Optimal probabilistic reconfiguration of smart distribution grids considering penetration of plug-in hybrid electric vehicles

Abstract

The capable incorporation of Plug-in Hybrid Electric Vehicles (PHEVs) in the upcoming transportation area presents several technical challenges to electrical distribution networks for example voltage drop and loss increase. The energy demand of these movable loads is stochastic naturally owing to the uncertainties accompanied by their location and amount of required energy. Accordingly, a new optimal stochastic reconfiguration procedure based on Gravitational Search Algorithm (GSA) is suggested that diminishes resistive loss and costs of radial distribution grids. The proposed technique is equipped with mutative operators to surpass the optimization process. Furthermore, a novel local smart charging pattern is recommended which lessens the congestion influence of PHEVs on system load curve successfully. The uncertainties of PHEV loads are modeled with Monte Carlo Simulation (MCS) and the proposed methodology is examined on Tai-power distribution network to validate its performance and robustness.

Keywords

Plug-in Hybrid Electric Vehicle (PHEV)smart charging Distribution Feeder Reconfiguration (DFR)Gravitational Search Algorithm (GSA)

Nomenclature

{accum}_{s} / {accum}_{s}^{initial}

Total/ initial accumulation factor of sth strategy.

Cost

Total network cost ($).

Cost _Charge

Weighted charging cost of PHEVs($).

D _ij

Euclidian distance between ith and jth individual.

Total energy demand of PHEVs (MWh).

F _i

Resultant forces acted on ith mass.

f_i/f_b/f_w

Objective value of ith/best/worst individual.

G ₀

Gravitational constant.

I_{i}^{t}

Current magnitude of ith branch at hour t (A).

I_{i}^{\max}

Maximum current of ith branch (A).

Iteration counter.

k _max

Maximum number of iterations.

M _i

Mass of ith individual in kth iteration.

N _p

Population size (number of bats).

N _mod

Number of individuals which select a strategy.

Prb _s

Normalized selection probability of sth strategy.

P_{grid}^{t}

Hourly imported power from upstream grid (MW).

P_{loss}^{t}

Hourly loss of the network (MW).

P_{g, i}^{t} / P_{d, i}^{t}

Hourly active power supply/ demand at bus i (MW).

Q_{g, i}^{t} / Q_{d, i}^{t}

Hourly reactive power supply/ demand at bus i

S_{i}^{t}

Apparent power flow of ith branch at hour t (MVA).

S_{i}^{\max}

Maximum apparent power flow of ith branch (MVA).

S _strategy

Set of modification strategies.

Planning horizon.

V_{i}^{t} / δ_{i}^{t}

Voltage magnitude/phase of bus i at hour t (V).

V_{i}^{\min} / V_{i}^{\max}

Minimum/maximum voltage at bus i (V).

V_{i}^{new} / V_{i}^{old}

New/old velocity of ith individual.

W _i

Weighting factor of ith individual.

X_{i}^{new} / X_{i}^{old}

New/old position of ith individual.

Y_ij/θ_ij

Magnitude/phase of impedance between bus i and j.

β _mut

Mutation level.

β_{PHEV}^{t}

Hourly existence coefficient of PHEVs.

Random number in roulette wheel process.

ρ _i

ith random number between [0, 1].

θ, α

Empirical constant factors.

λ_{m}^{t} / λ_{loss}^{t}

Hourly energy price/loss cost ($/MWh).

μ_{PHEV}^{\min / \max}

Minimum/maximum hourly PHEV demand (MW).

μ_{PHEV}^{t}

Hourly mean values of PHEV demand (MW).

Small constant number

1 Introduction

Latest cares to the subjects of fossil fuel emissions have resulted in electric vehicle technologies to diminution emissions from passenger vehicles and lessen faith to the fossil fuels. Plug-in Hybrid Electric Vehicles (PHEVs) show an interesting future in the private transportation part for meeting the main necessities concurrently [1]. Benefits of PHEV owners are less costs and emissions as per driving on electricity has been found to be less exclusive per mile and naturally yields less emissions than a conformist vehicle [2]. The popular accessibility of current charging organization in the form of 120/240 V outlets at households and offices is an alternative strong argument for a change to PHEVs, over other substitute vehicle technologies such as fuel cells. Besides, the unidirectional energy transaction between PHEVs and the grid (charge only) is unavoidable first owing to lack of required hardware [3].

While PHEV can provide good opportunities for the system, they can bring new challenges such as congestion on transmission and distribution lines and transformers, increase in energy loss, decreasing the reliability and power quality [4 –6]. In is shown in literature [7] that charging of PHEVs generally happen at the peak load hours. This issue can cause much costs for the system. In response to this issue, smart charging strategies should be considered to shift the charging hours to off-peak hours [8, 9].

