Probabilistic scheduling of smart electric grids considering plug-in hybrid electric vehicles

Nomenclature

1 Introduction

Feeder reconfiguration is defined as the process of changing the topology of the radial distribution system by the use of some remotely controlled switches (RCSs) [1]. Technically, RCSs can be either normally open switches called tie switches or normally closed switches that are called sectionalizing switches. Optimal distribution feeder reconfiguration can be a useful, precious and strategic technique in the hands of the distribution operator. Through the optimal feeder reconfiguration, the network situation is improved from different aspects such as reliability [2], loss [3], cost [4], voltage deviation [5], power quality and load balance [6]. Among these useful but objective functions, the most attention has been given to the active power losses. Some of the methods used for optimizing power losses through the reconfiguration can be named as neural network [7], optimum flow pattern [8], graph theory [9], brute-force approach [10], heuristic techniques [11], expert systems [12] and particle swarm optimization [13]. In addition, there have been some researches on the optimal reconfiguration of feeders in the presence of distributed generations (DGs) and renewable energy sources (RESs). The main purpose of these works is assessing the effect of DGs and RESs on the reconfiguration strategy. In this way, optimal distribution feeder reconfiguration is studied in the presence of fuel cell, photovoltaics and wind turbine in [14, 15]. Along with DGs, one of the new technologies that can greatly affect the reconfiguration strategy is electric vehicle (EV).

The recent statistical report forecast considerable portion for EVs in the future of the transportation system [16]. Plug-in hybrid electric vehicles (PHEVs) have attracted much interest due to their low pollution emissions and high fuel economy [17, 18]. In addition, the appearance of PHEVs in the network can support the idea of RESs by reducing the dependency on gas energy and increasing the dependency on electricity. However, the high penetration of PHEVs can cause some challenges regarding the supplement of their charging demand. Congestion or overloading of feeders and transformers, raise in energy loss, lessening the reliability and power quality are some of the unwanted technical challenges in front of the high integration of PHEVs in the grid [19–21]. In order to diminish their effects, smart charging management methods have been developed in recent years to manage the PHEVs’ charging demand [22–24]. The core idea of smart charging is to charge EVs at hours with lowest price and load consumption or when there is excess electricity capacity.

One of the strategies that can be affected by the appearance of PHEVs is the reconfiguration strategy. The high volatile and stochastic charging behavior of PHEVs can affect the optimal status of switches in the network greatly. In [25] multi-objective reconfiguration is solved considering the PHEV charging demand. However, neglecting the uncertainties of PHEVs and lack of a sufficient smart charging scheme are some of the main deficiencies of that work. In [3], the effect of electric vehicles (EVs) on the reconfiguration strategy is assessed. Nevertheless, there are some significant differences between [3] and this paper: 1) reference [3] considers Plug in EVs (PEVs) while this paper considers PHEVs. PEVs only use electricity while PHEVs can use both electricity and fossil fuels for movement. 2) Reference [3] considers both charging and discharging of PEVs using the V2G technology. But, this paper has focused only on the charging of demand of PHEVs in three different modes of controlled, uncontrolled and smart charging schemes. 3) Reference [3] solves the problem in a linear form neglecting the nonlinearity of the power flow equations (by making some assumptions on the formulations). But, our paper proposes a powerful optimizer based on TLO to solve the problem in its nonlinear form without loss of information or making any assumption. Therefore, the necessity of research on this area is sensed completely. Based on these discussions, the main purpose of this paper is to investigate the optimal operation of reconfiguration strategy in the networks with the high penetration of PHEVs. In this way, a sufficient stochastic framework based on point estimate method (PEM) is proposed to model the uncertainties of PHEVs’ charging demand, energy price and load consumption. PEM is an approximate method which makes use of some concentration points to model the uncertainty of the random variables or parameters [26]. The optimal reconfiguration is then solved for three different charging schemes including uncoordinated charging plan, coordinated charging plan and smart charging strategy. According to the recent reports [22], the unidirectional charging scheme is also a possible case due to the lack of hardware and present technology of chargers. In order to solve the problem optimally, a new optimization algorithm based on teacher learning optimization (TLO) algorithm is introduced too. TLO is a new meta-heuristic optimization algorithm which imitates the teaching process of the students in a class [27]. Also, a new two-stage modification method is proposed to increase the search ability of the algorithm when improving the balance between the local and the global searches. In order to check the performance of the proposed method, IEEE 86-bus network is used as the test system. To summarize, the main contributions of this paper can be named as: 1) investigating the role of charging of PHEVs on the optimal switching during the reconfiguration process, 2) introduction of a stochastic optimization problem based on 2 m PEM and TLO to solve the problem optimally, 3) introduction of a new modification method for TLO to increase its total search ability.

