergence. The important control parameter of the DE algorithm is also the population size NP, which is chosen by the user and fixed throughout the generations in classic DE. In order to provide higher population diversity at the beginning of the evolutionary process and decreased total number of function evaluation, a gradual population size reduction is proposed throughout the generations in Refs. [36], where the population size is reduced by half in each block of predefined generation numbers. In order to avoid the efforts needed to determine a proper set of control parameter, the proposed algorithm uses the self-adaptive control parameters F and CR. It also uses a novel method of varying the population size along the generation number. Population size is therefore referred to as dynamic population size. In this process the population can be decreased or even increased and this has a significant impact on the total number of function evaluations. When the algorithm increases the population size, new additional members are needed to fill up the new population size. In order to provide the additional members, the proposed algorithm uses a multi-core processor, where the population is
NP F w m dp
population size difference factor weight for objectives core number digit precision of float variable
Variables qi,k si,k vi,k pi,k G xi,G ui,G f dwk owi,k
discharge of hydro power plant i in hour k water spillage by reservoir i in hour k reservoir i storage in hour k power generated by hydro plant i in hour k denotes generation through algorithm steps value of i-th vector trial vector of i-th vector in generation G objective function demand energy in hour k sum of optimal energy production in hour k
separately and simultaneous processed by all cores. By the multiprocessing of the DE algorithm, the appropriate diversity of population is provided and the algorithm is more efficient in avoiding locally optimal solutions. According to the algorithm's increase population size, the missing members for the next generation are provided by the RSMM (random or the minimum to maximum sort strategy) described in Section 4.3. However, this paper presents a modified DE algorithm which was applied to hydro generation scheduling problem in order to satisfy the system demand and other objectives with fast convergence and decreased number of function evaluations, respectively. The hydro scheduling problem formulation is described in Section 2, the classic DE algorithm in Section 3, the modification of DE algorithm in Section 4, the experimental results in Section 5, and the conclusion in Section 6.
2. Hydro scheduling problem formulation In this paper, the hydro scheduling problem is formulated as a problem, where the main goal is to satisfy system demand by optimal scheduling across all HPPs. Cascade models of HPPs are used and therefore the production of downstream HPP is closely related with the upstream HPPs because of the upstream plant outflows' impact on downstream plant production. The system demand should be satisfied within the allowed deviation tolerance, usage of water quantity per electrical energy unit should be decreased and overflows should be also decreased and even eliminated, respectively. Since these goals should be achieved within acceptable CPU time, the hydro-scheduling problem can be solved by using the modified DE algorithm [32] with proposed dynamic population size and multi-processing. The final goal of the optimization problem is therefore conducted using three objectives, which are satisfied through single objective function by using the weighted sum method [37]. The optimal scheduling for connected HPPs can be obtained by searching the optimal schedule p, which is the function of flow through the turbine and the reservoir's volume,
p ¼ f ðv; qÞ;
(1)
where p is power, v volume of reservoir and q water discharge. The discharge q can be used as the optimization parameter, but within the allowed interval (2).
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
qi ; qi ¼ qi;k 2< ; qi qi;k qi ;
i ¼ 1; 2; 3; …; I;
(2)
k ¼ 1; 2; …; K: Changing the parameter q affects the variable v and consequently a different amount of electrical power (1) is obtained. The interval (2) consists of float variables, and the one digit precision (dp ¼ 10) has been selected in this paper. Therefore the interval for i-th HPP in hour k is conducted from zq variables:
qi qi $dp;
zqi;k ¼
i ¼ 1; 2; 3; …; I; k ¼ 1; 2; …; K:
(3)
The right mathematical approach should be used in order to evaluate the number of possible schedule combinations for test models I and II, as shown in Fig. 1a and b, respectively. The variation V with repetition [38] has been selected as an appropriate approach. It is used when the ordering of object matters and an object can be chosen from interval more than once,
The number of scheduled combinations according to Eq. (5) with the proposed one digit precision is 1.54 10696 for test model I and 1.06 10200 for test model II. Therefore, the HPPs' scheduling, where downstream plants operation depend on the upper plants' operation, in addition to a large solution space, is known as a complex optimization problem. It is impossible to obtain optimal schedule by calculating all combinations within reasonable CPU time. Therefore the modified DE optimization algorithm has been applied. 2.1. Objective function The optimization of HPPs' scheduling can be translated into simultaneous minimization of three different objectives:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 !!2 1 u X I X u@ K A$1=K : f1 ¼ t owi;k dwk
(4)
where z is the set of a size from which each r object can be chosen. The number of scheduled combinations for test models I and II with time interval K, can be obtained with:
f3 ¼
Vzr
¼
!K I Y qi qi $dp : i¼1
(5)
I X K X
qi;k $
i¼1 k¼1 I X K X
(6)
i¼1
k¼1
f2 ¼
Vzr ¼ zr ;
3
I X K X
!1 owi;k
:
(7)
i¼1 k¼1
si;k :
(8)
i¼1 k¼1
The first objective (6) evaluates the disagreement of system demand and energy obtained by the proposed algorithm within each individual hour k. The second objective (7) evaluates the used water quantity per electrical energy unit produced by summing up the discharge of all HPPs and by dividing this sum with the energy obtained. The last objective (8) evaluates water spillage. These three objective functions are merged by using the weighted sum method [37],
Minimize f ¼ w1 $f1 þ w2 $f2 þ w3 $f3 ;
(9)
where w1, w2, w3 are the weights which define priorities of each individual objective function. The weights selected by the authors were 0.65, 0.25 and 0.10. 2.2. Mathematical models and its constrains Two different mathematical models of HPP have been used in this paper. The first one represents the real parameters' models of eight cascade HPPs located in Slovenia (Dravske elektrarne Maribor) and the second one represents a test system used in several scientific papers [11,13e15,28,31] and consists of four HPPs. Both models are presented in Fig. 1. The model shown in Fig. 1 a consists of eight cascade HPPs where a1,k is the natural inflow of the first reservoir with volume v1,k in hour k. Each reservoir volume must be within the permitted limits
vi vi;k vi ; i ¼ 1; 2; 3; …; I; k ¼ 1; 2; …; K:
(10)
When the reservoir reaches the upper limit, the inflow ai,k of i-th HPP in hour k must be used as spillage si,k or can be used through turbines in order to produce energy. Water discharge of i-th HPP is also limited within minimal and maximal values.
