A Chaotic Firefly Algorithm Framework for Non ...

A Chaotic Firefly Algorithm Framework for Non-Convex Economic Dispatch Problem Bachir Bentouati1, Saliha Chettih1, Ragab El-Sehiemy2 1

LACoSERE Laboratory, Electrical Engineering Department, Amar Telidji University of Laghouat, Algeria. 2 Electrical Engineering Department, Faculty of Engineering, Kafrelsheikh University, Egypt

Abstract The aim of economic dispatch (ED) problem is to provide an efficient utilization of energy resources to produce economic and secure operating conditions for the planning and operation of a power system. ED is formed as a nonlinear optimization problem with conflicting objectives and it is subjected to both inequality and equality constraints. An efficient improvement of firefly algorithm (FA), a powerful metaheuristic method, has been introduced in this paper. FA is a bio-inspired optimization algorithm that is inspired by flashing patterns and behaviour of fireflies in nature, it has been introduced for solving non-convex economic dispatch problem due to valve-point effects. The proposed chaotic firefly algorithm (CFA) improvement is done by incorporating the chaos approach to FA algorithm for raising the global convergence speed and for enhancing its performance. The results show clearly the superiority of CFA in searching for the best cost value results when compared with well-known metaheuristic search algorithms. Keywords: firefly algorithm; chaotic maps; economic dispatch; valve-point effect Received: Dec, 29, 2016 To cite this article: Bachir Bentouati, Saliha Chettih, Ragab El-Sehiemy, "A Chaotic Firefly Algorithm Framework for Non-Convex Economic Dispatch Problem" in Electrotehnica, Electronica, Automatica (EEA), 2017, vol. 60, no. 1, pp. 172-179, ISSN 1582-5175.

1. Introduction In the last decades, planning and operation of modern power systems were more complex as result of the quick development of electricity demand, integration of networks and movement toward open markets in electricity around the world. The fundamentals of economic dispatch (ED) problem provides an optimum utilization of electrical power systems to produce economic and secure operating conditions for the planning and operation of a power system. Traditional ED has provided a tool to achieve such a task and has its first treaty as the cost of fuel only. Later, several technical and environmental targets were fused into the ED issue with the traditional economic objectives [1]. To optimize the objective functions, which include cost, the ED issue is dependent on two sorts of operational limitations. These limitations are recognized as equality and inequality constraints. The ED issue is a non-convex optimization problem, which requires huge computational effort if a large power system is considered. Essentially, the problem of ED is considered as static and non-linear that is the major operational functions of the innovative energy management framework. The approach of ED has gained more relevance in view of the increased availability of control devices and energy prices since its beginning point has demonstrated its efficiency in managing various issues. Recently, different techniques were explored in the literature to unravel the ED problem, which had been studied for over 20 years and many algorithms had been created to solve it. Various conventional optimization methods [1] were produced to tackle the ED problem as lambda-iteration method, gradient-method, linearprogramming-method and Newton’s method. Conventional

programming techniques are fast and reliable but often fail to have the best solution for solving highly complex nonlinear objective function. While applying the classical mathematical techniques, a generating unit of the fuel cost trademark is assumed to be smooth and to possess convex functions. These techniques are sensitive towards initial solutions and may fail due to initial improper values of variables. The practical power systems are difficult to solve using these classical mathematical techniques as result of their nonlinear attributes of limited operating zones, valvepoint effects, and piecewise quadratic cost functions. Therefore, an efficient strategy is highly required to deal with the non-convex, non-linear, and multi-modular power system problems. The drawbacks associated with these classical methods prompted the evolution of various artificial intelligence (AI) methods and their application to solve a practical ED problem. Although in general AI methods do not ensure the global ideal solution, they can produce feasible sub-optimal solutions with less computational time. Several AI techniques, like genetic algorithms (GAs) [2], particle swarm optimization (PSO) [2], differential evolution (DE) [3], cuckoo search (CS) [4], bat algorithm (BA) [5], teacherlearner-based-optimization (TLBO) [6], harmony search (HS) [7], artificial bee colony (ABC) [2], grey wolf optimizer (GWO) [2], biogeography-based optimization (BBO) [8], and flower pollination algorithm (FPA) [9] are documented in the literature for solving practical ED problem. Some attempts reported the use of hybrid approaches for solving ED problems, such as hybrid differential evolution with BBO [10], hybrid swarm intelligence based harmony search algorithm [11], hybrid genetic algorithm approach in view of differential evolution [12] and PSO embedded evolutionary programming technique [13], etc., to find the

ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ (EEA), 66 (2018), nr. 1 ideal allocation of power generation. The documented results showed a major improvement in accuracy when compared with the classical mathematical techniques. Therefore, applying techniques of metaheuristic to solve the complex ED problem for which deterministic mathematical technologies are not able to produce the required results is suggested. Optimization is becoming a field of inquiry for many researchers, especially since the competence of a particular system relies on gaining an optimal solution, which can be acquired by a good optimization method. This procedure finds the perfect solution after assessing the cost function, which indicates the association between the system framework and its constraints. Currently, metaheuristic algorithms have been formed in many areas such as hybridization, multi-objective version, binary version, training multi-layer perceptron, and improved several ways as Lévy flight, chaos theory, and genetic operator. Most of these improvements happened because the deterministic and evolutionary components are used [14]. A good combination of local and global search has intensive local exploration and global exploration [14]. In this paper, a chaotic optimization algorithm (COA) has been introduced based on firefly algorithm (FA) method. The COA is a type of random-based method using chaotic variables in place of random variables. Previously, many algorithms have been outlined with chaotic maps to solve different kinds of ED problems like chaotic sequence based DE [15], chaotic self-adaptive particle swarm optimization algorithm [16], chaotic ant swarm optimization (CASO) [17], chaotic differential bee colony optimization algorithm (CDBBC) [18] and chaotic bat algorithm (CBA) [19]. All these studies have indicated a better response and increased precision when compared with the original algorithm. An ED problem has more than one local optimum. While its selection may rely on an initial point result that has randomness nature, the optimal solution obtained may not really be the global optimum. Various adaptive and random variables are also integrated with FA to emphasize exploitation and exploration of the search space. Firefly algorithm (FA) was proposed by X.-S. Yang in 2010 [20] and because it performs well, many optimization strategies such as chaotic theory [21] quantum theory [22] and opposition-based learning [23] have been incorporated into it. Meanwhile, several state-of-the-art metaheuristic algorithms, such as particle swarm optimization [24], harmony search [25], krill herd [26] and pattern search [27] have been hybridized with the basic FA algorithm for the aim of further improving the performance of FA. Furthermore, FA is a novel population-based swarm intelligence algorithm [20] based on the flashing patterns and behaviour of fireflies. FA algorithm sometimes does not have the ability to avoid local optima [21]. For overcoming this problem, Gandomi et al. [21] proposed an improved algorithm by using chaotic sequences. Chaotic maps have been utilized to avoid local optima and get a global optimal solution, as well as less computational time to reach the optimum solution, local minima avoidance, and faster convergence, which make them appropriate for practical applications for solving various constrained optimization problems. In this study, an approach based on CFA followed by its mathematical model is proposed to solve the ED problem with the aim of minimizing the non-convex cost functions. In order to fulfil this task, the CFA method is simulated and tested on 5, 13 and 40 generating-units. The obtained

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results are compared with other relevant methods that have been reported in the literature. The rest of paper is structured as follows: The next Section describes the problem formulation of ED, while the presentation of CFA and its algebraic equations are given in Section 3. Section 4 shows the results of simulation and discussion. Section 5 contains the conclusion of this paper. 2. The formulation of ED problem The ED problem deals with the optimization of generation output of every available generating unit so that the general cost of generation of electric power is minimized while the constraints linked with the system is satisfied. The problem objective function and constraints are outlined as follows. 2.1 Objective function The objective of this problem is to lower the total fuel cost, which is formulated as the summation of the cost that is incurred on each available generating unit and it is expressed as: n

