A Convex Optimization Algorithm for Electricity Pricing of Charging Stations

algorithms

Article A Convex Optimization Algorithm for Electricity Pricing of Charging Stations

Jing Zhang 1, Xiangpeng Zhan 2,* , Taoyong Li 1, Linru Jiang 1, Jun Yang 2, Yuanxing Zhang 1, Xiaohong Diao 1 and Sining Han 2

1 Beijing Engineering Technology Research Center of Electric Vehicle Charging/Battery Swap, China Electric Power Research Institute Co., Ltd., Beijing 100192, China; [email protected] (J.Z.); [email protected] (T.L.); [email protected] (L.J.); [email protected] (Y.Z.); [email protected] (X.D.) 2 School of Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China; [email protected] (J.Y.); [email protected] (S.H.) * Correspondence: [email protected]; Tel.: +86-15827337902

Received: 27 July 2019; Accepted: 20 September 2019; Published: 1 October 2019

Abstract: The problem of electricity pricing for charging stations is a multi-objective mixed integer nonlinear programming. Existing algorithms have low eﬃciency in solving this problem. In this paper, a convex optimization algorithm is proposed to get the optimal solution quickly. Firstly, the model is transformed into a convex optimization problem by second-order conic relaxation and Karush–Kuhn–Tucker optimality conditions. Secondly, a polyhedral approximation method is applied to construct a mixed integer linear programming, which can be solved quickly by branch and bound method. Finally, the model is solved many times to obtain the Pareto front according to the scalarization basic theorem. Based on an IEEE 33-bus distribution network model, simulation results show that the proposed algorithm can obtain an exact global optimal solution quickly compared with the heuristic method.

Keywords: charging station; convex optimization; multi-objective programming; polyhedral approximation; scaling method

1. Introduction With the popularization of traffic electrification, the number of electric vehicles (EVs) is increasing year by year. It is estimated that the number of EVs will reach 11 million in 2025 and 30 million in 2030 [1]. In order to satisfy the charging demand of EVs, a large number of charging stations (CSs) have been built in the distribution network. Considering the charging behavior of EV closely related to people’s daily work and rest, EV users usually charge after work in the evening. In addition, the operation of the distribution network will be severely impacted by a large number of charging loads [2]. Therefore, the electric energy plan of CSs must be optimized to ensure the orderly charging of EVs [3]. Electric vehicle users are guided to change charging time by CSs setting charging prices [4]. The relationship among electric vehicle users, charging station operators, and distribution network operators (DSO), needs to be taken into consideration. However, existing algorithms are inefficient in solving this problem [5]. Therefore, it is very meaningful to develop the optimization algorithm for electric power planning of charging stations. For the time being, the research on charging price of EVs mainly focuses on time-of-use price [6]. In [7–9], the distribution locational marginal pricing (DLMP) method is applied to optimize the charging price of EVs to reduce the impact of charging load on distribution network. In [10–12], the electricity demand and travel characteristics of EV users are considered, and charging price is optimized by the

Algorithms 2019, 12, 208; doi:10.3390/a12100208 www.mdpi.com/journal/algorithms Algorithms 2019, 12, 208 2 of 14 price demand elasticity coeﬃcient. In [13], a Stackelberg leadership game model is applied to optimize the electricity prices for encouraging EV users to participate in the electricity market, improving their own interests, and promoting the development of smart energy systems. However, the problem of power planning for charging stations is a multi-objective mixed integer nonlinear programming (MINLP) and some constraints are nonconvex. Heuristic algorithms are usually used to solve this problem such as genetic algorithm (GA) in [14] and particle swarm optimization (PSO) in [15], which take a long time to solve and easily fall into local optimum. Aiming at getting the global optimal solution quickly, the main contribution of this paper is to propose a convex optimization algorithm for electricity pricing of charging stations. Firstly, the model is transformed into a convex optimization problem by second-order conic relaxation and Karush–Kuhn–Tucker optimality conditions. Secondly, a polyhedral approximation method is applied to construct a mixed integer linear programming, which can be solved quickly by branch and bound method. Finally, the model is solved iteratively to obtain the Pareto front according to the scalarization basic theorem. On the one hand, the optimal solution can be obtained in the strict sense provided with a convex optimization problem. The conic problem is approximated to a simple type based on polyhedral approximation in order to improve the speed of solution. In addition, the tripartite game problem is decomposed into a Stackelberg leadership game and a cooperative game. Besides, the multi-solution property of the multi-objective optimization problem is preserved under the scaling method. The remainder of the paper is organized as follows. Section2 introduces the model of charging station electric power planning. Section3 proposes the convex optimization algorithm based on second order conic relaxation (SOCR) and polyhedral approximation method. Simulations and analysis are carried out in Section4. Finally, a conclusion is drawn in Section5.