qi qi;k qi ; i ¼ 1; 2; 3; …; I; k ¼ 1; 2; …; K:
Fig. 1. (a) Test model I and (b) test model II.
(11)
Due to the reservoir ground, the test model I also has the denivelation speed limit which is expressed as the permissible discharge per hour,
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
4
vi;k vi;k1 dvi ; i ¼ 1; 2; 3; …; I; k ¼ 2; 3; …; K;
(12)
where dvi represents the water discharge allowed within 1 h. The 7th and 8th HPP are canal based and therefore two old river beds are considered throughout the biological minimum flows b7,k and b8,k. Model II shown as Fig. 1b consists of four HPPs with the input data and coefficients described in Refs. [31,39]. This model is simpler in comparison to model I and without considering the denivelation speed limits. Time delay ti between the first and the third reservoir is 2 h, between the second and the third it is 3 h and between the third and the fourth reservoirs it is 4 h. The volume of reservoir i in the hour k is,
vi;kþ1 ¼ vi;k þ ai;k qi;k þ
up X
qi;kti ; i ¼ 1; 2; 3; …; I;
m¼1
(13)
k ¼ 2; 3; …; K; where up means the number of upstream reservoirs for the observed reservoir. The other mathematical equations presented above for model I are also used for model II. The hydro generator power output in both models is expressed as,
2 2 pi;k ¼ ci;1 $ vi;k þ ci;2 $ qi;k þ ci;3 $vi;k $qi;k þ ci;4 $vi;k
(14)
þ ci;5 $qi;k þ ci;6 ;
3.1. Initialization The initialization step is defined as a randomly-chosen population. Each individual xj,G of the initial population xi,G is composed of j variables:
xi;G ¼ ½x1 ; x2 ; …xD ; i ¼ 1; …; NP;
(16)
xj;G ¼ xj;upp randð0; 1Þ$ xj;upp xj;low ; j ¼ 1; 2; …; D;
(17)
where xj,upp and xj,low are the upper and lower bounds defined for each variable. The population therefore consists of NP D-dimensional vectors. 3.2. Mutation The mutant vector vi,Gþ1 is created for each target individual xi,G according to the selected strategy, which have an important impact on creating new vectors. The strategy applied and used in this paper is DE/rand/1/bin, which is formulated as,
vi;Gþ1 ¼ xr3 ;G þ F$ xr1 ;G xr2 ;G ; i ¼ 1; 2; …; NP;
(18)
where indexes r1, r2 and r3 are randomly-chosen from the interval [1, NP]. The randomly-chosen indexes are also chosen to be different from the running indexes i.
where ci,1, ci,2, … ci,6 represent the hydro power generation coefficients, for test model I obtained from the Dravske elektrarne Maribor Company and for model II from Refs. [31,39]. The hydro generator power output for i-th HPP is limited to the minimal and maximal output powers.
According to the strategy used in this paper, the DE algorithm produces the new trial vector ui,Gþ1 for each mutant vector vi,Gþ1 via binomial operation as,
pi qi;k pi :
ui;Gþ1 ¼ u1i;Gþ1 ; u2i;Gþ1 ; …uDi;Gþ1 ; i ¼ 1; …NP;
(15)
3.3. Crossover
(19)
where each variable in trial vector is formed as in 3. Differential evolution algorithm The DE algorithm was proposed by R. Storn and K. Price [32]. This is one among evolutionary algorithms which mimic the natural evolution. The main steps of the DE algorithm are initialization, mutation, crossover, evaluation, and selection. The pseudo-code of standard DE is shown in Fig. 2.
