Effective Method for Wind and Solar Power Grid ...

Kimura, K. et al.

Paper:

Effective Method for Wind and Solar Power Grid Systems Based on Recurrent Neural Networks Keisuke Kimura, Takayuki Kimura, Takefumi Hiraguri, and Kenya Jin’no Nippon Institute of Technology 4-1-1 Gakuendai, Miyashiro, Minami-Saitama, Saitama 345-8501, Japan E-mail: {e1092204@estu., tkimura@, hira@, jinno@}nit.ac.jp [Received October 14, 2013; accepted June 18, 2014]

In this paper, the control method based on recurrent neural networks is proposed for optimizing large-scale wind and solar power generation systems. Recently, an optimal control method based on recurrent neural networks was proposed for wind and solar power generation systems. In this method, optimization problems are regarded as linear programming problems, which are solved by recurrent neural networks. Results suggest that this control method based on recurrent neural networks could be implemented in realworld systems. However, only small power generation systems were used to evaluate this control method in previous studies. Then, the method for power generation systems is evaluated by more realistic conditions. The results of our numerical experiments show that this control method delivers high performance with large-scale power generation systems. Furthermore, if the power generation systems has specific topologies, almost 20% of the supplying capacity is improved.

Keywords: complex networks, linear programming problems, recurrent neural networks, renewable energy systems

1. Introduction The development of a safe and secure society demands reductions in the use of conventional energy systems such as fossil fuel energy plants and nuclear power plants. A key technology that will facilitate the development of an ecologically sound society is the introduction of power generation systems based on renewable energy. Thus, power generation systems based on renewable energy have been studied intensively recently [1, 2]. However, the power generation systems based on renewable energy have major drawbacks because the availability of electric power depends on the environmental conditions. Thus, a sophisticated control method is required to use these power generation systems safely and reliably. To control the power generation systems including the renewable energy systems, Sharafi et al. proposed the control method for the power generation systems including the renewable energies by the particle swarm opti1034

mizer [3]. Zeng et al. proposed the control method by the improved genetic algorithm to reduce the cost of the power grid. Then, the quality of the electric power including the renewable energy systems is optimized [4]. The above optimization methods by the meta-heuristics can search the solution very fast but can not search exact solution. Recently, Gamez et al. proposed a control method based on recurrent neural networks for wind and solar power generation systems [5]. The problems of optimizing wind and solar power generation systems were treated as linear programming problems. These linear programming problems were then solved using recurrent neural networks. This control method is efficient for wind and solar power generation systems [5]. However, the control method by the recurrent neural networks [5] has been evaluated for the system in which only one customer is included. Then, in this study, we evaluated the control method by the recurrent neural networks [5] for the extended systems where the plural wind and solar power generation systems are connected, namely, the power gird systems. In addition, we newly define an evaluation parameter for showing an improvement rate of all the demands in the power grid. We constructed power grids for wind and solar power generation systems using complex network theory: star topology type networks and Watts and Strogatz type small-world networks [6]. The structure of the power grid is revealed by the complex networks analysis that the power gird has strong clustering property and small distance between any nodes. Namely, the power grid is one of the small-world networks [6]. Further, Pangani et al. shows that the small-world networks appear to have many appropriate characteristics according to a set of topological metrics defined for the power grids [7]. From above view points, we realize the power grid by the networks which has small-world property. Obtained results of our numerical simulations showed that this control method based on recurrent neural networks delivers excellent performance even if realistic conditions are introduced. This paper is organized as follows: in Section 2, we describe the optimization problems for wind and solar power generation systems [5]. In Section 3, we consider the linear programming problems for wind and solar power generation systems. In Section 4, we demonstrate the perfor-

Journal of Advanced Computational Intelligence and Intelligent Informatics

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Effective Method for Wind and Solar Power Grid Systems

power supplied to the utility grid so the power grid system can operate as cheaply as possible. Thus, the objective function for wind and solar power generation systems is defined as follows: min CT =

T

∑ FC(PG,t ).