2. The Model of Electricity Pricing of Charging Stations The benefits of DSO and EVs should be taken into account when CSs set electricity prices. The relationship among DSO, CSs, and EVs is shown in Figure1. Usually, optimizing the power plan of CSs can not only improve its own revenue, but also reduce the peak-valley difference of the distribution network. DSO and CSs can achieve a win-win situation, which constitutes a cooperative relationship [11]. In order to adjust the load curve, DSO guides CSs through marginal price to formulate an appropriate power plan. Charging fee is a cost for EVs but a revenue for CSs. Therefore, there is a conflict of benefits between EVs and CSs, which can be considered as a non-cooperative game relationship [16,17]. A more accurate description should be the Stackelberg game between EVs and CSs because there is a sequence of decisions between the two sides [18]. After charging station operators set charging service price, electric vehicle users decide charging mode according to the price. The goal of EVs is to reduce charging cost and thus EVs will charge when the charging price is low. When CSs formulate the power plan, guiding EVs to charge orderly by setting an appropriate charging price is necessary, and the revenue of CSs should also be maximized. Algorithms 2019, 12, x FOR PEER REVIEW 2 of 14 applied to optimize the electricity prices for encouraging EV users to participate in the electricity market, improving their own interests, and promoting the development of smart energy systems. However, the problem of power planning for charging stations is a multi-objective mixed integer nonlinear programming (MINLP) and some constraints are nonconvex. Heuristic algorithms are usually used to solve this problem such as genetic algorithm (GA) in [14] and particle swarm optimization (PSO) in [15], which take a long time to solve and easily fall into local optimum. Aiming at getting the global optimal solution quickly, the main contribution of this paper is to propose a convex optimization algorithm for electricity pricing of charging stations. Firstly, the model is transformed into a convex optimization problem by second-order conic relaxation and Karush–Kuhn–Tucker optimality conditions. Secondly, a polyhedral approximation method is applied to construct a mixed integer linear programming, which can be solved quickly by branch and bound method. Finally, the model is solved iteratively to obtain the Pareto front according to the scalarization basic theorem. On the one hand, the optimal solution can be obtained in the strict sense provided with a convex optimization problem. The conic problem is approximated to a simple type based on polyhedral approximation in order to improve the speed of solution. In addition, the tripartite game problem is decomposed into a Stackelberg leadership game and a cooperative game. Besides, the multi-solution property of the multi-objective optimization problem is preserved under the scaling method. The remainder of the paper is organized as follows. Section 2 introduces the model of charging station electric power planning. Section 3 proposes the convex optimization algorithm based on second order conic relaxation (SOCR) and polyhedral approximation method. Simulations and analysis are carried out in Section 4. Finally, a conclusion is drawn in Section 5.

2. The Model of Electricity Pricing of Charging Stations The benefits of DSO and EVs should be taken into account when CSs set electricity prices. The relationship among DSO, CSs, and EVs is shown in Figure 1. Usually, optimizing the power plan of CSs can not only improve its own revenue, but also reduce the peak-valley difference of the distribution network. DSO and CSs can achieve a win-win situation, which constitutes a cooperative relationship [11]. In order to adjust the load curve, DSO guides CSs through marginal price to formulate an appropriate power plan. Charging fee is a cost for EVs but a revenue for CSs. Therefore, there is a conflict of benefits between EVs and CSs, which can be considered as a non-cooperative game relationship [16,17]. A more accurate description should be the Stackelberg game between EVs and CSs because there is a sequence of decisions between the two sides [18]. After charging station operators set charging service price, electric vehicle users decide charging mode according to the price. The goal of EVs is to reduce charging cost and thus EVs will charge when the charging price is low. When CSs formulate the power plan, guiding EVs to charge orderly Algorithmsby setting2019 an, 12 ,appropriate 208 charging price is necessary, and the revenue of CSs should also3 of be 14 maximized.

Distribution System Charging Stations Electric Vehicle Operators Energy Charging Users Plan Time

Marginal Price Signal Goal: Reducing Load Fluctuation Price Goal: Increasing Income Goal: Reducing Charging Cost

Variable: Power of Charging Station Variable: Charging Price Variable: Charging plan

cooperative game Stackelberg leadership game Figure 1. TheThe relationship relationship among among distribution distribution network network operators operators (DSO), (DSO), charging charging stations stations (CSs), (CSs), and and electric vehicles (EVs). electric vehicles (EVs).

The electricity pricing problem of charging stations is a classical problem in power systems [2–11], which can be decomposed into the security checking problem of DSO, charging pricing problem of CSs, and charging mode optimization problem of EVs. In the security checking problem of DSO, the objective function of DSO is shown in Equation (1) and the constraints are shown in Equation (2), including power demand equation constraints, bus power balance constraints, bus voltage safety constraints, and line power transmission capacity constraints. The variables in this problem are n i PG,t, Pc,t, Vi,t, θij,t} and the purpose of this problem is to ensure the security of the power system [11].

min fDSO = PG,t PG 2, (1) i || − || Pc,t

= P ( i + i ) + P ( 2 + 2 ) s.t PG,t PL,t Pc,t Gij Vi,t Vj,t 2Vi,tVj,t cos θij,t , t T i I ij L − ∀ ∈ 1 ∈P ∈ PG = T PG,t t T Pi + Pi∈ = G V2 V V (G cos θ + B sin θ ), t T, i I L,t c,t ij i,t − i,t j,t ij ij,t ij ij,t ∀ ∈ ∀ ∈ (2) Qi = B V2 + V V (B cos θ G sin θ ), t T, i I L,t − ij i,t i,t j,t ij ij,t − ij ij,t ∀ ∈ ∀ ∈ V V Ve, t T, i I ≤ i,t ≤ ∀ ∈ ∀ ∈ (G2 + B2 )(V2 + V2 2V V cos θ + 2V V sin θ ) eI2 , t T, ij L ij ij i,t j,t − i,t j,t ij,t i,t j,t ij,t ≤ ij ∀ ∈ ∀ ∈ where fDSO is the deviation of load in distribution network; PG,t is the total active power of DSO in i i period t; PG is the average active power of DSO; PL,t is the basic active power at bus i in period t; QL,t is i the basic reactive power at bus i in period t; Pc,t is the charging power at bus i in period t; Gij is the conductance of line ij; Line ij stands for transmission line from bus i to bus j; Vi,t is the voltage at bus i in period t; θij,t is the power angle diﬀerence of line ij in period t; T is the set of dispatching time; Bij is the susceptance of line ij; I is the bus set of distribution network; L is the set of transmission lines of distribution network; Ve and V are the boundary of voltage. In the charging pricing problem of CSs, the objective function of CSs is shown in Equation (3) and the constraints are shown in Equation (4), including price boundary constraints and average price equality constraints in order to ensure the fairness of the market when the competition between CSs is n i n neglected [19]. The variables in this problem are πc,t, Pc,t, pc,t, Pb,t} and the purpose of this problem is to maximize the revenue of the CSs [18]. X X X i max fCSs = πc,t Pc,t πb,tPb,t, (3) πc,t − t T i I t T ∈ ∈ ∈ Algorithms 2019, 12, 208 4 of 14