( uji;Gþ1 ¼
vji;Gþ1 if randð0; 1Þ CR or j ¼ rnbrðiÞ xji;G if randð0; 1Þ > CR or jsrnbrðiÞ
) ;
(20)
i ¼ 1; 2; …; NP; j ¼ 1; 2; …; D: The rnbr(i) is a randomly-chosen index on interval [1, … D]. 3.4. Selection In the last step the DE evaluates trial vector ui,Gþ1 and target vector xi,G, commonly referred as the parent (target) vector,
xi;Gþ1 ¼
ui;Gþ1 if f ui;Gþ1
< f xi;G
; i ¼ 1; 2; …; NP; xi;G if f ui;Gþ1 f xi;G
(21)
where those with lower objective values occupy positions in the next generation (G þ 1). The DE algorithm steps restarts once again in the following order; mutation, crossover, evaluation, and selection. The abovementioned steps are repeated until the stopping criterions are reached and the optimal solution is obtained, respectively. 4. Modification of differential evolution algorithm
Fig. 2. Pseudo-code for classic DE algorithm.
The control parameters (F, CR and NP) of the classic DE algorithm have a significant impact on global or local search and convergence times. In order to avoid the efforts needed to determine a proper set
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
of control parameters, the self-adaptive parameters F and CR and dynamic changes in population size throughout the generations are proposed in this paper. The algorithm uses a multi-core processor where the modified DE algorithm is processed in parallel in such a way that the DE steps (mutation, crossover, and evaluation) simultaneously run on all cores, as shown in Fig. 3. After the steps of DE algorithm (except the selection) have been processed by each CPU core, the new trial vector n (29) is created and used for the selection step. According to the dynamic population size, the new population is created by the RSMM strategy described in Section 4.3. 4.1. Self-adaptive parameters F and CR Solving the optimization problems considered in this paper by using classic DE algorithm may not always lead towards a global solution, regardless of the effort made in order to choose the adequate control parameters. In Ref. [40] the authors conducted a parameter study for the DE algorithm by testing different parameters on two and five-dimensional spheres, Rosenbrock's and Rastrigin's functions. The conclusion from Ref. [40] was that the best choice for the population size NP is between 3 and 5-times the problem's dimension, the crossover constant CR between 0.9 and 1, the differential factor F to 0.6, and the reasonable choices for strategy were DE/best/2/bin and DE/rand/1/bin. If the control parameter F is chosen as a small value, the probability of escaping from a locally optimal solution decreases. The convergence can be speeded up by choosing a higher value of CR. The suggested selection of control parameters by Storn and Price [32] are: NP between 5D and 10D, F ¼ 0.5, CR between 0.9 and 1.
Fi;Gþ1 ¼
Frand if randð0; 1Þ < a1 ; i ¼ 1; 2; …; NP; Fi;G otherwise
(22)
CRrand if randð0; 1Þ < a2 CRi;G otherwise
; i ¼ 1; 2; …; NP;
(23)
where Frand and CRrand are randomly generated numbers at interval [0, 1] and the values for a1, a2 are fixed and set to 0.1. The important modification of the DE algorithm in order to decrease the number of function evaluations followed by the decreased CPU time is presented in Sections 4.2 and 4.3.
4.2. Dynamic population size throughout the generations The important parameter of the DE algorithm which affects the number of function evaluation is the population size NP. In the classic DE algorithm this parameter is fixed and selected by the user. The population size can be also variable throughout the generations and for instance it can be decreased by the gradual reduction as by Brest et al. [36] proposed. The population size here is reduced by half within each block of the predefined generation number. The downside of this idea is that the expected optimization duration should be known in advance. To avoid this problem and to allow the algorithm not just to decrease but also to increase the population size, this paper proposes approaches which dynamically varies the population size (26) along with the generation number. The idea behind the proposed solution is to calculate factor fpm for the observed generation as shown in Eq. (24). The factor fpm1 from the previous generation is also needed in order to calculate the difference dfpm between these two factors.
fpm
If the population converges prematurely, then F and/or NP should be increased [40]. In order to overcome these problems, Brest et al. [35] proposed the self-adaptive F and CR. For the initial generation both control parameters are selected by the user and vary along with the iteration number,
CRi;Gþ1 ¼
5
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!, u u f xbest;Gþ1
¼t 1 0:05; f xbest;G
dfpm ¼ fpm fpm1 :
(24)
(25)
The new population size is afterwards calculated as
9 8 NPG f pm $ðNPG NPmin Þ if dpfm < 0 > > > > > > > > > > > > NP þ f p $ðNP NP Þ if dpf > 0 > > m max m G = < G NPGþ1 ¼ ; G > $NPG if dpfm ¼ 0 > NP $0:9 þ 0:1$ 1 > > > > > > G 100 > > > > > > ; : NPG $0:9 if dpfm ¼ 0 and G 100
(26)
where NPmin is the minimal and NPmax is the maximal allowed population size. The authors selection of NPmin and NPmax was fixed to 1D, 5D and 1D, 10D. When the population size is increased according to Eq. (26), the problem of missing members is solved by parallel processing of the DE algorithm, as described in Section 4.3.
4.3. Parallel processing of DE with RSMM strategy
Fig. 3. The parallel processing of DE with RSMM strategy.
The proposed algorithm simultaneously runs the mutation, crossover and evaluation steps on the multi-core processor. All of the processor's cores run the DE mutation and the crossover step by using the data obtained for the previous generation, as shown in Fig. 3. When the separated and simultaneous DE steps are processed by each core, the populations from all cores are collected:
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
6
Fig. 4. RSMM strategy.