The rest of this paper is as follows: Section 2 describes the PHEV charging scheme. Section 3 explains the problem formulation. Section 4 describes the proposed modified TLO (MTLO) optimization algorithm. The stochastic method based on PEM is described in Section 5. The simulation results are explained in Section 6. Finally, the main remarks and conclusions are discussed in Section 7.

2 Molding of PHEV charging demand

PHEV is a hybrid EV that makes use of rechargeable batteries, or other energy storage device, that can be restored to full charge by connecting to an external electric power source. In addition, PHEVs can employ the fossil fuel to be supplied for driving long distances in the case of lack of charge in battery. In order to model the behavior of these devices, it should be considered that there are few effective parameters that determine the charging behavior of PHEVs including charger type, battery state of charge (SOC), battery capacity, number of PHEVs, charging type, charging time and length of charging. It is worth noting that SOC is defined as the ratio of available energy to maximum storable energy in the battery. Therefore, in an overall point of view, the charging demand of PHEVs either in a public station or in a residential community is uncertain. In this way, three different charging patterns are considered in this paper: 1) uncoordinated charging, 2) coordinated charging and 3) smart charging. In the uncorrelated charging pattern, PHEVs are charged at any time that they are plugged in to the charging point. According to the statistic reports, PHEVs experience two main travels during the day; one in the morning when leaving the home and one at the evening when returning to the home. Short travels are neglected in the hourly scheduling. In this case, PHEVs are plugged in around 6:00 p.m. when they arrive home. Therefore, the charging time can be modeled by a uniformly probability density function (PDF) with narrow range around 6:00 p.m. as follows [28]: f ( t start ) = 1 b - a a ≤ t start ≤ b a = 18 , b = 19(1)

In the coordinated charging pattern, EV owners prefer to connect their vehicles to the charging point during the off-peak load hours to keep away from coincidence with heavy-load hours with higher prices. This decision shifts the charging time after 9:00 p.m. This charging pattern is modeled by the use of the below equation: f ( t start ) = 1 b - a a ≤ t start ≤ b a = 21 , b = 24(2)

Finally, in the smart charging pattern, the charging of PHEVs should be done at hours that the electricity price is lowest or when there are extra electrical energy. This is a common idea among all smart charging patterns. In this paper, a normal PDF is employed to represent the complexity of employing various smart charging plans for determining the charging start time as follows [28]: f ( t start ) = 1 σ 2 π e ( - 1 2 ( t start - μ σ ) 2 ) μ = 1 , σ = 3(3)

As the PHEV is plugged in to the charging location, the battery starts charging. In order to calculate the amount of energy remained in the battery (SOC) after each travel, the data of the driven mile for the vehicle should be known. It is mentioned in the literature that the daily driven mile of a vehicle follows a log-normal PDF as below [28]: f ( m ) = 1 n σ 2 π e - ( ln ( n ) - μ ) 2 2 σ 2 n > 0(4)