P i s.t Pb,t = Pc,t, t T i I ∀ ∈ i ∈P n Pc,t = pc,t, t T, i I n N(i) ∀ ∈ ∀ ∈ ∈ (4) π πc,t eπc, t T c ≤ ≤ ∀ ∈ 1 P T πc,t = πc t T ∈ where fCSs is the income of CSs; πc,t is the charging price at period t; πb,t is the price of electricity n purchase Pb,t at period t; N is the set of electric vehicle users; pc,t is the charging power of EVn at period t; eπc and π are the boundary of charging price; πc is the average charging price. c In charging mode optimization problem of EVs, the objective function of EVs is shown in Equation (5) and the constraints are shown in Equation (6), including power demand equation constraints, maximum charging power constraints, and available charging time constraints. The variables in this n n problem are pc,t} and the purpose of this problem is to minimize the charging cost of EVs [18]. X n min fEV,n = pc,tπc,t, n N, (5) pn ∀ ∈ c,t t T ∈ P n n s.t pc,t = Eev, n N t T ∀ ∈ ∈ n (6) 0 pc,t pc, t T, n N n≤ ≤ ∀ ∈ ∀ ∈ p = 0, t < Te(n), n N c,t ∀ ∀ ∈ n where f EV,n is the charging cost of EVn; Eev is the power demand of EVn; pc is the maximum charging power; Te(n) is the set of rechargeable time of EVn. From the above formulas, there are three objective functions in the electricity pricing problem and there are correlations among variables, what lead to the complexity of the model. Hence a series of equivalent transformations must be applied to the model. According to the Stackelberg leadership game model, the game between CSs and EVs can be regarded as a bilevel optimization problem [20], and the subproblems can be expressed as Equation (7). X n n pc,t = argmin pc,tπc,t. (7) t T ∈ In order to obtain the optimal solution of the problem, the Lagrange function is shown in Equation (8). X X X X X n n n n n n n n n n Ln = p πc,t λ ( p E ) µ p µ (p p ) µ p , n N, (8) c,t − c,t − ev − lb,t c,t − ub,t c,t − c − a,t c,t ∀ ∈ t T t T t T t T t T ∈ ∈ ∈ ∈ ∈ n where, Ln is the Lagrange function of Equations (5) and (6); λ is the dual variable of power demand n n n equation constraints; µlb,t and µub,t are dual variables of maximum charging power constraints; µa,t is the dual variable of available charging time constraints. According to the linear duality theory [21], the dual expression of Equation (5) is shown in Equation (9). At the same time, the optimum condition is Equation (10). The optimization problem of electric vehicle users is transformed into a series of Karush–Kuhn–Tucker (KKT) optimality conditions. X min f = max( f ) = max(λnEn + µn p ), (9) EV,n − EV,n ev ub,t c t T ∈ Algorithms 2019, 12, 208 5 of 14

P n n pc,t Eev = 0, n N t T − ∀ ∈ 0∈ pn p ≤ c,t ≤ c ∂L = n n n n = ∂pn πc,t λ µ t µ t µa,t 0, t T, n N (10) c,t − − lb, − ub, − ∀ ∈ ∀ ∈ 0 µn pn , t T, n N ≤ lb,t⊥ c,t ∀ ∈ ∀ ∈ 0 µn (pn p ), t T, n N ≥ ub,t⊥ c,t − c ∀ ∈ ∀ ∈ However, Equation (10) includes some complementary constraints, which leads to the nonlinearity of the model. A big-M method is applied to relax the complementary constraints into mixed integer constraints, as Equation (11). At the same time, the main problem can be expressed as Equation (12) according to duality theory.

0 µn MXn, t T, n N ≤ lb,t ≤ t ∀ ∈ ∀ ∈ 0 pn M(1 Xn), t T, n N ≤ c,t ≤ − t ∀ ∈ ∀ ∈ (11) 0 p pn MYn, t T, n N ≤ c − c,t ≤ t ∀ ∈ ∀ ∈ M(Yn 1) µn 0, t T, n N t − ≤ ub,t ≤ ∀ ∈ ∀ ∈ X X X = ( n n + n ) max fCSs λ Eev µub,tpc πb,tPb,t, (12) πc,t − n N t T t T ∈ ∈ ∈ n n where M is a given big value; Xt and Yt are binary variables for relaxing Equation (10). So far, the charging station power planning model is a MINLP with integer variables and non-linear expressions. Additionally, the quadratic equality constraints in Equation (2) make the model non-convex. At the same time, the beneﬁts of DSO and CSs should be considered, which lead to multiple solutions of the model. In short, the problem of power planning for charging stations is a NP-hard problem and existing methods have diﬃculty in solving this problem.