0
x1;1 N¼@ « xm;1
… 1 /
1 x1;NP « A; xm;NP
(27) Fig. 5. Objective function values throughout the generations for test model I.
where xm,1, … xm,NP are the D-dimensional vectors obtained from the m-th core, and combined as
xs;i ¼ xs;1 ; xs;2 ; … xs;D ; i ¼ 1; 2; …; NP; s ¼ 1; 2; …; m;
(28)
changed, the vectors with lower (better) objective values occupy the positions in the G þ 1 population. Each i-th column in matrix N consists of m rows and therefore of m vectors. The vectors in each i-th column are sorted by their objective function values,
where s is the core number and the m is the total number of cores. The objective function values of the vectors (members) are compared at i-th position across all of the m cores. Therefore the vector which has the lowest objective function value occupies i-th place in vector n,
n ¼ ½z1 ; z2 ; …zNP ;
sort f x1;i ; f x2;i …; f xm;i ¼ f xpð1Þ;i ; f xpð2Þ;i ; …; f xpðmÞ;i ; i ¼ 1; 2; …; NP;
(29)
where zi 2 {x1,i, x2,i, … xm,i}. The vector n consists of NP Ddimensional vectors (31) where li 2 {1, 2, …, m}.
f xli ;i ¼ min f xs;i ; s ¼ 1; …; m ; i ¼ 1; 2; …; NP;
(30)
zi ¼ xli ;i ; i ¼ 1; 2; …; NP:
(31)
(32) where p : ℕm /ℕm is the permutation function which is subject to:
xpð1Þ;i xpð2Þ;i / xpðmÞ;i :
The vector n is used during the selection step for comparison purposes against the target vector x. If the population size is not
(33)
The sorted vectors from matrix N are collected into matrix S,
Table 1 Optimization results for test model I obtained with the proposed algorithm in comparison with the system demand from SCADA. Hour Total hydro generation in MW h
1 2 3 4 5 6 7 8 9 10 11 12 a b
MDEb
SCADA
542.57 543.16 537.87 532.66 542.35 557.64 517.70 557.47 529.88 526.09 524.13 542.69
542.80 543.40 538.10 532.90 542.60 557.90 517.70 557.70 530.10 526.10 524.30 542.90
Hour Total hydro generation in MW h
13 14 15 16 17 18 19 20 21 22 23 24
MDEb
SCADA
533.27 559.76 549.32 550.65 557.44 559.94 561.74 563.44 556.84 525.19 526.31 509.89
533.50 560.00 549.50 550.90 557.70 560.20 562.00 563.70 557.10 525.20 526.30 509.90
Hour Deviationa in
Hour Deviationa in
MW h % 1 2 3 4 5 6 7 8 9 10 11 12
0.23 0.24 0.23 0.24 0.25 0.26 0.00 0.23 0.22 0.01 0.17 0.21
0.04 0.04 0.04 0.05 0.05 0.05 0.00 0.04 0.04 0.00 0.03 0.04
13 14 15 16 17 18 19 20 21 22 23 24
Hour Total hydro discharge 104 m3
MW h %
MDEb
SCADA
0.23 0.24 0.18 0.25 0.26 0.26 0.26 0.26 0.26 0.01 0.01 0.01
1352.25 1368.03 1357.09 1334.97 1347.01 1392.87 1290.81 1387.03 1303.55 1300.84 1273.99 1343.30
1337.97 1344.07 1321.41 1341.65 1333.79 1394.67 1252.73 1413.30 1325.01 1306.45 1283.04 1341.07
0.04 1 0.04 2 0.03 3 0.05 4 0.05 5 0.05 6 0.05 7 0.05 8 0.05 9 0.00 10 0.00 11 0.00 12
Hour Total hydro discharge 104 m3
13 14 15 16 17 18 19 20 21 22 23 24
MDEb
SCADA
1311.33 1406.87 1372.50 1387.31 1410.48 1428.17 1431.84 1434.79 1415.20 1306.52 1315.14 1255.14
1319.27 1395.34 1355.27 1361.95 1413.72 1427.13 1431.79 1435.08 1405.74 1303.70 1298.23 1252.86
Deviation between total hydro generation obtained by MDE and SCADA. Modified Differential Evolution Algorithm.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
7
For the RSMM strategy purposes the vectors from each i-th column of matrix N are chosen in random sequence as
rand f x1;i ; f x2;i ; …; f xm;i ¼ f xpð1Þ;i ; f xpð2Þ;i ; …; f xpðmÞ;i ; i ¼ 1; 2; …; NP; (36) where p : ℕm /ℕm is the random generation of permutation. The vectors are then collected in the vector r.
i h r ¼ xpð1Þ;1 ; xpð1Þ;2 ; …; xpð1Þ;NP ; …; xpðmÞ;NP :
The vectors n, s, r are therefore used for RSMM strategy as shown in Fig. 4. When the population in the generation G þ 1 is increased, the new additional members are provided from vector s. When the population is decreased and the randomly-chosen number from interval [0, 1] is lower than the constant b1, the members for the new target vectors are obtained from vector r. The constant b1 is defined by the user and the authors' selection was 0.2. After the trial vectors are combined according to RSMM strategy and the selection step has already been processed, the population and other data are provided to m cores where the DE algorithm steps are repeated until the stopping criterion is met.