Therefore, at the plug-in time, the battery SOC can be evaluated by the use of the driven mile of the vehicle and its All-Electric Range (AER) as follows: SOC = { 0 n > AER AER - n AER × 100 % n ≤ AER(5) In terms of AER, there are different types of PHEVs including PHEV-20, PHEV-30, PHEV-40 and PHEV-60; wherein the numerical subscript shows the vehicle AER in miles. In this paper, PHEV-20 is considered in the analysis. It is clear that the method is general and can be applied to other types too. Therefore, the charging duration of the PHEV is computed by the use of the below equation: t D = C bat × ( 1 - SOC ) × DOD η × P(6)

Once PHEV arrives at home, it can be charged depending on its needs and minimum depth of discharge limit (here is considered 80%). But, the length of charging differs depending on the charger rating (P) and charger efficiency (η). Table 1 shows the charging rate as a function of charging level [29]. Charging levels 1 and 2 belong to the home charging locations. Level 3 is not considered since it belongs to the commercial and public transportations. With the purpose of considering variety among the PHEV parameters and battery capacities, they are divided into four different classes with special market shares and features as given in Table 2 [28].

Since the market share may be seen as a discrete distribution, the class of each PHEV is chosen at random in accordance with their market share. It is worth noting that the distribution of C _bat within the limited range of each class is regarded to have a normal PDF with parameters as follows [28]: μ C bat = MinC bat + MaxC bat 2 σ C bat = MaxC bat - MinC bat 4(7)

This section describes the proposed optimization problem including objective function and relevant constraints.

A. Objective Function

The objective function is the total cost of the network including the cost of energy supply for loads (both network load and charging demand of PHEVs), cost of power losses and cost of switching as follows: Minimize Cost = ∑ t = 1 T ( P grid t λ m t + P loss t λ loss t + C Switching )(8) The cost of power losses at each hour of scheduling is formulated as follows: P loss t = ∑ i = 1 N branch R i × | I i t | 2(9)

Also, the cost of switching operation is calculated as follows: C Switching = ∑ i = 1 N RCS ∑ t = 1 T | RCS k t - RCS k t - 1 | λ RCS(10)

The control variable vector includes the optimal status of the switches of the network as follows: X i t = [ RCS 1 t , RCS 2 t , … , RCS N RCS t ](11) where in the values 0 and 1 show the open and closed state of switches, respectively.

B. Constraints

The proposed optimization method includes some equality and inequality constraints that should be met at each hour of of scheduling as follows:

–Lower flow equations: P g , i t - P d , i t = ∑ j | V i t | | V j t | | Y ij | cos ( θ ij + δ i t - δ j t )(12)Q g , i t - Q d , i t = ∑ j | V i t | | V j t | | Y ij | sin ( θ ij + δ i t - δ j t )(13)

–Voltage limit constraints on buses: V i min ≤ V i t ≤ V i max(14)

–Maximum current flow in each feeder: | I i t | ≤ I i max(15)

–Maximum power injection in the main feeder: | S i t | ≤ S i max(16)

–Keeping the radiality of the network: The number of main loops before and after the reconfiguration is calculated as follows: N loop = N branch - N bus + 1(17)

–Maximum number of switching: ∑ t = 1 T | RCS k t - RCS k t - 1 | ≤ N switching max(18)

An evolutionary optimization algorithm based on TLO is presented in this section.

A. Original TLO

In recent years, the increasing nonlinearity and complexity of the engineering problems has increased the popularity of evolutionary algorithms as the optimization tool. One of the new powerful and successful evolutionary algorithms is TLO algorithm. The existence of some features such as low dependability on adjusting parameters, powerful local search mechanism, few setting parameters, easy implementation and ability of solving both continuous and discrete optimization problems makes the TLO algorithm a successful method amongst the others. TLO is inspired from the teaching process of students in the class [27]. Similar to any other algorithm, TLO starts with the generation of an initial random population. The main idea for updating the students’ population is constructed based on two phases:1) teacher phase and 2) student phase. Each of these phases is described in the rest:

–Teacher Phase: This phase simulates the attempts of the teacher for improving the knowledge of the students up to his/her knowledge. In this way, first the mean value of the population is calculated column-wise (M_D). Here, the best student in the class is chosen as X _Teacher . Then the whole population is moved to the teacher position as follows: X new = X old + θ 1 ( X Teacher - T F M D )(19)M D = [ m D , 1 , m D , 2 , m D , 3 , … , m D , N ](20)

If X _new is better than X _old (a student with higher knowledge) then replaces it.