. . . . . . . . . (1)

t=1

In Eq. (1), CT is the total energy cost, FC is the cost of the utility grid, T is the number of iterations, PG,t is the output power from the utility grid at the t-th time point. In addition, the right-hand size of Eq. (1) is redefined as: Fig. 1. Schematic diagram of a wind or solar power generation system.

mance of the proposed method for the house sizes of the wind and solar power generation system. In Section 5, we evaluate the proposed method for grid type wind and solar power generation systems. The conclusion is presented in Section 6.

2. Optimization Problems for Wind and Solar Power Generation Systems The problems of optimizing wind and solar power generation systems can be considered as linear programming problems, which may be solved using recurrent neural networks [5]. The simplex method [8] is the most well known and powerful tool for solving linear programming problems, but the number of variables increases if the size of the problem becomes large [9]. The interiorpoint method [10] is another algorithm that can handle large-scale linear programming problems. However, it is not clear whether the interior-point method is an effective algorithm [9]. Further, many real-time systems, such as massively interconnected electric power grids, require solving the large-scale linear programming problems in the real time. One of the key approaches is to use the neural networks for the optimization problem. Because of the inherent massive parallelism, the neural network can solve the large scale linear programming problems faster than the other algorithms. For wind and solar power generation systems, the optimal operation should be implemented as quickly as possible, depending on the environmental conditions. In such a system, sequential algorithms such as the simplex method are not efficient for solving linear programming problems rapidly. Thus, an optimization method based on recurrent neural networks was proposed [5]. Figure 1 shows a schematic diagram of a wind power generation system [5]. The system comprises a utility grid, a photovoltaic power generation system, a wind power generation system, a battery, and an electric vehicle. The first objective during the optimization of wind and solar power generation systems is to satisfy the demands of the customers of the power generation system, while the second objective is to minimize the total electric Vol.18 No.6, 2014

FC(PG,t ) = CG,t · PG,t , where CG,t is the cost of the utility grid at the t-th time point. In this study, we deal in optimization problem at the t-th time point, i.e., PG,t = PG and CG,t = CG . Furthermore, we set the following constraints on each power generator: PS + PW + PG + PB + PEC = L, . . . . . . . (2) where PS is the output power of a photovoltaic power system, PW is the output power of the wind power system, PB is the storage power of the batteries, PEC is the storage power of electric vehicles, and L is the power demand of customers. The output power of each power generator is defined as follows [5]. •

Solar power generation systems The maximum output power by the solar power generation system, PSMax , is defined as, ⎧ η ⎨ c (G)2 0 < G < Gstd , (3) PSMax (G) = Gstd ⎩ ηc G Gstd ≤ G, where ηc is the energy conversion efficiency of a solar cell, G is the solar radiation energy, and Gstd is the output power in the standard environment.

•

Wind power generation systems The maximum output power by the wind power generation system, PW Max , is defined by, ⎧ v < v1 or v ≥ v3 , ⎪ ⎨ 0 2 3 (4) PW Max (v) = 0.5 C p ρπ r v v1 ≤ v < v2 , ⎪ ⎩ Pr v2 ≤ v < v3 , where C p is the coefficient of the generator, ρ is the air density, r is the radius of blades of the generator, v is the wind speed, v1 is the cut-in wind speed, v2 is the rated wind speed, v3 is the cut-out wind speed, and Pr is the constant output power.

•

Battery power and electric vehicle power To avoid over-discharging and over-charging, we set constraints when charging the electric power supplied for a battery and an electric vehicle. The storage power of these generators is defined by, PBMin ≤ PB ≤ PBMax ,


. . . . . . . . (5) 1035

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PECMin ≤ PEC ≤ PECMax . . . . . . . . (6) In Eq. (5), PBMin is the minimum power of the storage battery and PBMax is the maximum power of the storage battery. In Eq. (6), PECMin is the minimum power of the electric vehicle and PECMax is the maximum power of the electric vehicle. Then, PBMin and PECMin take the negative values when the battery and the electric vehicle are charged. •

The utility grid The output power of the utility gird is defined by, 0 ≤ PG ≤ PGMax , . . . . . . . . . . (7) where PGMax is the maximum power usage.