Fig. 6. Population size variations throughout the generations for test model I.
0
xpð1Þ;1 « S¼@ xpðmÞ;1
… 1 /
1 xpð1Þ;NP A; « xpðmÞ;NP
(37)
(34) 5. Experimental results
where the vectors in each i-th column are sorted from minimal to maximal objective function value and afterwards the vectors are arranged in vector s.
i h s ¼ xpð1Þ;1 ; xpð1Þ;2 ; …; xpð1Þ;NP ; …; xpðmÞ;NP :
The performance of the proposed self-adaptive DE algorithm was implemented within the MATLAB and evaluated using a Intel Core i7, 3.50 GHz computer and two HPPs test models. The input data for test model I was obtained from the Dravske elektrarne Maribor Company and the HSE Group, respectively. The input data for test model II were obtained from Refs. [31,39].
(35)
Table 2 Experimental results for test model I averaged over 10 independent runs of the modified and classic DE by different sets and self-adaptive F and CR. Algorithm
Proposed modified DE
Population size
Fixed
Type F, CR Best objective value [p.u.] nfevala CPU time [s]
1D 3D Self-adaptive 0.24608 0.24485
5D
[1D, 5D]
0.24483
1,381,440 159
6,907,200 615
a
Classic DE Dynamic
4,144,320 366
Reduction by Brest et al. [36]
Fixed
Reduction by Brest et al. [36]
[1D, 10D]
5D/2.5D/1.25D
1D
3D
5D
5D/2.5D/1.25D
0.24443
0.24416
0.24472
0.24761
0.24739
0.24767
0.24674
2,181,450 325
3,993,870 391
3,955,200 440
1,381,440 241
4,144,320 691
6,907,200 1002
6,907,200 582
Number of function evaluations.
Table 3 Experimental results for test model I averaged over 10 independent runs of the classic DE by fixed population size and different F and CR values. Algorithm
Classic DE
Population size
Fixed
Type F,CR Best objective value [p.u.] nfevala CPU time [s]
1D 0.1, 0.1 0.25184
3D
5D
0.25109
1,381,440 96
4,144,320 630
a
3D
5D
0.25066
1D 0.5, 0.5 0.25319
3D
5D
0.25306
1D 0.5, 0.9 0.24805
0.25317
6,907,200 823
1,381,440 129
4,144,320 395
3D
5D
0.25244
1D 0.9, 0.9 0.28255
0.25072
0.29142
0.29592
6,907,200 1003
1,381,440 118
4,144,320 461
6,907,200 1148
1,381,440 357
4,144,320 1149
6,907,200 2313
Number of function evaluations.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
0.005 0.010 0.016 0.002 0.002 0.014 0.011 0.010 0.008 0.014 0.000 0.013 13 14 15 16 17 18 19 20 21 22 23 24
436.37 406.35 447.19 424.23 484.20 494.16 446.35 486.33 483.31 511.12 499.91 461.62
436.39 406.39 447.26 424.22 484.21 494.23 446.40 486.38 483.35 511.19 499.91 461.68
System demand [31] in MW h Total hydro generation in MW h Hour
0.017 0.012 0.013 0.013 0.014 0.009 0.003 0.010 0.007 0.013 0.007 0.007
Deviationa in % 4 4
Deviation between total hydro generation obtained by modified Differential Evolution Algorithm and system demand [31]. a
117.92 116.73 118.63 116.90 117.32 118.99 121.60 128.01 136.12 142.71 150.34 169.98
3 2
89.22 92.21 93.48 94.31 93.07 89.54 89.15 87.63 86.68 83.67 81.21 70.00 148.13 126.41 111.86 101.02 104.75 100.38 106.22 106.89 107.92 108.79 109.74 112.94
3 2
80.39 79.86 82.09 82.07 83.09 83.50 83.50 84.50 85.66 85.45 86.77 88.65 98.56 98.08 97.72 94.73 92.09 92.26 93.04 94.87 96.82 98.98 102.04 106.28 1 2 3 4 5 6 7 8 9 10 11 12
1
109.63 99.00 87.60 74.60 91.57 108.38 125.38 142.31 145.47 156.25 157.81 159.95
13 14 15 16 17 18 19 20 21 22 23 24
109.54 115.26 118.70 123.06 123.65 122.80 123.22 121.95 121.84 120.15 120.21 120.00
1
159.66 159.91 159.89 160.00 159.70 159.98 159.97 159.99 158.03 153.72 149.04 140.00
1 2 3 4 5 6 7 8 9 10 11 12
352.49 335.55 305.55 309.47 346.42 318.60 373.81 391.06 409.39 452.41 436.36 405.48
352.55 335.59 305.59 309.51 346.47 318.63 373.82 391.10 409.42 452.47 436.39 405.51
System demand [31] in MW h Total hydro generation in MW h Hour Hydro storage volume 104 m3 of reservoir no. Hour
The system demand from Ref. [31] was successfully satisfied using the adapted DE algorithm and the dynamic population size. The hydro storage volumes and the comparison between the total hydro generations against the system demand [31] are shown in Table 4. The final states of the reservoirs were also satisfied by using the adapted DE. The optimal hydro generation by each HPP and the associated hydro discharges obtained by the adapted DE algorithm and dynamic population size at intervals [1D, 10D] are shown in Table 5. The comparison between the objective function values' convergence obtained with the adapted DE algorithm and the different types of populations, are shown in Fig. 7. The best objective function value was obtained with adapted algorithm and dynamic changes of population sizes at intervals [1D, 10D]. The variations in the population sizes throughout the generations with different population types are shown in Fig. 8. The gradual reduction sizes of the population were set to 5D, 2.5D and 1.25D in the 1st, 300th, and 600th generation blocks. The numerical comparisons between final results, the objective function evaluations and the CPU times obtained by the adapted and classic DE algorithms by different sets and self-adaptive F and CR, are shown in Table 6. Experimental tests with the classic DE algorithm were also made using fixed population size and different fixed F and CR values. The associated results are shown in Table 7. When the classic DE algorithm with dynamic and fixed population sizes is compared against the adapted algorithm by using
Hydro storage volume 104 m3 of reservoir no.