3. Convex Optimization Algorithm In this section, the second order conic relaxation method is proposed to transform the model into a convex optimization problem and the scaling algorithm is proposed to obtain the Pareto front.

3.1. SOCR Method In [22], a method of equality transformation is proposed to promotion dimension of model like Equation (13). R = V V cos θ , t T, ij L ij,t i,t j,t ij,t ∀ ∈ ∀ ∈ Sij,t = Vi,tVj,t sin θij,t, t T, ij L ∀ ∈ ∀ ∈ (13) W = V2 , t T, i I i,t i,t ∀ ∈ ∀ ∈ R2 + S2 = W W , t T, ij L ij,t ij,t i,t j,t ∀ ∈ ∀ ∈ where Rij,t, Sij,t, Wi,t are temporary variables in order to replace the original variables, which have no practical physical signiﬁcance. n i o After this transformation, decision variables have become PG,t, Pc,t, Wi,t, Rij,t, Sij,t in the security checking problem of DSO. Additionally, Equation (2) becomes a linear form like Equation (14). In this process, variables are projected into a higher dimensional space. In order to ensure the accuracy of the original problem, a new set of equality constraints need to be added as Equation (13), which Algorithms 2019, 12, 208 6 of 14 constitutes a set of rotating conical spaces. In order to transform the model into convex optimization form, Equation (15) is applied to relax the feasible region to some second order cones.

= P ( i + i ) + P ( + ) PG,t PL,t Pc,t Gij Wi,t Wj,t 2Rij,t , t T i I ij L − ∀ ∈ ∈ ∈ Pi + Pi = G W (G R + B S ), t T, i I L,t c,t ij i,t − ij ij,t ij ij,t ∀ ∈ ∀ ∈ Qi = B W + (B R G S ), t T, i I (14) L,t − ij i,t ij ij,t − ij ij,t ∀ ∈ ∀ ∈ V2 W Ve2, t T, i I ≤ i,t ≤ ∀ ∈ ∀ ∈ (G2 + B2 )(W + W 2R + 2S ) eI2 , t T, ij L ij ij i,t j,t − ij,t ij,t ≤ ij ∀ ∈ ∀ ∈

R2 + S2 W W , t T, ij L. (15) ij,t ij,t ≤ i,t j,t ∀ ∈ ∀ ∈ So far, the original problem has been relaxed into a mixed integer second order conic programming (MISOCP). The existing commercial solver can solve this convex optimization problem. However, the existing convex optimization algorithms are weak in solving this problem because the problem contains a large number of variables [21]. In order to improve the efficiency of solving this problem, a polyhedral approximation method is proposed to construct a mixed integer linear programming (MILP), which can be solved quickly by general algorithms like the branch and bound method [23]. At the same time, the literature [22] has confirmed that the polyhedral approximation method can accurately control model errors and has high efficiency in power flow problems. At first, Equation (15) is rewritten as Equation (16). And Equation (17) is applied to relax Equation (16) to a polyhedral space. s 2 Wi,t Wj,t Wi,t + Wj,t R2 + S2 + ( − ) , t T, ij L, (16) ij,t ij,t 2 ≤ 2 ∀ ∈ ∀ ∈ s 2 Wi,t Wj,t Wi,t + Wj,t R2 + S2 + ( − ) (1 + ε)( ), t T, ij Lm (17) ij,t ij,t 2 ≤ 2 ∀ ∈ ∀ ∈ where ε is a relaxation scalar and ε (0, 0.5]. ∈ Secondly, the rotating cones are equivalent to two standard cones like Equations (18) and (19). q E R2 + S2 , t T, ij L, (18) ij,t ≥ ij,t ij,t ∀ ∈ ∀ ∈ s 2 Wi,t + Wj,t Wi,t Wj,t ( − ) + E2 , t T, ij L. (19) 2 ≥ 2 ij,t ∀ ∈ ∀ ∈ Finally, the smooth edges of cones are replaced with polyhedrons by Equations (21)–(23). The dimensions of a polyhedron depend on the accuracy required in Equation (20).

1 ε(υ) = 1. (20) cos( π ) − 2υ+1

The boundary of a polyhedron is calculated by recursion. The initial boundary can be calculated by Equation (21) and the recursive expression is shown in Equation (22). These expressions form a set of linear spaces when υ is given.

ς0 R , t T, ij L ij,t ≥ | ij,t| ∀ ∈ ∀ ∈ 0 ηij,t Sij,t , t T, ij L ≥ | W | W∀ ∈ ∀ ∈ (21) ξ0 i,t− j,t , t T, ij L ij,t ≥ | 2 | ∀ ∈ ∀ ∈ σ0 E , t T, ij L ij,t ≥ | ij,t| ∀ ∈ ∀ ∈ Algorithms 2019, 12, x FOR PEER REVIEW 7 of 14

