Day-Ahead Scheduling for an Electric Vehicle PV-Based Battery Swapping Station Considering the Dual Uncertainties

A day-ahead economic scheduling method based on chance-constrained programming and probabilistic sequence operation is proposed in this paper for an electric vehicle (EV) battery swapping station (BSS), considering the dual uncertainties of swapping demand and photovoltaic (PV) generation. First of all, a BSS day-ahead scheduling model that can deal with the uncertainties is established by using the chance-constrained programming. The optimization objective is to minimize the cost of electricity purchased from the utility grid with the chance constraints of swapping demand satisfaction and the confidence level of the minimum cost. Then, the deterministic transformation of chance constraints is implemented based on probabilistic sequences of stochastic variables. Thereafter, the feasible solution space of the proposed model is determined based on the battery controllable load margin, and then the fast optimization method for the BSS day-ahead scheduling model is developed by combining the feasible solution space and genetic algorithm (GA). In order to evaluate the solution quality, a risk assessment method based on the probabilistic sequence for day-ahead scheduling solutions is proposed. Finally, the efficiency and applicability of the proposed method is verified through the comparative analysis on a PV-based BSS system. Results illustrate that the model can provides a more reasonable charging strategy for the BSS operators with different risk appetite.


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A sign that a negative number returns to zero. The unit penalty for not meeting the swapping demand / the underestimation of electricity cost

INTRODUCTION
Recently, with the increase in the number of electric vehicles (EVs), the charging service has been widely concerned. The battery swapping mode not only enable the rapid battery energy replenishment, but also enables orderly charge management of batteries, so it could be considered as the most economical and efficient alternative mode. The addition of PV output can more effectively reduce energy cost [1][2][3].
The day-ahead economic scheduling of PV-based BSSs needs to take into account uncertain factors during the actual operation process. At present, scholars have carried out some initial relevant research work, including short-term forecasting, Monte Carlo simulation and multi-scene technology to deal with the uncertainties of swapping demand or PV output [3]- [5]. Most studies have used the typical cases of uncertain factors to represent uncertainties. However, one major drawback of these methods is that the timing combination of the actual values of uncertain factors inevitably deviates from the above typical cases, thus affecting the economy and applicability of day-ahead scheduling results in actual operation.
In order to solve the above problem, the probabilistic sequence is used to describe the dual uncertainties of swapping demand and PV output in this paper, which can cover the timing combination of any possible value of uncertain factors in actual operation. The obtained scheduling scheme has stronger applicability to the changes of uncertain factors.
Considering different risk appetite of operators, the chance-constrained programming with flexible choice of confidence level is adopted to establish the economic operation model of the BSS in this paper, and then the relationship between the economy and risk is measured.
For the problem of solving the BSS scheduling model, the algorithm used to solve this problem is compared in [6], and the results indicate that the performance of the GA and differential evolution algorithm is better than particle swarm optimization. In [7], GA is used to solve the economic operation model of the BSS, and removed the new individuals which did not meet the conditions generated by the genetic operation. But this operation is detrimental to the diversity of the population. Due to the complexity and timing correlation of the BSS scheduling model, how to ensure the feasibility of the solution and remain the efficiency of the algorithm are the key issues needed to be paid attention to in the research. In [8], for the dynamic classification of EVs, the energy limitation calculation model is proposed to determine the upper and lower limits of the controllable energy. The results were used to design the charging power allocation algorithm. The controllable load margin of the batteries in the BSS proposed in [9] has great guiding significance for the BSS operator to formulate the corresponding charging plan according to the grid operation requirements. By using the controllable load margin of batteries in the BSS proposed in [9] to determine the feasible solution space, a fast optimization method for the BSS day-ahead scheduling model is proposed.

Objective function
The objective function is to minimize the cost of electricity purchased from the utility grid, as follows:

Constraints
The probability of the actual swapping demand satisfaction should not be lower than the given confidence level  .
The probability that the electricity cost of the dayahead scheduling plan for the actual operation does not exceed the objective value should not be lower than the given confidence level  .
In this paper, batteries charging load should not exceed the upper and lower limits proposed in [9].

Probabilistic sequence and its operation theory
In [10], the probabilistic sequence is defined as: a discrete sequence   (1) Addition-type convolution operation . The operation defined by Eq (5) Where ya NN  . The operation defined by Eq (7)

Probability serialization of PV generation
The PV generation value can be expressed by the sum of the short-term predicted value and predicted error as is shown below, , and the standard deviation is 10% of the predicted value. The predicted value is used as the mean value to construct the probability density function (PDF) of PV output.
PV output timing multi-state probabilistic sequence represents the probability of different PV output states in each period, which can be calculated as Eq (12).

Probability serialization of swapping demand
The Monte Carlo simulation result of swapping demand in [9] is used as the predicted value. The timing multi-state probabilistic sequence of swapping demand is calculated in the same way as Pi.