5.2. Test model II
Table 4 Optimization results for test model II obtained with the proposed algorithm in comparison with the system demand from Ref. [31].
The test model I is combined of eight HPPs with real parameters and constraints. The main aim of optimization was to satisfy the real system demand obtained from SCADA (Supervisory control and data acquisition). The deviation between the system demand and the schedule obtained by the proposed algorithm was within the allowed tolerance and therefore the system demand was successfully satisfied. The optimal generation scheduling obtained by the proposed algorithm and the comparison with SCADA is shown in Table 1. The adapted algorithm with dynamic population size was compared with the gradual reduction [36] and the fixed population sizes throughout the generations. The comparison is shown in Fig. 5. The gradual reducing population method must include predefined blocks of generations as each population size will be reduced. The authors prepared for gradual reductions being made in such a way that the population sizes 5D, 2.5D and 1.25D were applied to the 1st, 300th and 500th generations. The best result was obtained by the adapted algorithm with the dynamic population size. The population sizes' variations for the tested algorithms during the optimization process are shown in Fig. 6. The impact of varying the population sizes throughout generation number was evident from the total number of function evaluations. The comparison of the results obtained with the adapted and classic DE algorithms by the different sets and self-adaptive control parameters F and CR are shown in Table 2. The experimental results obtained by the classic DE algorithm, but with the fixed population sizes and different fixed control parameters F and CR are shown in Table 3. When the results in Tables 2 and 3 for the adapted and classic DE algorithms are compared, it can be seen that the adapted algorithm provided better final results with at least a 3-times lower total number of function evaluations.
Deviationa in %
5.1. Test model I
Hour
8
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
9
Table 5 Optimal hydro generations and hydro discharges for test model II obtained with the proposed algorithm. Hour
1 2 3 4 5 6 7 8 9 10 11 12
Hydro generation in MW h for HPP no.
Hour
1
2
3
4
91.17 82.81 76.59 83.93 76.19 64.88 67.83 68.13 74.41 79.78 81.33 60.17
59.88 64.98 55.50 69.02 57.48 55.17 51.00 51.57 57.99 72.07 64.08 54.66
0.00 0.00 0.00 0.00 34.29 0.00 38.01 37.67 40.36 41.23 40.66 41.36
201.44 187.76 173.46 156.52 178.46 198.55 216.97 233.69 236.63 259.33 250.29 249.29
13 14 15 16 17 18 19 20 21 22 23 24
Hydro generation in MW h for HPP no.
Hour
1
2
3
4
75.93 65.84 76.14 61.07 82.68 85.36 69.11 74.35 73.17 89.91 85.54 92.61
63.84 55.66 67.99 64.77 70.84 76.11 63.56 74.91 76.52 83.29 75.48 80.95
36.92 34.96 42.70 35.66 41.87 44.63 45.29 48.43 49.28 50.98 52.57 0.00
259.68 249.89 260.36 262.73 288.81 288.06 268.39 288.64 284.34 286.94 286.32 288.06
1 2 3 4 5 6 7 8 9 10 11 12
Hydro discharge 104 m3 of reservoir no. 1
2
3
4
11.44 9.48 8.36 9.99 8.64 6.83 7.22 7.17 8.05 8.84 8.95 5.75
7.61 8.53 6.77 9.02 6.98 6.60 6.00 6.00 6.83 9.21 7.68 6.12
29.97 29.82 30.00 29.93 16.16 25.12 14.83 15.14 13.78 13.30 14.09 14.47
13.17 13.03 13.00 13.00 13.00 13.01 13.00 13.00 13.00 14.35 13.27 13.00
Hour
13 14 15 16 17 18 19 20 21 22 23 24
Hydro discharge 104 m3 of reservoir no. 1
2
3
4
7.75 6.27 7.56 5.65 8.41 8.84 6.59 7.26 7.11 9.69 8.94 10.21
7.43 6.01 7.74 7.17 8.24 9.53 7.39 9.51 9.95 12.01 10.46 15.00
17.18 17.62 14.97 17.44 15.14 13.72 13.97 11.67 10.00 10.07 10.00 0.00
14.07 13.06 14.12 14.36 17.48 17.34 14.97 17.42 17.10 18.03 18.65 20.71
An adapted DE algorithm for dynamic population sizes is proposed for optimal dispatching of cascade HPPs by considering all constraints and the final states of the reservoirs. The classic DE algorithm has control parameters which are selected by the user. Choosing the proper values for all control parameters can be a challenging task because different control parameters have different impacts on the functions being optimized. Therefore the
control parameters F and CR are used as self-adaptive throughout the generation number. In order to decrease the number of function evaluations and consequently to decrease the convergence times, this paper presented a novel approach which dynamic population size. In order to ensure the missing population's members when expending it's the proposed algorithm uses a novel RSMM strategy. This strategy ensures new members by using parallel processing of DE algorithm's steps on multi-core processors. This implementation enables global search with fast convergence. The results show that the proposed adapted DE algorithm by parallel processing and self-adaptive F and CR with RSMM strategy achieves satisfactory system demand by decreasing the water quantity used per electrical energy unit produced and also the number of function evaluations in comparison with the other tested algorithm. The proposed algorithm is appropriate for use in the real-time applications. This is evident from the results section where the algorithm's results are shown for the application on real parameter's model as well as on standard test model used in many scientific publications.