3. Linear Programming Problems for Wind and Solar Power Generation systems To solve the linear programming problems for wind and solar power generation systems, we use a recurrent neural network [11]. In general, the linear programming problem is defined as follows: min f (x) = cT x, s.t. Ax = b, 0 ≤ xi ≤ xiMax (i = 1, . . . , n), x∈R , c∈R , A∈R n

n

m×n

(8)

, b∈R . m

In Eq. (8), f (·) is the objective function, x is the n-dimensional column vector of the decision variables, xiMax is an n-dimensional column vector that decides maximum value of x, c is the coefficient vector of the objective function, A is the constraint coefficient matrix, and b is the constraint coefficient vector. The optimization problem for wind and solar power generation systems is defined as follows: min CG PG , s.t. PG + PW + PS + PB + PEC = L, min PG − PW − PS , max PW , max PS , min PB , min PEC , 0 ≤ PG ≤ PGMax , PBMin ≤ PB ≤ PBMax , PECMin ≤ PEC ≤ PECMax , 0 ≤ PW ≤ PW Max , 0 ≤ PS ≤ PSMax .

(9)

The objective function defined by Eq. (9) works to minimize the cost of the power from the utility gird, PG . Also, maximizing PW and PS should be considered to maximize 1036

the power from the renewable energies and minimizing PB and PEC , to possibly charge the batteries and the ones of the electric vehicles, to PG . To solve the linear programming problems by the recurrent neural networks, each variable must have the positive value. Then, the variables of PS , PW , PB , PEC and PG defined by Eq. (9) are replaced by ones defined by Eq. (10). In Eq. (10), PBout and PECout are the output power from the battery and the electric vehicle, PBin and PECin are the input power into the battery and the electric vehicle, PBMax and PECMax are the maximum output power from the battery and the electric vehicle, PBMin and PECMin are the maximum input power into the battery and the electric vehicle. In the numerical simulations, we set CG = 1. min CG PG , s.t. PG + PW + PS + PBout − PBin + PECout − PECin = L, min PG − PW − PS , min PB , min PEC , max PW , max PS , 0 ≤ PG ≤ PGMax , 0 ≤ PS ≤ PSMax , 0 ≤ PW ≤ PW Max , 0 ≤ PBout ≤ PBMax , (10) 0 ≤ PBin ≤ |PBMin |, 0 ≤ PECout ≤ PECMax , 0 ≤ PECin ≤ |PECMin |, PBout − PBin = PB , PECout − PECin = PEC .

In Eq. (10), PBout and PBin are the output and input power of the battery, respectively, while PECout and PECin are the output and input power of the electric vehicle, respectively. The values PBMin and PECMin in Eqs. (5) and (6) take the negative values and these negative values correspond to the charging mode of the batteries. However, if the recurrent neural network is applied to solve the linear programming problems, all of the variables have to the positive values. Then, the absolute values are added to PBMin and PECMin in Eq. (10). The recurrent neural network comprises n neurons. An internal state and output of the recurrent neural network are defined as follows : du(t) = −α AT Ax(t) + α AT b − β e−η t c, dt xiMax xi (t) = (i = 1, ...n), 1 + e−ξ u(t)

(11)

u ∈ Rn , x ∈ Rn , c ∈ Rn , A ∈ Rm×n , b ∈ Rm . α , β , η , ξ ≥ 0,


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In Eq. (11), x(t) is an n-dimensional column of activation states that correspond to the decision variable vector ([x1 (t), . . . , xn (t)]T ), u(t) is an n-dimensional column vector of instantaneous inputs for neurons, and α , β , η , and ξ are parameters that take positive values. The constraint coefficient matrix, A, is realized by the variables of the objective function and the constraints defined by Eq. (10). The advantage point is that we can adjust the ratios of the output power produced by the power generation systems using the settings of elements in the constraint coefficient matrix, A, using the recurrent neural networks. In addition, the constraint coefficient vector, b, is realized by the total demands of the system, L, in Eq. (10). The upper bound power from the solar power generation system PSMax (G) and the one from the wind power generation system PW Max (v) are first calculated by the Eqs. (3) and (4). Using these obtained power, the recurrent neural network automatically decides how much the power from the renewable energies are required to satisfy the demand of customer depending on the values of the elements in the constraint coefficient matrix A in Eq. (11). In other words, this automatic controlling corresponds to the max PS and max PW in Eq. (10); the recurrent neural network automatically calculates the maximum power from the renewable energies, max PS and max PW , to minimize the power from the utility grid PG . The first two terms (−α AT Ax(t) + α AT b) enforce the penalization of violation of constraint and the third term (−β exp(−η t)c) enforce the minimization of the object function. The output has infimum and supremum of the activation state. Each output of the neuron takes [0, xiMax ]. In the numerical simulations, we set α = 1, 000, β = 1, 000, η = 100, and ξ = 10.