Algorithms 2019, 12, 208 7 of 14 ς υ ≤∀∈∀∈ ij,, tEtTijL ij t ,, k π k 1 π k 1 ς = cos( + )ς − + sin(π + )η − , k 1, ... , υ, t T, ij L ij,t 2k 1 ηςijυυ,t ≤∀∈∀∈tan(2k )1 ,ij,ttT∈ , ijL∀ ∈ ∀ ∈ ηk sin( π )ijς, t k 1 + cos2υ+(1 π )ηk 1 , k 1, ... , υ, t T, ij L ij,t ≥ | − 2k+1 ij−,t 2k+1 ij−,t | ∈ ∀ ∈ ∀ ∈ k π k 1 + π k 1 . (22) ξ = cos( )ξυ +WWsinit,,( jt)σ , k 1, ... , υ, t T, ij L (23) ij,t 2k+1 ξij−,t ≤∀∈∀∈2k+1 ,,ijtT−,t ∈ ijL ∀ ∈ ∀ ∈ k π ij, t k 1 2 π k 1 σ sin( k+1 )ξ − + cos( k+1 )σ − , k 1, ... , υ, t T, ij L ij,t ≥ | − 2 ij,t π 2 ij,t | ∈ ∀ ∈ ∀ ∈ σξυυ≤∀∈∀∈tan( ) ,tT , ijL Finally, the upper bound of theij,,polyhedron t2υ+1 is ij t deﬁned by Equation (23).

ςυ E , t T, ij L The details of polyhedral approximationij,t ≤ ij,t ∀ method∈ ∀ are∈ described in Algorithm 1. ηυ tan( π )ςυ, t T, ij L ij,t 2υ+1 ≤ W +W ∀ ∈ ∀ ∈ (23) ξυ i,t j,t , t T, ij L Algorithm 1: Polyhedral approximationij,t ≤ 2 algorithm∀ ∈ ∀ ∈ συ tan( π )ξυ , t T, ij L 1: υ: = 0, obtain ε by Equation ij(20),,t ≤ construct2υ+1 modelij,t ∀ ∈ by Equations∀ ∈ (11), (12), and (14) -7 2: The while details ε>10 of polyhedral do approximation method are described in Algorithm 1. 3: υ++, obtain ε by Equation (20) 4: endAlgorithm while 1: Polyhedral approximation algorithm 5: for1: t =υ 1: =to0, T obtain do ε by Equation (20), construct model by Equations (11), (12), and (14) 7 2: while ε > 10− do 6: for ij = 1 to L do 3: υ++, obtain ε by Equation (20) 7: 4: addend Equations while (21) and (23) into model 8: 5: for fort =k 1= to1 toT doυ do 6: for ij = 1 to L do 9: 7: add add Equations Equation (21) (22) and into (23) intomodel model 10: 8: endfor dok = 1 to υ do 9: add Equation (22) into model 11: end do 10: end do 12: end11: do end do 13: solve12: MILPend do by branch and bound method 13: solve MILP by branch and bound method 14: return PG,t, Pc,t, πc,t, Rij,t, Sij,t, Wi,t, fDSO, fCSs and fEV,n 14: return PG,t, Pc,t, πc,t, Rij,t, Sij,t, Wi,t, fDSO, fCSs and fEV,n

TheThe relaxation relaxation process process is is shown shown in in Figure Figure 22.. The original non-convex feasible region is ﬁrstfirst relaxedrelaxed to to a a second second order order cone cone and and then then to to a a series series of of polyhedrons, polyhedrons, which which can can be be solved solved quickly quickly by by simplexsimplex method. method.

optimum solution

S SOCR polyhedral approximation S S

original solution domain (nonconvex) simplex Figure 2. Relaxation process. Figure 2. Relaxation process. The original solution variables are projected into high-dimensional space by SOCR, so it is necessaryThe original to get thesolution original variables variables are by projected returning in theto mapping.high-dimensional Firstly, the spac voltagee by canSOCR, be returnedso it is necessaryby Equation to (24).get the original variables by returning the mapping. Firstly, the voltage can be p returned by Equation (24). V = W , t T, i I. (24) i,t i,t ∀ ∈ ∀ ∈ Algorithms 2019, 12, 208 8 of 14

Then the power angle diﬀerence of the line can be obtained by Equations (25) and (26). Based on the agreement of θ0,t = 0, all angles can be obtained by solving the matrix.

Sij,t tan(θi,t θj,t) = tan θij,t = , t T, ij L, (25) − Rij,t ∀ ∈ ∀ ∈

S 1 ij,t θi,t θj,t = tan− , t T, ij L. (26) − Rij,t ∀ ∈ ∀ ∈ The details of return mapping method are described in Algorithm 2.

Algorithm 2: Return mapping algorithm

1: Rij,t, Sij,t, and Wi,t: calculated by Algorithm 1 2: for t = 1 to T do 3: for i = 1 to N do 4: Vi,t: calculated by Equation (24) 5: end do 6: θN0,t:=0 7: for ij = 1 to L do 8: θi,t: calculated by Equation (26) 9: end do 10: end do 11: return Vi,t and θi,t

In order to verify the accuracy of relaxation, relaxation errors are deﬁned as Equation (27) to check whether constraints are satisﬁed.

gap = R2 + S2 W W , t T, ij L. (27) ij,t | ij,t ij,t − i,t j,t| ∀ ∈ ∀ ∈ 3.2. Scaling Algorithm DSO and CSs are cooperative game relations, but both sides have their own beneﬁts. The problem of power planning for charging stations is a multi-objective programming, and the Pareto front can balance the beneﬁts of DSO and CSs well. According to the Pareto front, all the cooperation results can be obtained, and then the appropriate cooperation decision can be selected [24]. Therefore, the key to the problem is how to get the Pareto front of the model. According to the scalarization basic theorem [21], the Pareto front can be obtained by transforming the objective function into a set of constraints and solving the model many times. In this paper, the objective function of DSO is transformed into a constraint like Equation (28). By changing DL, the model is solved many times to obtain the Pareto front. P P DL. (28) || G,t − G||2 ≤ In order to simplify the model, standard deviation as Equation (29) is used to simplify Equation (28), which can be rewritten as a set of linear constraints.