Fig. 7. Objective function values throughout the generations for test model II.
Fig. 8. Population size variations throughout the generations for test model II.
results from Tables 6 and 7, it is obvious that the adapted algorithm achieved better objective function values at a lower number of function evaluations. Consequently, in comparison to other algorithms this means that the optimal solution is obtained at lower CPU times. The adapted algorithm with the dynamic population size on interval [1D, 10D] obtained better objective function value comparing to interval [1D, 5D]. However, these values are relatively close to each other, while the numbers of function evaluations are notably lower for narrower interval.
6. Conclusion
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11
10
Table 6 Experimental results for test model II averaged over 10 independent runs of the modified and classic DE by different sets and self-adaptive F and CR. Algorithm
Proposed modified DE
Population size
Fixed
Type F, CR Best objective value [p.u.] nfevala CPU time [s]
1D 3D Self-adaptive 0.20929 0.20884
5D
[1D, 5D]
0.20851
460,320 37
2,301,600 104
a
Classic DE Dynamic
1,380,960 63
Reduction by Brest et al. [36]
Fixed
Reduction by Brest et al. [36]
[1D, 10D]
5D/2.5D/1.25D
1D
3D
5D
5D/2.5D/1.25D
0.20818
0.20810
0.20821
0.21629
0.21077
0.21103
0.20846
1,021,674 69
1,728,666 100
1,437,600 86
460,320 10
1,380,960 27
2,301,600 56
2,301,600 65
Number of function evaluations.
Table 7 Experimental results for test model II averaged over 10 independent runs of the classic DE by fixed population size and different F and CR values. Algorithm
Classic DE
Population size
Fixed
Type F, CR Best objective value [p.u.] nfevala CPU time [s]
1D 0.1, 0.1 0.21711
3D
5D
0.21306
460,320 9
1,380,960 21
a
3D
5D
0.21629
1D 0.5, 0.5 0.27155
3D
5D
0.27378
1D 0.5, 0.9 0.21959
0.27108
2,301,600 41
460,320 14
1,380,960 37
3D
5D
0.23042
1D 0.9, 0.9 0.37816
0.22244
0.40844
0.39627
2,301,600 70
460,320 10
1,380,960 31
2,301,600 66
460,320 20
1,380,960 31
2,301,600 70
Number of function evaluations.
References [1] Olabi AG. State of the art on renewable and sustainable energy. Energy 2013;61:2e5. [2] Ferreira HL, Garde R, Fulli G, Kling W, Lopes JP. Characterisation of electrical energy storage technologies. Energy 2013;53:288e98. [3] Nazari ME, Ardehali MM, Jafari S. Pumped-storage unit commitment with considerations for energy demand, economics, and environmental constraints. Energy 2010;35:4092e101. [4] Ostergaard PA. Reviewing optimisation criteria for energy systems analyses of renewable energy integration. Energy 2009;34:1236e45. [5] Olabi AG. The 3rd international conference on sustainable energy and environmental protection SEEP 2009 e Guest Editor's Introduction. Energy 2010;35:4508e9. [6] Carton JG, Olabi AG. Wind/hydrogen hybrid systems: opportunity for Ireland's wind resource to provide consistent sustainable energy supply. Energy 2010;35:4536e44. [7] Lund H. Renewable energy strategies for sustainable development. Energy 2007;32:912e9. [8] Olabi AG. Developments in sustainable energy and environmental protection. Energy 2012;39:2e5. [9] Catalao JPS, Pousinho HMI, Contreras J. Optimal hydro scheduling and offering strategies considering price uncertainty and risk management. Energy 2012;37:293e8. [10] Dursun B, Alboyaci B, Gokcol C. Optimal wind-hydro solution for the Marmara region of Turkey to meet electricity demand. Energy 2011;36:864e72. [11] Yuan X, Wang L, Yuan Y. Application of enhanced PSO approach to optimal scheduling of hydro system. Energy Convers Manag 2008;49:2966e72. [12] Narang N, Dhillon JS, Kothari DP. Multiobjective fixed head hydrothermal scheduling using integrated predatoreprey optimization and Powell search method. Energy 2012;47:237e52. [13] Qin H, Zhou J, Lu Y, Wang Y, Zhang Y. Multi-objective differential evolution with adaptive Cauchy mutation for short-term multi-objective optimal hydrothermal scheduling. Energy Convers Manag 2010;51:788e94. [14] Wang Y, Zhou J, Mo L, Zhang R, Zhang Y. Short-term hydrothermal generation scheduling using differential real-coded quantum-inspired evolutionary algorithm. Energy 2012;44:657e71. [15] Zhang H, Zhou J, Fang N, Zhang R, Zhang Y. Daily hydrothermal scheduling with economic emission using simulated annealing technique based multiobjective cultural differential evolution approach. Energy 2013;50:24e37. [16] Basu M. Artificial immune system for fixed head hydrothermal power system. Energy 2011;36:606e12. [17] Farhat IA, El-Hawary ME. Optimization methods applied for solving the shortterm hydrothermal coordination problem. Electr Power Syst Res 2009;79: 1308e20.