One of the technologies that is affected by the PHEV charging demand is Distribution Feeder Reconfiguration (DFR). DFR is a suitable tool for increasing the penetration of PHEVs using some normally open or closed switches called tie and sectionalizing, respectively [10]. DFR can improve the load balance [11], reduce the voltage deviation [12] and emission [13, 14, 13, 14] and enhance the reliability [15, 16]. Nevertheless, the most attention has been paid to the power loss minimization using some well-known methods such as Artificial Neural Networks (ANN) [17], Optimum Flow Pattern (OFP) [18], Graph Theory (GT) [19], Brute-Force Approach (BFA) [20], heuristic techniques [21], expert systems [22], Particle Swarm Optimization (PSO) [23] and quantum-inspired binary PSO [24]. According to our research, there is one work that has considered DFR effect on the PHEV in [24].

According to the above discussion, this paper aims to investigate the DFR strategy and its role in increasing the penetration of PHEVs. Since PHEVs are mobile loads, their stochastic characteristics for both location and energy demand should be modeled in the problem. Here we make use of Monte Carlo Simulation (MCS) method to model the uncertainties of the problem when optimizing the power losses and network total costs. Since the problem has a complex and nonlinear nature, a new optimization algorithm based on self-adaptive modified gravitational search algorithm (MGSA) is proposed [25]. Finally, the feasibility and satisfying performance of the proposed method are examined on the IEEE 86- bus distribution test system. It is shown that proposed smart charging plan has been fruitful in flattening the load curve and the proposed stochastic DFR solution can efficaciously diminish PHEV influences in both coordinated and uncoordinated charging frameworks.

2 PHEV smart charging

Transition to electric transportation structure re-quires the setting up of charging apparatus in both houses and public garages. In fact, because of the low battery capacity, PHEVs need frequent charging in the above places [26]. The charging characteristics of a PHEV in the residential or public stations are determined based on the charging voltage, charging current, start/end time of charging and their locations. According to the recent researches, the overall charging demand of PHEVs in the residential and public stations can be modeled using the normal and Weibull probability density functions (PDFs) [27]. For charging, the electric vehicle can work in either uncontrolled or smart charging scheme. In the uncontrolled mode, the PHEV is charged as soon as it reaches the home. But in the smart charging, the network operator determines the charging demand and its time [28]. In this way, this paper proposes a new smart charging methodology to minimize the total network costs due to each PHEV charging demand as follows: $\begin{matrix} \begin{matrix} Minimize & {Cost}_{charg} = \sum_{t = 1}^{T} (\frac{λ_{m}^{t}}{β_{PHEV}^{t}} \times μ_{PHEV}^{t}) \\ \begin{matrix} subject & to \end{matrix} \end{matrix} \\ \sum_{t = 1}^{T} μ_{PHEV}^{t} = E \\ μ_{PHEV}^{min} \leq μ_{PHEV}^{t} \leq μ_{PHEV}^{max} \end{matrix}$ (1)

The above Linear Programming (LP) will set the hourly optimal value of PHEV charging demand in the residential and public stations. As it can be seen, the hourly energy price will determine the PHEV charging time. Also, the variable β _PHEV shows the possibility of a PHEV to be in the relevant place at the specific time. Figure 1 shows the typical curve of β _PHEV for a PHEV charged at home or public station.

3 Proposed stochastic DFR formulation

Optimal DFR is a precious strategy that can improve the network by changing its topology using somepre-determined switches. In order to make use of DFR, we have devised a single-objective formulation to optimize the cost of power losses and cost of power supply by the main grid when meeting the operation and security constraints. In the below formulation, Equation 2 is the objective function and Equations 3 to 8 are the limitations. $\begin{matrix} \begin{matrix} Minimize & Cost = \sum_{t = 1}^{T} (P_{grid}^{t} λ_{m}^{t} + P_{loss}^{t} λ_{loss}^{t}) \end{matrix} \end{matrix}$ (2) $\begin{matrix} Subject to \\ | S_{i}^{t} | \leq S_{i}^{max} \end{matrix}$ (3) $| I_{i}^{t} | \leq I_{i}^{max}$ (4) $V_{i}^{min} \leq V_{i}^{t} \leq V_{i}^{max}$ (5) $P_{g, i}^{t} - P_{d, i}^{t} = \sum_{j} | V_{i}^{t} | | V_{j}^{t} | | Y_{ij} | cos (θ_{ij} + δ_{i}^{t} - δ_{j}^{t})$ (6) $Q_{g, i}^{t} - Q_{d, i}^{t} = \sum_{j} | V_{i}^{t} | | V_{j}^{t} | | Y_{ij} | sin (θ_{ij} + δ_{i}^{t} - δ_{j}^{t})$ (7) $Preserving Radial structure of network topology$ (8)

One significant limitation in the DFR problem is preserving the radiality of the network. In this regard, each time that a tie switch is closed in the network, a sectionalizing switch should be opened randomly to keep the structure as radial. Considering the above nonlinear programming formulation in the stochastic framework, a powerful optimizer is required to solve it optimally. Therefore, the next section is paid to the optimization algorithm.