1 X P P DL. (29) T | G,t − G| ≤ t T ∈ The details of scaling method are described in Algorithm 3. Algorithms 2019, 12, x FOR PEER REVIEW 9 of 14

Algorithm 3: Scaling algorithm

1 DL: = +∞, PG,t: obtain by Algorithm 1, fDSO: obtain by Algorithm 1, fCSs: obtain by Algorithm 1 1 − Algorithms2 D2019L: = , 12, 208||PPGt, G , t: = 0, obj1[t]: = fDSO, obj2[t]: = fCSs 9 of 14 T tT∈ 3 repeat Algorithm 3: Scaling algorithm 4 solve MILP by algorithm 1, fDSO: obtain by Algorithm 1, fCSs: obtain by Algorithm 1 1 D : = + , P ,t: obtain by Algorithm 1, f : obtain by Algorithm 1, f : obtain by Algorithm 1 5 L if fCSs ==G obj2[t] then DSO CSs 1∞P 2 DL: = T PG,t PG , t: = 0, obj1[t]: = fDSO, obj2[t]: = fCSs 6 t breakT | − | 3 repeat ∈ 7 else then 4 solve MILP by algorithm 1, fDSO: obtain by Algorithm 1, fCSs: obtain by Algorithm 1 85 if fCSs ==t: t++obj 2[t] then 6 break 9 obj1[t]: = fDSO, obj2[t]: = fCSs 7 else then 108 t: tendif++ 119 until obj1 [t]:the= problemfDSO, obj 2is[t]: infeasible= fCSs 10 end if 12 return obj1 and obj2 11 until the problem is infeasible 12 return obj1 and obj2 4. Case Studies 4. Case Studies 4.1. Data Generation 4.1. DataThe case Generation uses an IEEE 33-bus system [25], and the urban area is divided into residential area, commercialThe case area, uses and an industry IEEE 33-bus area system[26]. The [25 total], and load the is urban3715 kVA area and is divided there are into three residential types of area,EVs incommercial the distribution area, and network. industry The area [structure26]. The totalof th loade distribution is 3715 kVA network and there and are threethe location types of EVsof the in chargingthe distribution station network.are shown The in structureFigure 3 ofand the the distribution electricity networkprice of andDSO the is shown location in of Figure the charging 4. The datastation of areelectric shown vehicles in Figure can3 andbe obtained the electricity in [20] price. The of DSObattery is showncapacity in Figureof EVs4 .is The assumed data of to electric be 42 kWh,vehicles and can the be initial obtained state in of [20 charge]. The battery(SOC) capacityis 30%. There of EVs are is assumed60 charging to be piles 42 kWh, in the and CS the and initial the maximumstate of charge charging (SOC) powe is 30%.r of There each are charging 60 charging pile is piles 30 inkW. the In CS order and theto simplify maximum the charging problem, power this paperof each assumes charging that pile there is 30 are kW. three In order types to of simplify EVs in distribution the problem, network this paper and assumes their parking that there time are is shownthree types in Figure of EVs 5. in Additionally, distribution networkthe number and of their thr parkingee types timeof electric is shown vehicles in Figure is 200,5. Additionally, 150, and 50 respectively.the number ofAll three simulations types of electricare im vehiclesplemented is 200,on 150,a laptop and 50with respectively. Intel® Core(TM) All simulations i5-7300HQ are [email protected] GHz, on 8 GB a laptop memory, with and Intel the® algorithmsCore(TM) i5-7300HQare compiled [email protected] by MATLAB. GHz, 8YALMIP GB memory, tools and[23] theare usedalgorithms to model are compiledand CPLEX by MATLAB.[27] is called YALMIP to solve tools the [23 MILP] are usedmodel to by model branch and and CPLEX bound [27] method. is called Branchto solve and the bound MILP modelmethod by is brancha common and method, bound method.whose process Branch only and briefly bound desc methodribed isin athis common article canmethod, be seen whose in [28]. process only brieﬂy described in this article can be seen in [28].

Business Area Industry Area Residence Area 18 19 20 21

012 3 4 5 6 789 10 11 12 13 14 15 16 17

25 26 27 28 29 30 31 32 22 23 24

Figure 3. IEEEIEEE 33-bus 33-bus system. system. Algorithms 2019, 12, x FOR PEER REVIEW 10 of 14

AlgorithmsAlgorithms2019 2019,,12 12,, 208 x FOR PEER REVIEW 1010 ofof 1414 0.12 0.12 0.1 0.1 0.08 0.08 Base price($) Base 0.06 Base price($) Base 0.06 0.04 00:00 05:00 10:00 15:00 20:00 24:00 0.04 Time 00:00 05:00 10:00 15:00 20:00 24:00 Time Figure 4. The basic price of DSO. Figure 4. The basic price of DSO. Figure 4. The basic price of DSO.