4. Performance Evaluations First, we evaluated the control method using recurrent neural networks [5] as the house sizes of the wind and solar power generation systems. We used five states for the battery chargers in these numerical experiments because the states of the battery chargers mainly determine the supply levels of wind and solar power generation systems. The five states of the battery chargers are shown in Table 1. In Table 1, the battery and the electric vehicle are fully charged in Ex. 1, the battery and the electric vehicle are partially charged in Ex. 2, the battery and the electric vehicle are fully discharged in Ex. 3, the battery and the electric vehicle are fully charged and total power demand is less than the other examples in Ex. 4, while the battery and the electric vehicle are partially charged and no power supply is provided by the utility grids in Ex. 5. The results obtained using the five states of the battery chargers (Table 1) are shown in Table 2. In Table 2, although all of the demands are satisfied adequately in Ex. 1, Ex. 2, Ex. 3, and Ex. 4, electric power from the utility grid is required to supply all of the demands, with the exception of Ex. 1. In Ex. 4, the power from the PS Vol.18 No.6, 2014

Table 1. Experimental conditions used to evaluate the method.

PSMax PW Max PGMax PBout PBin PECout PECin L

Ex. 1 3.60 2.80 30.00 2.00 0.00 2.00 0.00 10.00

Ex. 2 3.60 2.80 30.00 1.00 1.00 1.00 1.00 10.00

Ex. 3 3.60 2.80 30.00 0.00 2.00 0.00 2.00 10.00

Ex. 4 3.60 2.80 30.00 2.00 0.00 2.00 0.00 3.00

Ex. 5 3.60 2.80 0.00 1.00 1.00 1.00 1.00 10.00

Table 2. Experimental results.

PG PS PW PB PEC Total L

Ex. 1 0.00 3.60 2.80 1.80 1.80 10.0 10.0

Ex. 2 1.60 3.60 2.80 1.00 1.00 10.0 10.0

Ex. 3 3.60 3.60 2.80 0.00 0.00 10.0 10.0

Ex. 4 0.00 1.69 1.31 0.00 0.00 3.00 3.00

Ex. 5

No Solution

and PW take large values and the power demand is satisfied by the power from these renewable energies only. In addition, the surplus power is charged to the batteries in the system. The demands are not supplied for Ex. 5 because the system itself does not have sufficient electric power. Fig. 2 shows the outputs of the neurons in the five types of experimental conditions for the house sizes in the wind and solar power generation systems. Fig. 2 shows that each neuron converges and that the control method searches for the global optima. However, the recurrent neural networks cannot search for the global optima in Ex. 5. The control method using recurrent neural networks delivered better performance for the house sizes in the wind and solar power generation systems, but it is necessary to compare the performance of this control method with conventional methods. Table 3 shows the experimental results obtained using the simplex method. In Table 3, the simplex method obtained the global optima in Ex. 1, Ex. 2, Ex. 3, and Ex. 4. However, the demands were not satisfied in Ex. 5 because the system had insufficient electric power. Compared with the results obtained using the control method based on recurrent neural networks (Table 2), we can see that the power output of each generator was biased with the simplex method, although all of the demands were satisfied. For example, in Table 3, PB = 1.60 and PEC = 2.00 in Ex. 1, and PS = 0.20 and PW = 2.80 in Ex. 4. Given the possibility of failure or the maintenance of power generation systems, the simplex method is not a good strategy for the house sizes of the wind and solar power generation systems.


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Table 3. Experimental results obtained using the simplex method. PG PS PW PB PEC Total L

Ex. 1 0.00 3.60 2.80 1.60 2.00 10.0 10.0

Ex. 2 1.60 3.60 2.80 1.00 1.00 10.0 10.0

Ex. 3 3.60 3.60 2.80 0.00 0.00 10.0 10.0

Ex. 4 0.00 0.20 2.80 0.00 0.00 3.00 3.00

Ex. 5

No Solution

Fig. 3. Star topology type power grid.

networks [6]. The power grid comprises nodes and links. Each node represents a wind and solar power generation system (Fig. 1), and each link represents a power line.