4 Optimization procedure

4.1 Original gravitational search algorithm

Gravitation is defined as the inclination of masses to move toward each other. In the Newton law of gravity,each particle fascinates the surrounding masses in a diverse relationship with the distance. According to this simple and significant idea, a new meta-heuristic optimization algorithm called gravitational search algorithm (GSA) is introduced by the researchers [25]. In order to describe the GSA, first a population of masses should be produced in the different locations of the space. Then, the position of each mass is updated using the below equations: $\begin{matrix} X_{i}^{new} = X_{i}^{old} + V_{i}^{new} \\ V_{i}^{new} = ρ_{1} \times V_{i}^{old} + \frac{F_{i}}{M_{i}} \end{matrix}$ (9)

The acting force (F _i) on each mass is created by the gravitational forces of other mases in the population which is calculated as follows: $F_{i}^{k} = \sum_{j = 1, j \neq i}^{Np} ρ_{2} G_{0} e^{(- α \frac{k}{k_{max}})} \frac{M_{i}^{k} M_{j}^{k}}{D_{ij}^{k} + ɛ} [X_{i}^{k} - X_{j}^{k}]$ (10) where G ₀ and α are empirical constants set to 100 and 20 respectively. Also, ɛ is a small constant added to the Euclidian distance between two masses (D _ij) to evade the singularity. Each mass will be updated as said by the objective values of current best (f _b) and worst (f _w) masses as follows: $\begin{matrix} M_{i}^{k} = \frac{m_{i}^{k}}{\sum_{j = 1}^{Np} m_{j}^{k}} & where & m_{i}^{k} = \frac{f_{i} - f_{w}}{f_{b} - f_{w}} \end{matrix}$ (11)

4.2 Self-adaptive modified gravitational search algorithm

Similar to other metaheuristic algorithms, GSA is a random population based optimization algorithm. Therefore, we have devised a two-stage modification method for improving the overall performance of this algorithm. Each of these modifications can improve the system in special way that is described in the rest.

Strategy 1: This strategy will increase the diversity of the mass population using the croosover and mutation operators. Therefore, for mass X _i, three dissimilar individuals are used from the population randomly such that (X ₁ ≠ X ₂ ≠ X ₃ ≠ X _i) to produce a random solution as follows: $X_{mut} = X_{1} + ρ_{3} (X_{2} - X_{3})$ (12)

Now, three test individuals are produced as follows: $\begin{matrix} X_{Test 1, j} = {\begin{matrix} \begin{matrix} X_{mut, j} & ρ_{4} \end{matrix} \leq β_{mut} \\ \begin{matrix} X_{G} & otherwise \end{matrix} \end{matrix} \\ X_{Test 2, j} = {\begin{matrix} \begin{matrix} X_{mut, j} & ρ_{5} \leq β_{mut} \end{matrix} \\ \begin{matrix} X_{j} & otherwise \end{matrix} \end{matrix} \\ X_{Test 3, j} = ρ_{6} X_{G} + ρ_{7} (X_{G} - X_{r}) \end{matrix}$ (13)

Strategy 2: This modification simulates a kind of consultation between two solutions to increase their knowledge. Mathematically, this interaction is formulated as follows: $\begin{matrix} \begin{matrix} if & f_{i} \end{matrix} < f_{j} \\ X_{Test 4, i} = X_{i} + ρ_{8} (X_{i} - X_{j}) \\ \begin{matrix} elseif & f_{j} < f_{i} \end{matrix} \\ X_{Test 4, i} = X_{i} + ρ_{9} (X_{j} - X_{i}) \\ End \end{matrix}$ (14)

In order choose between the above two strategies, a probability is designated to each modification method that is initially equal to 0.5. Then, the population is sorted in a descending order such that X ₁ would become the best solution and X _Np would become the worst solution. Now each solution would gain a weighting factor: $W_{i} = \frac{log (f_{Np} - f_{i} + 1)}{\sum_{j = 1}^{Np} log (f_{j})}$ (15)

Then, the accumulative weighting factor of each strategy is updated as follows: $\begin{matrix} {accum}_{s} = {accum}_{s}^{initial} + \frac{\sum_{i = 1}^{N_{mod}} W_{i}}{N_{mod}} & \forall s \in S_{strategy} \end{matrix}$ (16)