EV3 EV3

EV2 EV2

EV1 EV1 00:00 05:00 10:00 15:00 20:00 24:00 Time 00:00 05:00 10:00 15:00 20:00 24:00 Figure 5. The parkingTime time of EVs. Figure 5. The parking time of EVs. 4.2. Discussion Figure 5. The parking time of EVs. 4.2. Discussion The Pareto front obtained is shown in Figure6a, and an equilibrium solution is selected to show the4.2. results.TheDiscussion Pareto As shownfront obtained in Figure is6 b,shown charging in Figure price 6a, is higher and an in equilibrium the peak period solution and is lower selected in the to valleyshow theperiod, results.The so Pareto asAs to shown guide front obtained electricin Figure vehicle is 6b, shown charging users in Figure to chargeprice 6a, is inand higher the an valley equilibrium in the period. peak solution However,period isand selected charging lower toin prices show the valleyarethe theresults. period, same As asso shown inas manyto guide in periodsFigure electric 6b, of vehicletime, charging in users order price to tocharge is avoid higher in users the in valleythe concentrating peak period. period However, on and charging lower charging atin thethe priceslowvalley price are period, the period, same so as resulting as to in guide many in electric newperiods load vehicle of peaks. time, users in Under order to chargethe to avoid guidance in the users valley of concentrating charging period. price,However, on thecharging chargingcharging at thepowerprices low ofare price di theﬀerent period,same electric as resultingin many vehicles periodsin isnew shown of load time, in peaks. Figure in order 6Underc. to It canavoid the be usersguidance seen thatconcentrating chargingof chargingload on chargingprice, is mainly the at chargingconcentratedthe low powerprice in period, theof different night resulting time electric and in some vehicles new electric load is shown vehiclespeaks. in Under Figure charge the6c. in the Itguidance can afternoon, be seen of duechargingthat to charging their price, parking load the ischaracteristics.charging mainly concentrated power The of different load in curvethe electricnight of distribution time vehicles and someis networkshown electric in after Figure vehicles orderly 6c. chargeIt chargingcan be in seen the of electricafternoon,that charging vehicles due load to is theirshownis mainly parking in Figureconcentrated characteristics.6d. Under in the The the night guidanceload time curve and of of charging distsomeribution electric price, network vehicles charging after charge load orderly mainlyin the charging afternoon, concentrates of electric due on to vehiclesthetheir period parking is ofshown lowcharacteristics. powerin Figure consumption, The6d. loadUnder curve which the of greatlyguidance distribution reduces of networkcharging the load after price, ﬂuctuation orderly charging charging of the load distribution of mainly electric concentratesnetwork.vehicles Itis can onshown bethe seen period in thatFigure of the low cooperative6d. power Under consumptio gamethe guidance betweenn, which DSOof charginggreatly and CSs reduces canprice, achieve thecharging load win-win fluctuation load situation mainly of theandconcentrates distribution maximize on social network.the period welfare. It of can low be power seen consumptiothat the cooperativen, which greatly game reducesbetween the DSO load and fluctuation CSs can of achievethe distribution win-win situation network. and It canmaximize be seen social that welfare. the cooperative game between DSO and CSs can achieve win-win situation and maximize social welfare. Algorithms 2019, 12, 208 11 of 14

Algorithms 2019, 12, x FOR PEER REVIEW 11 of 14

140 Equilibrium Solution

135

130

Income125 of CS($)

120 100 120 140 160 180 200 220 240 Standard deviation of load(kW) (a) 0.13

0.12

0.11

0.1 Chariging price($) 0.09

0.08 00:00 05:00 10:00 15:00 20:00 24:00 Time (b) 1000

800 EV1 EV2 600 EV3

400

200 Charging load(kW) Charging 0 00:00 05:00 10:00 15:00 20:00 24:00 Time (c) 5000 Base load Charging load 4000 Total load

3000

2000 Load(kW)

1000

0 00:00 05:00 10:00 15:00 20:00 24:00 Time (d)

Figure 6. Simulation results. (a) shows the Pareto front. (b) shows the charging prices. (c) shows the charging load of charging stations. (d) shows the total load of distribution system. Algorithms 2019, 12, x FOR PEER REVIEW 12 of 14 Algorithms 2019, 12, x FOR PEER REVIEW 12 of 14 Figure 6. Simulation results. (a) shows the Pareto front. (b) shows the charging prices. (c) shows the AlgorithmsFigure2019 6. ,Simulation12charging, 208 load results. of charging (a) shows stations. the Pareto (d) showsfront. ( theb) shows total load the chargingof distribution prices. system. (c) shows the 12 of 14 charging load of charging stations. (d) shows the total load of distribution system. The voltage distribution in the distribution network is shown in Figure 7. It can be seen that the The voltage distribution in the distribution network is shown in Figure7. It can be seen that the returnThe voltagemapping distribution algorithm in can the correctlydistribution restore network the isoriginal shown invariables. Figure 7. At It canthe be same seen time,that the the return mapping algorithm can correctly restore the original variables. At the same time, the minimum returnminimum mapping voltage algorithm of distribution can correctly network restore is 0.9 th921e originalpu, which variables. occurs atAt 9 thea.m. same on bustime, 31. the The voltage of distribution network is 0.9921 pu, which occurs at 9 a.m. on bus 31. The minimum operating minimumminimum voltage operating of distributionvoltage allowed network in the is 0.9distribution921 pu, which network occurs is 0.93 at 9pu. a.m. In onother bus words, 31. The the voltage allowed in the distribution network is 0.93 pu. In other words, the voltage security of the minimumvoltage security operating of thevoltage distributi allowedon network in the distributioncan be guaranteed network unde is r0.93 the schedulingpu. In other strategy words, in the this distribution network can be guaranteed under the scheduling strategy in this paper. As shown in voltagepaper. security As shown of thein Figure distributi 8, theon relaxationnetwork can error be ofguaranteed MILP is larger unde thanr the thatscheduling of MISOCP strategy but bothin this are Figure8, the relaxation error of MILP is larger than that of MISOCP but both are less than 10 7. paper.less than As shown 10–7. Compared in Figure 8,with the the relaxation numerical error value of MILP of decision is larger variables, than that this of MISOCPpart of the but error both can are− be Compared with the numerical value of decision variables, this part of the error can be neglected. lessneglected. than 10–7 It. Comparedcan be seen with that the the numerical second order value co ofne decisionrelaxation variables, has high this accuracy. part of Atthe the error same can time, be It can be seen that the second order cone relaxation has high accuracy. At the same time, polyhedral neglected.polyhedral It canapproximation be seen that will the notsecond greatly order reduce cone therelaxation accuracy has of highthe model. accuracy. At the same time, approximation will not greatly reduce the accuracy of the model. polyhedral approximation will not greatly reduce the accuracy of the model. 1.05 1.05 1.04 1.06 1.04 1.061.05 1.03