FC 

 F P  i

(1)

i

i 1

where Fi(Pi) is a fuel cost of the unit ith generating unit, and n is the total number of generating units. It represents the simplest cost characteristic function of a generating unit of an ED problem that can be expressed as a single quadratic cost function:

 a P n

Fi (Pi) 

i i

2

 bi Pi  ci



(2)

i 1

where Pi is the real power generation of the ith generator unit i. ai , bi , and ci are fuel cost coefficients of generating for unit i in [$/MW2h], [$/MWh] and [$/h], respectively, and n is the total number of generating units. In thermal power plants, the generating units power output is controlled by multiple valves. When inlet steam valve is opened, a sudden rise in the losses is observed which gives rise to the formation of ripples in the cost characteristic curve. The occurrence of this phenomenon is called valve-point loading effects. This accounts for the inclusion of multiple non-differential points in the cost characteristic function and thus, turns it into a non-smooth function. It shall be expressed as a quadratic and a sinusoidal function given below:

 a P n

Fi (Pi) 

2

i i



 bi Pi  ci  d i  (ei  ( Pgimin  Pgi )

(3)

i 1

where di and ei are the coefficients that represent the min

valve-point loading effects and Pi is the minimum output power generation of the ith generating unit. 2.2 Constraints The power balance and generation limits of different units without taken into account transmission line losses are considered for obtaining optimal power dispatch in this work. The summation of total power developed by each generating unit must be equal to the total load demand Pd

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there should be power balance in the system as given below.

P  P i

d

(4)

where, PG is the total power generation of the system,

Pd is the total demand of the system, and PL is the total transmission loss of the system. The transmission loss PL can be expressed as: n

has

three

important

The attractiveness of a firefly by another firefly can be represented by the function β(r) using the formula:

 PL

i 1

PL 

algorithm

Attractiveness:

n

PG 

The firefly characteristics:

n



n

Pi Bij Pj 

i 1 j 1

B

0i Pi

 B00

(5)

i 1

where, Pi and P j are the real power generated by the ith and jth generation unit, Bij , B0i , and B00 are the transmission loss coefficients. The optimum power generation of a generating unit must be within its minimum and maximum power generation.

Pgimin  Pgi  Pgimax min

where Pgi

(6) max

and Pgi

are the minimum and maximum

real power generations of the ith generating unit, respectively. 3. Firefly algorithm Firefly Algorithm is a novel swarm intelligence global optimization technique developed by Yang [20]. It is classified as a stochastic, nature-inspired, meta-heuristic algorithm for solving NP-hard problems. It is based on the flashing lights of fireflies in nature. Fireflies are interesting insects due to their spectacular courtship displays. They are characterized by their flashing lights emitted through a process known as bioluminescence, which serve as courtship signals for mating and warning of potential predators. The firefly algorithm simulates the social behaviour of a population of fireflies. Each firefly has brightness (or intensity) of its emitting bioluminescent light. A firefly searches for partners in its environment, and is attracted by other brighter fireflies. The attractiveness of a Firefly to another Firefly depends on the brightness of the latter and the distance between the two. Once a particular brighter firefly is targeted, the searching firefly moves towards it, hence updating its position and brightness based on an objective function. Ultimately, the fireflies in a given population will converge to the brightest fireflies.  The three main rules used to construct the standard algorithm are the following:  The sex of the fireflies is unimportant, such that any firefly can be attracted to any other brighter one independent of her sex;  The attractiveness of a firefly is specified by its brightness, which is specified by its inherent objective function;  The attractiveness of a firefly is directly proportional to its brightness but decreases with distance. If there were no brighter than a particular firefly, it would randomly move.

 (r )   0 e  r

2

(7)

where β0 is the attractiveness (brightness or intensity) at origin, r is the distance between two fireflies, and γ is the coefficient of light absorption which depends on the fireflies environment. Distance: The distance between two fire flies i and j being at positions Xi and Xj respectively can be calculated using the formula:

rij  X i  X

j





d k 1

( xi ,k  x j ,k ) 2

(8)

where xi ,k is the k-th component of the i-th firefly (Xi). X is a vector representing a firefly’s position in a ddimensional space:

X  ( x1 , x 2 ,..., x d )

(9)

Movement: When a firefly moves, it changes its position. The movement of a firefly i being attracted by a brighter firefly j can be calculated using:

X i  X i  0e

rij2

1  ( X j  X i )    rand   2 

(10)

where the second term is the increment due to attractiveness and the third term induces stochasticity,  is the randomization parameter and rand - a random real number between 0.0 and 1.0 inclusive. The Standard Firefly Algorithm [20] is shown in Figure 1.