Fig. 2. Output of each neuron and the objective value.

5. Optimal Operation for Power Grid Type Wind and Solar Power Generation Systems The control method for the power grid which has the renewable energy generators, such as the photovoltaic power generation systems or the wind turbine power generation systems, pay much attention recently because each customer can share the electric power to prevent the blackout. If we apply the control method based on recurrent neural networks to power grids in the real world, it is important to consider the topologies of the power grids as well as the number of power generation systems. We are also interested in how the performance of the optimization method changes when the power grid has various topologies. Therefore, we implemented star topology type power grids and Watts and Strogatz type small-world 1038

5.1. Star Topology Type Power Grid First, we evaluated the control method based on recurrent neural networks using the star topology type power grid. Fig. 3 shows the star topology type power grid. In Fig. 3, every node is connected to the central management system. Then, each node delivers the following information, PSMax , PW Max , PGMax , PBin , PBout , PECin , and PECout to the central management system. Then, the management system determines the final output power of the each generator using these collected information to satisfy all the demand. In these numerical simulations, each customer had the state classified in Table 4. In Table 4, the state of a customer is represented by the text in parentheses. Table 5 shows the experimental conditions used in these numerical simulations. Table 6 shows the results of the numerical simulations using the star topology type power grids. In Table 6, although the electric power was not supplied in Ex. 5 according to the previous results (Table 2), the one was supplied by electric power lines (Table 6) even when the wind and solar power generation systems have no sustainable power available. These results indicate that the control method based on recurrent neural networks works well for power grids with the star topology.


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Table 4. Experimental conditions used to evaluate each power generation system.


A(Ex. 1) 3.60 2.80 30.00 2.00 0.00 2.00 0.00 10.00

B(Ex. 2) 3.60 2.80 30.00 1.00 1.00 1.00 1.00 10.00

C(Ex. 3) 3.60 2.80 30.00 0.00 2.00 0.00 2.00 10.00

D(Ex. 4) 3.60 2.80 30.00 2.00 0.00 2.00 0.00 3.00

E(Ex. 5) 3.60 2.80 0.00 1.00 1.00 1.00 1.00 10.00

Table 7. Experimental conditions used to evaluate largescale power grids.


H = 10 36.00 28.00 240.00 12.00 12.00 12.00 12.00 86.00

H = 100 360.00 280.00 2400.00 120.00 120.00 120.00 120.00 860.00

H = 1000 3600.00 2800.00 24000.00 1200.00 1200.00 1200.00 1200.00 8600.00

Table 8. Results obtained for large-scale power grids. Table 5. Experimental conditions.


All Houses 18.00 14.00 120.0 6.00 6.00 6.00 6.00 43.00

Table 6. Results obtained using the experiment conditions shown in Table 5. PG PS PW PB PEC Total L

All Houses 0.00 18.00 14.00 5.50 5.50 43.0 43.0

Next, we evaluated the method using large-scale star topology type power grids. We constructed star topology type power grids with 10 customers (H = 10), 100 customers (H = 100), and 1, 000 customers (H = 1, 000). Table 7 shows the experimental conditions. The results for the large-scale star topology type power grids are shown in Table 8. Table 8 shows that all of the demands were supplied by the control method based on recurrent neural networks. In addition, the control method shows similar performance when the number of customers increased. Fig. 4 shows the convergence time for the neurons in the recurrent neural networks with different numbers of star topology type power grids. In Figs. 4(a), (b), and (c), the convergence time increased with the number of customers.