1.051.04 1.03 1.02 1.041.03 1.02 1.03 Volta/pu 1.02 1.01

Volta/pu 1.021.01 1.01 1.011 1 Minimum voltage:0.9921pu 1 At 9 a.m. in Node 31 1 0.99 Minimum voltage:0.9921pu 0.99 35 30 At25 9 a.m. in20 Node 3115 10 5 0 Node 35 30 25 20 15 10 5 0 Node Figure 7. Voltage distribution in the distribution network. Figure 7. Voltage distribution in the distribution network. Figure 7. Voltage distribution in the distribution network. x 10-7 1 -7 x 10 MISOCP 1 0.8 MILP MISOCP 0.8 MILP 0.6

0.6gap 0.4 gap 0.4 0.2 0.2 0 0 5 10 15 20 25 30 0 0 5 10 15Line 20 25 30 Figure 8. RelaxationLine error of the model. Figure 8. Relaxation error of the model. In this paper, the proposedFigure algorithm 8. Relaxation was compared error of the with model. GA [14] and PSO [15]. As can be seen In this paper, the proposed algorithm was compared with GA [14] and PSO [15]. As can be seen from Table1, the performance of GA and PSO is not ideal in solving this problem. In contrast, MISOCP from Table 1, the performance of GA and PSO is not ideal in solving this problem. In contrast, andIn MILP this paper, can obtain the proposed the optimal algorithm solution was in acompared very short with time. GA Besides, [14] and MISOCP PSO [15]. and As MILP can be get seen the MISOCP and MILP can obtain the optimal solution in a very short time. Besides, MISOCP and MILP fromglobal Table optimal 1, the solution performance according of toGA the and convex PSO optimization is not ideal theory. in solving Moreover, this problem. MILP is moreIn contrast, efficient get the global optimal solution according to the convex optimization theory. Moreover, MILP is MISOCPthan MISOCP and MILP without can losingobtain accuracy.the optimal Therefore, solution thein a polyhedral very short approximationtime. Besides, MISOCP algorithm and proposed MILP more efficient than MISOCP without losing accuracy. Therefore, the polyhedral approximation getin thisthe paperglobal canoptimal effectively solution improve according the effi tociency the convex of solving optimization this problem. theory. Moreover, MILP is morealgorithm efficient proposed than MISOCP in this paper with outcan effectivelylosing accuracy. improve Therefore, the efficiency the polyhedralof solving this approximation problem. algorithm proposed in this paper can effectively improve the efficiency of solving this problem. Algorithms 2019, 12, 208 13 of 14

Table 1. Comparisons of calculation results.

Algorithm fDSO (kW) fCSs ($) The Sum of fEVs ($) CPU Time (s) GA 183.5642 133.1157 1207.11 892.4 PSO 178.7535 127.5048 1244.34 411.5 MISOCP 177.1124 135.1261 1142.67 10.843 MILP 177.1121 135.1372 1143.12 0.172

5. Conclusions In this paper, a convex optimization algorithm was proposed to solve the problem of power planning for charging station. Second-order conic relaxation and Karush–Kuhn–Tucker optimality conditions were proposed to transform the model into a convex optimization problem. Further, a polyhedral approximation method was applied to construct a MILP and the model was solved iteratively to obtain the Pareto front according to the scalarization basic theorem. The simulation results of IEEE 33-bus test system show that some conclusions can be drawn as follows:

(1) The proposed second-order conic relaxation is accurate, which can transform the original model into a convex optimization model in order to obtain the global optimal solution. At the same time, the original variables can be easily restored. (2) The proposed polyhedral approximation method can speed up the solution of the model. Compared with other algorithms, the polyhedral approximation algorithm can get the global optimal solution with the minimum CPU time. (3) The proposed scaling algorithm can obtain the Pareto front by solving the problem iteratively, which can preserve the multi-solution of the model and obtain the equilibrium solution.

However, this study ignores the competitive relationship between charging stations, which may lead to an unfair market. Therefore, the game between charging stations needs further consideration, which is the focus of our future work.

Author Contributions: Conceptualization, J.Z. and T.L.; Methodology, X.Z. and S.H.; Validation, L.J. and Y.Z.; Formal Analysis, J.Y. and X.D. Funding: This research was funded by Open Fund of Beijing Engineering Technology Research Center of Electric Vehicle Charging/Battery Swap, China Electric Power Research Institute Co., Ltd. (YD83-19-003). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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