Sequence operation results of the objective function
According to [5], the probabilistic sequence of the all-day electricity cost in the objective function is obtained by sequentially using the subtraction-type convolution, sequence multiplication, and addition-type convolution operations as is shown below

Deterministic transformation of chance constraints
The cumulative probability corresponding to different values of stochastic variables can be calculated by probabilistic sequence of stochastic variables. According to the cumulative probability, the value satisfying the chance constraint can be obtained, so that the chance constraint can be transformed into a deterministic constraint. The specific methods are described below.
In Section 3.1.3, the possible value of swapping demand in the t-th period is divided into , +1 Nneed t N states, and the probability corresponding to each state is Then, the cumulative probability corresponding to each state of swapping demand in the t-th period is In order to satisfy the chance constraint in Eq (2), it is assumed that the minimum number of available batteries in the t-th period is

Risk of unmet swapping demand
Due to the uncertainty of swapping demand, the actual value of swapping demand may be more or less than the number of batteries available in the BSS. There is a certain probability that the chance constraint in Eq (2) is not satisfied. Therefore, considering the economy and user satisfaction, the following rules for the BSS actual operation are formulated: 1) The number of batteries available in the BSS is updated every time period. 2) If the current swapping demand is lower than the number of available batteries, the number of available batteries in the next period increases the number of currently remaining available batteries. 3) If the current swapping demand is higher than the number of available batteries, the extra swapping demand will not be met.
The probabilistic sequence of the unmet swapping demand in the t-th period can be calculated by sequence operation as is shown below (20) The mathematical expectation of unmet swapping demand can be calculated as Eq (12).

Risk of underestimation of costs
Due to the influence of PV output on the forecast error, the electricity cost may be underestimated, and the mathematical expectation of the undervalued part can be expressed by the following equation. Where,

The penalty cost corresponding to the risk of dayahead scheduling
The penalty cost corresponding to the risk of dayahead scheduling can be calculated as Eq (24).

FAST OPTIMAIZATION METHOD BASED ON DETERMINING FEASIBLE SOLUTION SPACE
In this paper, the batteries charging load of each period is the decision variable in the day-ahead scheduling model. By using the controllable load margin of batteries in the BSS proposed in [9] to determine the feasible solution space, the operation of eliminating the infeasible solution in GA is avoided, and the optimization efficiency of GA can be improved. In order to represent the position of batteries charging load in the feasible solution space, 24 stochastic numbers   [0,1] xt are set as decision variables in GA.

CASE STUDY
It is assumed that there are 1,000 EVs in the service area of the BSS. The number of batteries in the BSS is 600. The number of chargers is 350. The average charging power of a battery is 5kW with 50kWh battery capacity. The minimum and maximum charge capacities of the battery are 20%, and 90% respectively. The TOU price for electricity purchased from the utility grid refers to the data in [7].
The PV installation capacity of the BSS is 240 kW. The typical summer PV output data is selected as the PV output prediction parameter. The discretization step size is 2.5 kW. Time-series multi-state probabilistic sequences of PV output and swapping demand are shown in Fig. 2, which represents the probability of different values in each period.
Case 1: multi-scene technology is used to deal with uncertain factors in the BSS day-ahead scheduling. The feasible solution space determined according to [9] is adopted to solve the model. Case 2: probabilistic sequence is used to deal with uncertain factors in the BSS day-ahead scheduling. The feasible solution space determined according to [9] is adopted to solve the model. Case 3: probabilistic sequence is used to deal with uncertain factors in the BSS day-ahead scheduling. The feasible solution space determined according to [9] is not adopted to solve the model.

Comparison of processing methods for uncertain factors
Charging plans for the above three cases are separately formulated. When  Table 1. There is little difference in electricity cost between the two cases, which proves the effectiveness of the proposed method in this paper. In addition, the penalty cost of Case 2 is much lower than that of Case 1. This is because the scenario reduction of multi-scene technology sacrifices the comprehensiveness of considering the uncertainty, which makes it more likely to underestimate the electricity cost or fail to meet the swapping demand compared with the method proposed in this paper.  Fig. 3. presents the probability of different electricity costs in each period in Case 2. It can be seen from Fig. 3 that the uncertainty of PV output is large in the 11am-3pm with a large average PV output. The wider the range of possible charging costs, the greater the uncertainty is. At 2pm, due to the lower charging load of the BSS, the PV output can fully bear the battery charging load in a certain probability, and the BSS does not need to purchase electricity from the grid. As a result, the charging cost is zero during this period. On the contrary, during the time when the average PV output is small, the cost floatation is smaller when the range of possible charging becomes narrow.

Comparison of optimization methods
In order to illustrate the superiority of the proposed fast determining the feasible solution space based optimization method, the undefined feasible solution space method and the proposed method are used to calculate the above case, respectively. The GA parameters of the two methods are consistent, and the corresponding iterative curve is shown in Fig. 4. It can be seen from Fig. 4 that the proposed optimal value based on the determination of the feasible solution space has stabilized at 400 generations, and the optimal average value is approached. The convergence speed of Copyright © 2019 ICAE the unconstrained feasible solution space is different from that of the proposed method, and the method falls into local optimum many times. Compared with the method that does not define the feasible solution space, the proposed method has improved the optimization ability, which indicates that the proposed method has advantages. From this research, the comparative method still does not converge at 500 generations, and there is still a certain gap in the optimal value between the two methods. This is because that the introduction of the feasible solution space makes the individuals generated by the genetic operation feasible, and avoids the damage to the population diversity due to the elimination of the infeasible individuals. In the case of the same population size, the proposed method improves the global diversity of the algorithm and achieves the goal of searching the optimization solution efficiently.

CONCLUSION
Due to the influence from the uncertainty of swapping demand and PV output on the day-ahead dispatching of the BSS, the probabilistic sequence is adopted to describe the stochastic variables. The chance-constrained programming is introduced into the economic operation model of the BSS. Compared with the multi-scene technology method, the advantages of using the probabilistic sequence are analyzed. A fast optimization method for the BSS scheduling model has been developed. The simulation results show that the proposed method has a higher tolerance to uncertainties and has the ability to improve the solving efficiency. Results illustrate that the model can provides a more reasonable charging strategy for the BSS operators with different risk appetite.

ACKNOWLEDGEMENT
This work was supported by the National Natural Science Foundation of China (Grant NO. 51477116).