Firefly Algorithm begin Define the objective function f(x); x = (x1, …, xd) in d-dimension Generate initial population of fireflies xi, (i = 1, 2, …, n) Determine the light intensity Ii = f(xi) for each firefly xi for t =1 : MaxGenerations for i = 1 : n (all n fireflies) for j = 1 : n (all n fireflies) if (Ij > Ii) Move firefly i towards j in d-dimension end if Attractiveness varies with distance r via exp-γr Evaluate new solutions and update light intensity end for j end for i Rank the fireflies and find the current best end for t Post process results and visualization end Figure 1. Firefly algorithm

ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ (EEA), 66 (2018), nr. 1 3.1 Chaos Firefly Algorithm In any optimization algorithm, random solutions are first generated. Then, the initial random solutions are discovered fitness functions based on predefined maximum iterations. The search process in all metaheuristic algorithms is similar and can be divided into two phases: exploration and exploitation. The exploration phase finds out a search space as largely as possible, providing more solutions in the process. The exploitation phase, on the other hand, refers to the convergence towards the global optimum as quickly as possible. Exploration and exploitation are two complex factors used to obtain global optimum value when the balance between both phases represents an accurate solution in a search space. However, a real challenging optimization problem in economic load dispatch problem is the unknown search space. Therefore, we cannot find a transient time between exploration and exploitation. Several random and adaptive variables are also integrated with FA to focus on exploration and exploitation of the search space. An ED problem has more than one local optimum, the result may depend on the selection of an initial point that

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has randomness nature, and the obtained optimal solution may not necessarily be the global optimum. The chaotic maps (CM) employment is one of the effective approaches to adjust some of the static parameters in metaheuristic algorithms and conquer early convergence. This domain has swiftly grown up to be a new research area in the recent optimization literature. Chaos is one of the mathematical approaches that have recently been used to improve both exploration and exploitation [21]. Chaos theory is part of the best methods to promote the evolutionary algorithms performance in terms of avoidance of local optima and convergence speed. There is no random component in any chaos. In [21], twelve chaotic maps are used to enhance the performance of the FA algorithm and tuning the attractiveness coefficient,  , of FA. The best chaotic FA is a combination of Gauss map in place of  , as is clear in [21] and is used to solve the problem of ED in this paper. For further explanation, the flowchart of the proposed application of CFA algorithm to solve the problem is shown in Figure 2.

Figure 2. Flowchart of the application of CFA algorithm for the ED problem.

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4. Application and results To elucidate the robustness, performance, and applicability of CFA, three different case studies are implemented for solving non-convex ED problem. The outcome obtained have been contrasted with different well-known algorithms suggested in the literature. Metaheuristic algorithms are always based on certain stochastic distribution. Therefore, independent runs of about 50 are implemented to get the most representative results.

The suggested algorithm was developed and implemented using MATLAB R2013b and computed under Intel (R) Core (TM), 2.40 GHz computers with 8 GB RAM. Case Study 1: 5-Unit System The small test power system contains 5-generating unit is considered. The attributes of the five thermal units such as fuel cost coefficients and generation limits are given in [2]. The results of 5-generating units systems is tested for load demand of 730 MW and effectiveness of CFA for 5generating unit system is compared with lambda iteration method [2], Genetic Algorithm (GA) [2], Particle Swarm Optimization [2], APSO [2], Artificial Bee Colony (ABC) [2] and Evolutionary Programming (EP) [2] as shown in Table 1.

Table 1. Economic Load Dispatch Problem for 5-Generating Units (Load Demand=730 MW) Method Lambda Iteration GA PSO APSO EP ABC CFA

Load Demand 730 MW 730 MW 730 MW 730 MW 730 MW 730 MW 730 MW

U1 218.028 218.0184 229.5195 225.3845 229.803 229.5247 229.520

U2 109.014 109.0092 125 113.02 101.5736 102.0669 102.992

From this table, we observed that it was respecting the inequality constraint, and that all of the powers generated belongs to the interval [Pmin, Pmax]. These results corroborate the applicability of the suggested approach, which have the ability to reach the global optimum for the cost function. Depending on these results, it can be stated that FA approach with the integration of chaos map can produce feasible and good solutions.