Using the accumulation value, the possibility of each strategy is updated as follows: ${prb}_{s} = (1 - θ) {prb}_{s} + θ \times \frac{{accum}_{s}}{M}$ (17)

In this paper, the constant parameter θ is set to 0.142 empirically to control the learning speed of the algorithm. In each iteration, the probabilities are normalized as follows: ${prb}_{s} = \frac{{prb}_{s}}{\sum_{k \in S_{strategy}} {prb}_{k}}$ (18)

5 Simulation results

In this section, the Taiwan Power Company (TPC) distribution test system with 86 buses is employed as the case study [29]. Figure 2 shows the single-line diagram of the test system. It incorporates 13 tie switches shown by dotted lines and 85 sectionalizing switches shown by solid lines. The network should supply 28350 kW and 20700 kVAr total active and reactive loads neglecting the PHEV charging demands. In order to consider the PHEVs, three charging locations on buses 9, 38 and 72 are assumed. Also, it is supposed that the residential charging can take place on all buses expect those that have public charging stations.

Table 1 shows the hourly forecast value of energy price $λ_{loss}^{t}$ .

For better comparison, three different cases are defined: (I) neglecting PHEV demands, (II) uncontrolled charging and (III) smart charging. Regarding the optimization algorithm, the initial size of the algorithm is 25 and the termination criterion is 100 iterations.

5.1 Neglecting PHEVs

Since no PHEV is considered in the network, this part is solved in the deterministic framework. Table 2 shows the results of single-objective optimization of the power losses. According to this table, the proposed MGSA could reach the best optimal solution which was found by Ahuja et al. [29]. Also, the statuses of the open switches are shown in this table.

As it can be seen from Table 2, the minimum voltage of the network is 0.9532 per unit which is in the acceptable range of voltage. In order to see the convergence of the algorithm, Fig. 3 shows the convergence diagram of the proposed MGSA in comparison with GA, PSO and original GSA. As it is seen, the proposed MGSA could converge in the first place which shows it s good search ability.

The simulation results of optimal reconfiguration in the 24-hour time interval are given in Table 3. In this way, the 24-hour total load curve of the network is shown in Fig. 4. According to Table 3, the optimal DFR could reduce the network costs in all hours of the day. The open switches are also shown in the last column of Table 3.

5.2 Uncontrolled charging

In the second case, the ability of DFR on eliminatingthe uncontrolled charging of PHEVs is assessed. In the uncontrolled mode, all PHEVs start charging as they reach to the parking. Therefore, it will coincide with the peak load hours. This event shows a “worst case” scenario as shown in Fig. 4. According to this figure, the peak load of the network is increased effectively.

The results of the 24-hour analyses are given in Table 4. As it can be seen from this table, the uncontrolled charging of PHEVs has resulted in incremental cost at the peak load hours. Nevertheless, yet the DFR strategy could reduce the network costs at most of the hours with respect to the system constraints.

5.3 Smart charging

In the last case, the smart charging of PHEVs is considered in the network. Here, by the use of Equation 1,the charging demands of PHEVs are determined in both residential and public charging places as shown in Fig. 5. As it can be seen from Fig. 5, the smart charging has shifted the average demand of PHEVs to off peak or inexpensive hours.

Using the charging pattern suggested by the smart charging method, the load curve of the network would become as shown in Fig. 6. According to Fig. 6, the aggregated load demand of the network is shifted to the first hours of the day with lower cost.

As explained before, both types of PHEV charging points incline to be charged throughout cheap hours. Using the new load pattern, Table 5 shows the effect of DFR on the network before and after the reconfiguration. In comparison with the uncontrolled charging scheme, the hourly cost of the network is reduced using the smart charging pattern. Similarly, the positive effect of DFR can be deduced from this table clearly.

In order to have a better comparison, Fig. 7 shows the hourly cost of the network in three cases of 1) DFR neglecting PHEVs, 2) DFR with uncontrolled charging of PHEVs and 3) DFR with smart charging of PHEVs. The usefulness of the proposed smart charging pattern is deduced from this figure clearly. Finally, Table 6 shows the daily cost of the network for the above mentioned three cases. The positive role of DFR can be deduced from these results too.

6 Conclusion

The Ability of DFR in minimizing the influence of PHEV charging in the forthcoming smart distribution grids was studied in a stochastic framework. In this regard, dissimilar charging schemes for PHEVs were regarded to check the DFR strategy. Furthermore, the stochastic performance of PHEVs was modeled to offer more representative conditions. The suggested stochastic DFR was solved using a new self-adaptive optimization method based on MGSA. Relative results showed that uncontrolled charging can overlap the network peak hours and thus increases the total network costs. Nevertheless, DFR has been successful in reducing the network costs in all charging patterns. Ultimately, considering DFR along with the smart charging strategy can decrease the loading impacts of PHEVs properly.

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