–Learner Phase: The learner phase simulates the discussion between the students for improving their own knowledge. Students can improve their knowledge by talking together and sharing their knowledge as follows: For i = 1 : N class If F ( X i ) ≺ F ( X j ) X new , i = X old , i + θ 1 ( X i - X j ) If F ( X i ) ≺ F ( X j ) X new , i = X old , i + θ 2 ( X j - X i ) End if End for(21) where F (X) is the objective function value of the student X.

In order to model the uncertainty effects in the loads flow, the use of stochastic methods is necessary. Technically, the stochastic methods can be categorized in three main groups [30]: 1) Monte Carlo Simulation (MCS) 2) Analytical methods and 3) Approximate methods. MCS is the most accurate method but suffers from the high computation efforts. The analytical methods are computationally sufficient but require some simplification assumptions to solve the problem. Finally, in the last group are approximate methods that have overcome both of the above deficiencies. One of the most successful methods of approximate group is 2m-PEM wherein m is the number of uncertain variables. This method first replaces each uncertain parameter by a suitable probability density function (PDF) and then extracts two concentration points from the PDF function to solve the problem. The main characteristic of PEM is the use of little information from the PDF including the mean, variance, skewness, and kurtosis. In order to describe the 2m-PEM, the general load flow can be supposed as a nonlinear function with an input vector z (including active/reactive load, network structure, branch data, etc) and an output vector S (including bus voltage, line active/reactive power flow) as follows: S = F ( z )(24)

From the above equation it is seen that the input uncertainty is transferred to the output variables through the power flow equations. The main concept is to determine the moments of S by the use of few deterministic load flows. Figure 2 shows the idea of 2m-PEM. For each uncertain variable z _l , a probability function named f _zl is determined. 2m-PEM will replace f _zl with two probability concentration points matching the mean, variance and skewness coefficient of f _zl [31]. z l , k = μ z l + ξ l , k . σ z l ; k = 1 , 2(25) where ζ_l,k is the standard location and is evaluated as follows [31]: ξ l , k = λ l , 3 2 + ( - 1 ) 3 - k m - ( λ l , 3 2 / 2 ) 2 , k = 1 , 2(26)

Also, the skewness coefficient (λ _l , 3) as the third central moment is evaluated as follows: λ l , 3 = E [ ( z l - μ z l ) 3 ] ( σ z l ) 3(27) where the word E denoted the expected value. Also the standard deviation value of S _i can be evaluated as follows: σ = var ( S i ) = E ( S i 2 ) - [ E ( S i ) ] 2(28)E ( S i j ) = ∑ l = 1 m ∑ k = 1 2 ( ω l , k × S i j ( μ z 1 , μ z 2 , … , z l , k , … , μ zm ) ) ω l , k = 1 2 m(29)

In this part, the proposed method is examined on the IEEE 86-bus test system. The nominal voltage of this network is 12.66 and the single-line diagram is shown in Fig. 3. The sectionalizing switches are shown by solid line and tie switches are shown by dotted line. The total hourly load of the network is shown in Fig. 4. The complete data of the network are given in [32]. The hourly market price data are provided in Table 3. In terms of the PHEVs, the charging demand is distributed on all buses uniformly. To model the uncertainty effects, normal PDF with the mean values equal to the forecast data and the standard deviations equal to the percentages of the mean value are considered. The average and standard deviation values of PHEV daily driven distance are supposed 33 miles and 20.4 miles, correspondingly [28].