U3 147.535 147.5229 175 109.4146 113.7999 113.4005 112.674

U4 28.38 28.37844 75 73.11176 75 75 75

U5 272.042 227.0275 125.4804 209.0692 209.8235 210.0079 209.816

Cost ($/hr) 2412.709 2412.538 2252.572 2140.97 2030.673 2030.259 2029.666

Figure 3 presents the CFA convergence; from this figure, the competence of the approach is mainly shown in a rapid convergence with a global solution.

Case Study 1: 13-Unit System This medium system comprises of 13-generating units with valve-point loading as given in [1]. Therefore, this system has non-convex solution spaces and many local minima as result of valve-point effects. The detail of this system as shown is gotten from [1]. The results of 13-generating units systems is tested for load demand of 2520 MW, as shown in Table 2. Table 2. Economic Dispatch Problem for 13-Generating Units [load demand=2520 MW] ED using CFA Unit Power(MW) No. 1 628,317897 2 299,198008 3 294,559296 4 159,731991 5 159,733283 6 159,732778 7 159,732042 8 159,732991 9 159,732199 10 77,3960051 11 77,3461823 12 92,388952 13 92,3983753

Test System- 2 Comparison with others algorithms Method

Cost ($/hr)

SA[28] GA[28] GA-SA[28] EP-SQP[28] PSO-SQP[28] UHGA[12] GA-MU[29] IGAMU[29] ACO[30] HGA[12] EDSA[29]

24970.9 24398.2 24275.7 24266.4 24261.0 24172.2 24170.7 24169.9 241169 24169.9 24169.9

CFA

24164.18

These results are in comparison with other algorithms. The proposed CFA offer a lower cost than GA, GA-SA, EPSQP, PSO-SQP, UHGA, GA-MU, IGAMU, ACO and EDSA while respecting the constraints of the system.

Figure 3. Convergence characteristics of system 2

These results because of the perfection of FA global search by CM, which worthily accomplished the best solution. In addition, CM presides the algorithm to converge quickly to its global optimality without improvisation and outfits a switch ploy for FA mechanism in the exploration phase that prohibits to be stuck in fake solution. Case Study 3: 40-Unit System A large system, which contains 40 generating unit was considered in this case study. In the system, the cost function is non-convex because of valve-point effects and the global minimum is very difficult to ascertain. The loading effect coefficients of the valve-point is contained in [18]. The data of the system are listed in [18]. The power demand (PD) used is 10500 MW. After the proposed approach was applied, all of the agents reached (Pi) consensus on the solution as shown in Table 3 under CFA.

ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ (EEA), 66 (2018), nr. 1 Table 3. Results obtained by CFA for test system 3 Unit 1 2 3 4 5 6 7 8 9 10 11 12 13

Generation 110,110995 114 119,713715 187,023563 82,1859346 81,4774122 300 300 297,813607 136,059592 361,427906 154,683398 125

Unit 21 22 23 24 25 26 27 28 29 30 31 32 33

Unit 14 15 16 17 18 19 20

Generation 550 550 550 550 550 419,855184 11,2303944 10 10 97 168,825151 109,055185 177,22634

Generation 419,649934 252,370838 312,312144 488,051542 380,431059 550 550 Cost ($/hr)

Unit 34 35 36 37 38 39 40 121413.62

177 Generation 200 200 200 104,600713 63,3314487 106,563912 550

Table 4 displays a comparison between the results obtained using the CFA and other algorithms in the literature for the 40 units’ system.