5.2. Small-World Topology Type Power Grids Next, we evaluate the performance using more realistic network topologies. In Section 5.1, the proposed method is evaluated for the star topology type power grids. As we have already mentioned that the networks which have Vol.18 No.6, 2014

PG PS PW PB PEC Total L

H = 10 0.00 36.00 28.00 11.00 11.00 86.00 86.00

H = 100 0.00 360.00 280.00 110.00 110.00 860.00 860.00

H = 1000 0.00 3600.00 2800.00 1100.00 1100.00 8600.00 8600.00

small-world property appears to have many appropriate characteristics according to a set of topological metrics defined for the power grids [7]. From the above view point, we realized the power grid as the same manner shown in [6]. Thus, we constructed power grids based on the complex network theory. If the number of nodes in the power grid increases, the calculation cost for determining the amount of the output power by the central management system becomes large. In addition, because of large calculation cost, the demand of the customer may not be satisfied in the worst case. From this view point, we realize the decentralized power grid systems; each node autonomously determines the amount of power to deliver the other nodes. Then, we used the following equations to determine the states of each customer. Pstate = (PSMax − PS ) + (PW Max − PW ) +(PBout − PB ) + (PECout − PEC ) − L.

(12)

In Eq. (12), if Pstate takes a large value, the customer has surplus power. On the other hand, if Pstate takes a small value, the demand of a customer is not satisfied by the power grids. In these simulations, we use two methods for assigning the states of the battery chargers to the nodes in the power grids. The first method is that the states of the battery chargers (Table 4) were assigned randomly to the nodes. The second one is that the states of the battery chargers (Table 4) depended on the spatial structure of the power grid. The state of the node according to the spatial structure of the power grid is shown in Fig. 5. In Fig. 5, white nodes have surplus power whereas black nodes have power shortages. Thus, the blue nodes require electric power to satisfy their demands. In addition, we introduced


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(a) 10 customers

(b) 100 customers

Fig. 6. Distance constraints on sending surplus power. In this example, nodes 1 and 3 have power shortages. Thus, only the adjacent nodes, i.e., nodes 2, 3, 4, can send surplus power to node 1 to satisfy its demands.

1

L, C

0.8

(c) 1,000 customers

Fig. 4. Relationship between the number of iterations and the output of each neuron in the case that (a) the number of customers was 10, (b) the number of customer was 100, and (c) the number of customer was 1, 000.

0.6 0.4 0.2

L(Rp)/L(0) C(Rp)/C(0)

0 0.0001

0.001 0.01 0.1 Rewiring plobability Rp

1

Fig. 7. Relationship between the rewiring probability R p , the average distance L(R p ), and the clustering coefficient C(R p ). In this figure, the number of nodes is 1, 000.

Fig. 5. States of the nodes in a power grid. White nodes have surplus power, whereas black nodes have power shortages.

a distance constraint for sending surplus power. Fig. 6 shows an example of the distance constraint on sending power. We evaluated the control method using an improvement rate in these simulations. The improvement rate, η , is defined by Eq. (13). Ka f ter η = 1− × 100 [%]. . . . . . . . (13) K In Eq. (13), K is the number of initial nodes with power shortages and Ka f ter is the number of nodes with power shortages after operation. If η takes a large value, the 1040

power demands are improved in the power grids. To construct the topologies of the power grids, we used Watts and Strogatz type small-world networks [6]. A Watts and Strogatz small-world network is constructed in the manner proposed by Watts and Strogatz [6]. First, N nodes are placed in a closed one-dimensional ring and each node is connected to its K-th nearest neighbors. Next, each link is wired randomly with the probability Rp. Watts and Strogatz type small-world networks are characterized by the average distance between the nodes and the clustering coefficient. The average distance between nodes and the clustering coefficient are defined by Eqs. (14)–(16). L(R p ) =

N N 1 di j , ∑ ∑ N(N − 1) i=1 j=1, j=i

. . . . . (14)

1 N ∑ Ci, . . . . . . . . . . . (15) N i=1 Ti Ci = . . . . . . . . . . . (16) ki (ki − 1) 2

C(R p ) =


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80 Improvement rate η [%]

Improvement rate η [%]

80 70 60 50 40 30 20 10 0 0.0001

0.001 0.01 0.1 Rewiring probability Rp

70 60 50 40 30 20 10 0 0.0001

1


(a) 20 nodes

(a) 20 nodes 80 Improvement rate η [%]


80 70 60 50 40 30 20 10 0 0.0001


70 60 50 40 30 20 10 0 0.0001

1


(b) 100 nodes 80 Improvement rate η [%]


1

(b) 100 nodes

80 70 60 50 40 30 20 10 0 0.0001

1


1

70 60 50 40 30 20 10 0 0.0001


(c) 1,000 nodes

1

(c) 1,000 nodes

Fig. 8. Relationship between the rewiring probability R p and the improvement rate in the case that (a) the power grids had 20 nodes, (b) the power grids had 100 nodes, and (c) the power grids had 1, 000 nodes. In these simulations, the state of the battery charger on each node was assigned randomly.