For the optimization algorithm, the initial size of the students is 30 and the termination criterion is supposed to be 200 iterations. The reason for this is that it was seen that there is not much improvement in the objective function value after about 200 iterations. For better understanding of the problem, four different scenarios are defined here: Scenario 1: This scenario solves the reconfiguration for the peak load data neglecting PHEV charging demand. Scenario 2: This scenario solves the reconfiguration for 24-hour operation considering uncoordinated charging of PHEVs. Scenario 3: This scenario solves the reconfiguration for 24-hour operation considering coordinated charging of PHEVs. Scenario 4: This scenario solves the reconfiguration for 24-hour operation considering smart charging of PHEVs. Table 4 shows the results of single-objective optimization of the power losses before and after reconfiguration in Scenario 1. Here the analysis is done neglecting uncertainty effects and PHEV charging demand. According to these results, optimal reconfiguration could reduce the network losses when improving the voltage level of the network at the same time. The statuses of the switches after the reconfiguration are shown in Table 4. Also, the proposed optimization method could reach the best optimal solution found by the other well-known methods in the area.

Table 5 shows the simulation results of the network in Scenario 2. In this scenario, the charging demand of PHEVs in the uncoordinated pattern is modeled in the network. It is worth noting that this scenario considers the PHEVs to start charging between 6 p.m. to 7 p.m. Also, three different penetration levels of 30%, 60 % and 90% are considered to check the PHEV charging demand on the network. The hourly cost of the network before and after the reconfiguration for 30% penetration level of PHEVs is shown in Table 5. As it can be seen from this table, optimal reconfiguration has reduced the cost of the network suitably. The higher cost of the network at mid-day hours roots in the higher electricity cost in the market. Figure 5 shows the total cost of the network for different penetration levels before and after the reconfiguration, comparatively. According to this figure, the reconfiguration strategy could reduce the cost of the network for all penetration levels of PHEVs.

The simulation results for Scenario 3 are shown in Table 6 and Fig. 6. In Table 6, the comparative hourly cost data of the network before and after the reconfiguration are shown. As it can be seen from this table, the cost of the network is reduced at almost all hours through the reconfiguration. In comparison to the second scenario, the charging start time is shifted to 21–24 to reduce the total costs. Similarly, the positive effect of the reconfiguration in reducing the cost value for the three different penetration levels are shown too.

The simulation results for smart charging in Scenario 4 are shown in Table 7 and Fig. 7. In this scenario, the charging demands of PHEVs are shifted to hours with lower energy costs or off-peak hours. This is a beneficial strategy for using the excess capacity of the network to charge the PHEVs. As the result, the total cost of the network is reduced greatly. The positive effect of reconfiguration can be deduced from both Table 7 and Fig. 7 clearly. This effect can be seen for different PHEV penetration levels too.

To have better comparison, the costs of the different charging strategies of PHEVs are shown in Table 8. The results show that uncoordinated charging has higher costs for the network due to coincidence with peak load hours. On the other hand, the controlled charging scheme has lower costs since it shifts the charging of PHEVS to off-peak load hours. Finally, the smart charging scheme has the lowest cost for the network. In comparison with different topologies of the network, it is deduced that considering DFR can help the network to have lower costs due to reduction in the active losses and providing new paths for feeding the electric loads.

7 Conclusion

This paper proposed a new method for investigating the optimal operation and management of the reconfiguration strategy in the smart grids considering the PHEV charging demand. In this way, a stochastic method based on 2m-PEM and MTLO algorithm was devised to solve the problem suitably. In order to understand the problem deeply, four different scenarios with different charging penetration levels were defined to highlight the effects of different charging strategies. In addition, three different penetration levels were considered in the simulations. The simulation results on the IEEE 86-bus network showed that the proposed method could make use of smart charging scheme for PHEVs when reducing the cost of the network even for higher penetration levels. In addition, it was seen that optimal reconfiguration can be a useful strategy for improving the network situation by minimizing the costs. Therefore, it can support the idea of PHEVs through cost reduction properly. Last but not least, the proposed MTLO algorithm showed suitable performance in solving the problem in hand.