Table 4. Comparison of fuel costs obtained with different algorithms for test system 3 Best fuel cost ($/hr) CFA 121413,62 CBA [19] 121,412.5468 SQPSO [31] 121,412.57 MABC [32] 121,412.5918 DE [33] 121,412.68 CSO [34] 121,461.6707 EP-SQP [13] 122,323.97 PSO [35] 121,735.47 BBO [8] 121,426.95 BF [36] 121,423.63 Algorithm

Mean fuel cost ($/hr) 121414,234 121,418.9826 121,455.7 121,431.5763 121,439.89 121,936.1926 122,379.63 122,513.91 121,508.03 121,814.94

The smallest total fuel cost is obtained by the CBA and SQPSO, which gives us 121,412.5468 $/hr and 121,412.57 $/hr, respectively. Moreover, we can note that on the average the CFA algorithm performs better than the CBA and SQPSO algorithms, the lowest maximum value is supplied by the CFA while the highest maximum value is produced by the SQPSO algorithm. The results show that SQPSO solutions are very sensitive and more volatile. The average execution time of the CFA for this test system is 1.345s, which is again a good computation time when compared with the other methods. Additionally, it can be noticed that the effectiveness of the CFA approach appears in the speed and the smooth convergence of the algorithm, resulting in the global optimum as shown in Figure 4.

Figure 4. Convergence characteristics of system 3

Max. fuel cost ($/hr) 121414,758 121,436.15 121,709.5582 121,493.1885 121479.63 NA NA 123,467.40 121,688.66 NA

Standard deviation 0.569592 1.611 49.8076 18.16 NA 32 NA NA NA 124.876

Time (s) 1.354 1.55 47.24 1.92 min 31.5037 s NA NA NA 11.74 NA

5. Conclusion The Gauss chaotic map has been integrated with FA algorithm to regulate the attractiveness coefficient,  , of the FA and mainly enhance the performance of the standard FA. Chaotic firefly algorithm (CFA) has the ability to find an optimum solution for the economic dispatch problem. The problems have been derived from three test cases consisting of 5, 13 and 40-generating units. The conclusions and findings of this work can be summarized as follows:

The proposed CFA based approach is found to be simple, robust and easy to understand, and has no difficulty in when to tune the parameters, and hence, it can be applied to any complex computational problem. A comparison of the results of the CFA algorithm with those by well-known optimization algorithms shows the robustness of the suggested method. It can be implied that chaotic map is efficient in producing a feasible set of solutions and increases the convergence rate with less improvisation. CPU times of the proposed CFA algorithm are shorter than other algorithms, which due to the integration of chaos map is mainly one of the CM advantages. Thus, the advantages linked with the suggested algorithm applied to the solution of the ED problem can be listed as: The combination of chaotic maps with FA confers early convergence; The chaotic maps (CM) ensure best performance thus best results; CM generates another search space distinguished from the original FA operations; there are different chaotic maps, each of which has its own property according to its formulation. For future scope, we have used chaos theory, with different recent devolved and efficient algorithms and the FA will be combined with other search strategies such as opposition-based learning, Levy flights strategy, Adaptive

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Acknowledgment Authors would like to thank the referees for suggestions that have improved the content of this work. Special thanks for Dr. GaiGe Wang for his valuable comments and support.

ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ (EEA), 66 (2018), nr. 1 Biography Bentouati Bachir was born in Laghouat, Algeria,on August 22, 1990. He received license and master degrees in Electrotechnic and Electrical Power System in 2010 and 2012 respectively from Laghouat University. He is Member in LACoSERE Laboratory, University of Laghouat, Algeria. His areas of research include optimal power flow, Artificial intelligence and optimization techniques. e-mail: [email protected] Chettih Saliha was born on 11 may 1971 in Laghouat/Algeria. She earned his PhD degree in Electro-technology from the university of Amar Telidji. She joined the faculty of the Technology at the University of Amar Telidji, Laghouat (Algeria), in 2008 where she is currently a professor.

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Her research interests optimal power flow, Artificial intelligence and optimization techniques. Ragab El- Sehiemy was born at Minoufiya, 1973. He received the B.Sc., M.Sc., and Ph.D. degrees in 1996, 2005, and 2008, respectively. He is an Associate Professor in the Department of Electrical Engineering, Faculty of Engineering, Kafrelsheikh University, Kafr elSheikh, Egypt. His research interests involve power system operation, control, and planning, applications of modern optimization techniques for variant electric power systems applications, renewable sources, and smart grid.