Fig. 9. Relationship between the rewiring probability R p and the improvement rate in the case that (a) the power grids had 20 nodes, (b) the power grids had 100 nodes, and (c) the power grids had 1, 000 nodes. In these simulations, the state of the battery charger on each node depended on the spatial structure of the power grid topologies.

In Eqs. (14)–(16), N is the number of nodes in the networks, di j is the shortest distance between the node i and the node j, Ci is the clustering coefficient of the node i, Ti is the number of the triangles that contain the node i, and ki is the degree of the node i. Figure 7 shows the relationship between the rewiring probability R p , the average distance L(R p ), and the clustering coefficient C(R p ). If the rewiring probability is 0, the network has a regular topology. In addition, if the rewiring probability is 1, the network has a completely random topology. Furthermore, if the rewiring probability is between 0 and 1, the network has a small-world topology. Results obtained with the improvement rates (Eq. (13)), where the states of the battery chargers were assigned randomly to the nodes shown in Fig. 8. In Fig. 8(a), the improvement rate takes 50% when the rewiring probabil-

ity R p is in between 0 and 1. However, if the number of nodes in the power grid, N, becomes 100 (Fig. 8(b)) or 1, 000 (Fig. 8(c)), we confirm that the improvement rate is increased as the rewiring probabilities R p becomes large. Figure 9 shows the results for the improvement rates when the states of the battery chargers depended on the spatial structure of the power grids. Fig. 9 shows that the improvement rate was 0 % when the rewiring probability was less than 0.01 in all the cases (Figs. 9(a), (b) and (c)). In addition, the improvement rate increased with the rewiring probability. Also, if the rewiring probability, R p , takes 1, the improvement rate where N = 100 (Fig. 9(b)) and N = 1, 000 (Fig. 9(c)) cases drastically increases as compared to N = 20 case (Fig. 9(a)). From these results, we confirmed that all the demands in the power grid are greatly improved when the connections between the nodes in the the power grid have strong randomness.

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Kimura, K. et al.

6. Conclusion Name:

In this study, we evaluated a control method based on recurrent neural networks for wind and solar power generation systems. First, we compared the performance of the control method based on recurrent neural networks with the simplex method, for the house size of the wind and solar power generation systems. We showed that the performance of the simplex method was biased as compared to the control method based on recurrent neural networks. We also evaluated the control method using realistic power grids. We constructed power grids with star topology type networks and Watts and Strogatz type small-world networks [6]. We also proposed two methods for assigning the states of the battery chargers to the nodes in the power grids. According to the results of the numerical experiments, the control method based on recurrent neural networks showed better performance with various topology type power grids. In this paper, we dealt with the static power grid system; the output power produced by the power generation systems and the demands of the customers have static amounts. Then, we optimized the amounts of the power to the customers using the recurrent neural networks. In the real-world system, the output power produced by the power generation system depends on the environmental circumstances. Then, it is important to evaluate the time-dependent power grid systems using our proposed method for implementation into the realworld system. Thus, we consider to evaluate the dynamic power grid system in the future works. In addition,we aim to develop more efficient optimization method for power grids. References: [1] F. P. Sioshansi, “Smart grid: Integrating renewable, distributed & efficient energy,” Academic Press, 2011. [2] T. Hikihara, K. Tashiro, and Y. Kitamori, “Power packetization and routing for smart management of electricity,” Proc. of the Int. Energy Conversion Engineering Conference, Vol.2012-3732, pp. 1-6, 2012. [3] M. Sharafi and T. Y. ELMekkawy, “Multi-objective optimal design of hybrid renewable energy systems using pso-simulation based approach,” Renewable Energy, Vol.68, pp. 67-71, 2014. [4] J. Zeng, M. Li, J. F. Liu, J. Wu, and H. W. Ngan, “Operational optimization of a stand-alone hybrid renewable energy generation system based on an improved genetic algorithm,” IEEE Trans. on Power and Energy Society General Meeting, pp. 1-6, 2010. [5] M. E. Gamez, E. N. Sanchez, and L. J. Ricalde, “Optimal operation via a recurrent neural network of a wind- solar energy system,” Proc. of the Int. Joint Conf. on Neural Networks, Vol.69, pp. 22222228, 2011. [6] D. J. Watts and S. H. Strogatz, “Collective dynamics of smallworld networks,” Nature, Vol.393, pp. 440-442, 1998. [7] G. A. Pagani and M. Aiello, “Power grid complex network evolutions for the smart grid,” Physica A, Vol.396, pp. 248-266, 2014. [8] G. B. Dantzig, “Linear programming and extensions,” Princeton Univ. Press, 1998. [9] W. Li, “A new neural network approach of linear programming,” Proc. of the 7th Int. Conf. on Machine Learning, pp. 723-727, 2008. [10] N. Karmaekar, “A new polynomial-time algorithm for linear programming,” Combinatorica, Vol.4, pp. 373-395, 1984. [11] J. Wang, “Analysis and design of a recurrent neural network for linear programming,” IEEE Trans. on Circuits and Systems, Vol.40, pp 613-618, 1993.