[18] Catalao JPS, Pousinho HMI, Mendes VMF. Hydro energy systems management in Portugal: profit-based evaluation of a mixed-integer nonlinear approach. Energy 2011;36:500e7. [19] Wong SYW. Hybrid simulated annealing/genetic algorithm approach to shortterm hydro-thermal scheduling with multiple thermal plants. Electr power & Energy Syst 2001;23:565e75. [20] Yu B, Yuan X, Wang J. Short-term hydro-thermal scheduling using particle swarm optimization method. Energy Convers Manag 2007;48:1902e8. [21] Soares J, Silva M, Sousa T, Vale Z, Morais H. Distributed energy resource shortterm scheduling using signaled particle swarm optimization. Energy 2012;42: 466e76. [22] Wood AJ, Wollenberg BF. Power generation, operation and control. 2nd ed. New York: Wiley; 1996. [23] Martins LSA, Soares S, Azevedo AT. A nonlinear model for the long-term hydro-thermal generation scheduling problem over multiple areas with transmission constraints. Power Systems Conference and Exposition; 2009. pp. 1e7. [24] Michalewicz Z, Schoenauer M. Evolutionary algorithms for constrained parameter optimization problems. Evol Comput 1996;4:1e32. [25] Khanmohammadi S, Amiri M, Haque MT. A new three-stage method for solving unit commitment problem. Energy 2010;35:3072e80. [26] Christober C, Rajan A. Hydro-thermal unit commitment problem using simulated annealing embedded evolutionary programming approach. Electr Power Energy Syst 2011;33:939e46. [27] Hong YY, Lin JK. Interactive multi-objective active power scheduling considering uncertain renewable energies using adaptive chaos clonal evolutionary programming. Energy 2013;53:212e20. [28] Yuan X, Zhang Y, Wang L, Yuan Y. An enhanced differential evolution algorithm for daily optimal hydro generation scheduling. Comput Math Appl 2008;55:2458e68. [29] Caponetto R, Fortuna L. Chaotic sequences to improve the performance of evolutionary algorithms. IEEE Trans Evol Comput 2003;3:289e304. [30] Lu Y, Zhou J, Qin H, Wang Y, Zhang Y. An adaptive chaotic differential evolution for the short-term hydrothermal generation scheduling problem. Energy Convers Manag 2010;51:1481e90. [31] Lakshminarasimman L, Subramanian S. Short-term scheduling of hydrothermal power system with cascaded reservoirs by using modified differential evolution. Generation, Transm Distrib, IEE Proc 2006;153:693e700. [32] Storn R, Price K. Differential evolution e a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 1997;11:341e59. [33] Mandal KK, Chakraborty. Differential evolution techniques-based short-term economic generation scheduling of hydrothermal systems. Electr Power Syst Res 2008;78:1972e9. [34] Mandal KK, Chakraborty N. Parameter study of differential evolution based optimal scheduling of hydrothermal system. Hydro-Environ Res 2013;7:72e80.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004
A. Glotic et al. / Energy xxx (2014) 1e11 [35] Brest J, Greiner S, Boskovi c B, Mernik M, Zumer V. Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans Evol Comput 2006;10:646e57. [36] Brest J, Maucec MS. Population size reduction for the differential evolution algorithm. Appl Intell 2008;29:228e47. [37] Vahidinasab V, Jadid S. Joint economic and emission dispatch in energy markets: a multiobjective mathematical programming approach. Energy 2010;35:1497e504.
11
[38] Mazur DR. Combinatorics: a guided tour. United States of America; 2010. pp. 1e15. [39] Orero SO, Irving MR. A genetic algorithm modeling framework and solution technique for short term optimal hydrothermal scheduling. IEEE Trans Power Syst 1998;13:501e18. [40] Gamperle R, Muller SD, Koumoutsakos P. A parameter study for differential evolution. WSEAS International Conference on Advances in Intelligent Systems, Fuzzy Systems, Evolutionary Computation; 2002. pp. 293e298.
Please cite this article in press as: Glotic A, et al., Optimization of hydro energy storage plants by using differential evolution algorithm, Energy (2014), http://dx.doi.org/10.1016/j.energy.2014.05.004