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Keisuke Kimura

Affiliation: Nippon Institute of Technology

Address: 4-1-1 Gakuendai, Miyashiro, Minami-Saitama, Saitama 345-8501, Japan

Brief Biographical History: 2012 Received his B.Eng. degree from Nippon Institute of Technology 2012- Master’s degree course, Nippon Institute of Technology

Main Works:

• Optimization using neural networks

Name: Takayuki Kimura

Affiliation: Assistant Professor, Nippon Institute of Technology


Brief Biographical History: 2003, 2005 Received his B. Eng. and M. Eng. degrees from Kinki University, respectively 2008 Received his Ph.D. degree in Engineering from Saitama University 2008-2009 Postdoctoral fellow at Hong Kong Polytechnic University 2009-2011 Assistant Professor at Nagasaki University 2011- Assistant Professor in the Department of Electrical and Electronic Engineering at Nippon Institute of Technology

Main Works:

• Analysis and design of optimal networks for traffic networks, optimization using chaotic dynamics, etc.

Membership in Academic Societies:

• The Institute of Electrical and Electronics Engineers (IEEE) • The Institute of Electronics, Information and Communication Engineers (IEICE) • Research Institute of Signal Processing (RISP)


Vol.18 No.6, 2014


Name: Takefumi Hiraguri

Affiliation: Nippon Institute of Technology


Brief Biographical History: 1999, 2008 Received his M.E. and Ph.D. degrees from the University of Tsukuba, respectively 1999- Joined the NTT Access Network Service Systems Laboratories, Nippon Telegraph, and Telephone Corporation in Japan

Main Works:

• MAC protocols for high speed and high communication quality in wireless systems


• The Institute of Electrical and Electronics Engineers (IEEE)

Name: Kenya Jin’no

Affiliation: Professor, Nippon Institute of Technology


Brief Biographical History: 1991, 1993, 1996 Received his B.E., M.E., and Ph.D. degrees in Electrical Engineering from Hosei University, respectively 1994-1996 Research Fellow of the Japan Society for the Promotion of Science 1996 Received research encouragement awards from IEICE 1996-1998 Assistant Professor in the Department of Electrical and Electronics Engineering at Sophia University 1998-2004 Associate Professor in the Department of Electrical and Electronics Engineering, Nippon Institute of Technology 2004-2008 Associate Professor in the Department of Networks and Multimedia Engineering, Kanto-Gakuin University 2008-2009 Research Fellow on the JST ERATO Aihara Complexity Modeling Project 2009- Professor, the Department of Electrical and Electronics Engineering, Nippon Institute of Technology

Main Works:

• Artificial Neural Networks, Soft Computing, High-Dimensional Nonlinear Systems


• The Institute of Electrical and Electronics Engineers (IEEE) • The International Neural Network Society (INNS) • The Institute of Electronics, Information and Communication Engineers (IEICE) • Information Processing Society of Japan (IPSJ) • Research Institute of Signal Processing (RISP)

Vol.18 No.6, 2014


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