# Model Predictive Control for Smart Grids with Multiple Electric-Vehicle Charging Stations

## Abstract

Next-generation power grids will likely enable concurrent service for residences and plug-in electric vehicles (PEVs). While the residence power demand profile is known and thus can be considered inelastic, the PEVs’ power demand is only known after random PEVs’ arrivals. PEV charging scheduling aims at minimizing the potential impact of the massive integration of PEVs into power grids to save service costs to customers while power control aims at minimizing the cost of power generation subject to operating constraints and meeting demand. The present paper develops a model predictive control (MPC)-based approach to address the joint PEV charging scheduling and power control to minimize both PEV charging cost and energy generation cost in meeting both residence and PEV power demands. Unlike in related works, no assumptions are made about the probability distribution of PEVs’ arrivals, the known PEVs’ future demand, or the unlimited charging capacity of PEVs. The proposed approach is shown to achieve a globally optimal solution. Numerical results for IEEE benchmark power grids serving Tesla Model S PEVs show the merit of this approach.

## 1Introduction

Electrical vehicles (EVs) have emerged as a promising solution to resolve both the economic and environmental concerns in the transportation industry [1]. Using a smart power grid in concurrently serving residences and charging EVs constitutes one of the most important applications of the smart grid technology. However, the massive integration of plug-in EVs (PEVs) into the grid causes many potential impacts such as voltage deviation, increased load variations and power loss of the grid [2], which requires different strategies for load shifting and energy trading and storage in the grid [3]. The main difficulty in scheduling of PEV charging to manage the cost and impact of PEV integration is that individual PEVs randomly arrive for charging with their individual demands on charging load and deadlines, which cannot be known before hand. In other words, the future charging demand of PEVs cannot be known a priori. Many existing works consider a simple smart grid with a single charging station (CS) to exclusively serve PEVs. For instance, [7] sets no charging deadlines for PEVs, whose arrival process follows a probability distribution, while [8] assumes that the future load demand is perfectly known a priori. The future load demand is also assumed to be known in [9] as all PEVs are assumed to arrive at the same time with no charging deadline. It is assumed in [10] that only statistics of demand are known but the PEVs can be fully charged in a single time slot [10]. It should be realized that serving PEVs is typically considered during a -hour time period (for instance from 8:00 pm to 8:00 am), where the integration of a massive number of PEVs has a sizable effect on the power grid, and as such, the length of a time slot is rationally set by minutes or one hour. In other words, the charging scheduling should be considered over a finite horizon of - time slots, but not over an infinite horizon as considered in [11]. Due to their physical limitations, PEVs are rarely able to be fully charged just during a single time slot.

In this paper, we consider joint PEV charging scheduling and power control to save service costs for PEVs and the power generation costs in meeting both residential and PEV power demands. Such a problem was considered in [12] but only a small number of PEVs and with each CS serving only one PEV, whose power demand is very small compared with the inelastic demand, so that the integration of PEVs into the grid has almost no effect on the grid. Note that the optimal power flow problems posed in [12] cannot be solved exactly by semi-definite programming relaxation (SDR) [13]. Therefore, it is not known if the objective in PEVs charging scheduling is convex and as such, it is not known if its proposed valley-filling solution is optimal. Larger PEVs’ penetration in a few CSs was considered in [14] under the assumption of known arrival and departure times of PEVs. In the present paper, we are interested in more practical scenarios of a massive number of PEVs arriving randomly at different CSs. No assumption on the probability distribution of their arrival is made, so the conventional model predictive control (MPC) [16] is not applicable. Our contribution is to develop a novel MPC-based approach to address this problem.

The rest of the paper is structured as follows. Section II is devoted to the system modeling for this problem and analyzing its computational challenges. An online computational solution using the proposed MPC-based approach is developed in Section III. An off-line computational solution is considered in Section IV, which is then compared with the online computational solution in Section V to show the optimality of the later. Section VI concludes the paper.

*Notation.* The notation used in this paper is standard. Particularly, is the imaginary unit, is Hermitian transpose of a vector/matrix , for a Hermitian symmetric matrix means that it is positive semi-definite, and are the rank and trace of a matrix , respectively. and are the real and imaginary parts of a complex quantity, and for two complex numbers and is componentwise understood, i.e. and . The cardinality of a set is denoted by .

## 2Problem statement and computational challenges

Consider an eletricity grid with a set of buses connected through a set of flow lines , i.e. bus is connected to bus if and only if . Accordingly, is the set of other buses connected to bus . There is a subset , whose elements are connected to distributed generators (DGs). Any bus is thus not connected to DGs. Any bus also has a function to serve PEVs and in what follow is also referred to CS . By defining , there are CSs in the grid. Denote by the set of those PEVs that arrive at CS . Accordingly, is the -th PEV that arrives at CS . Figure 1 presents a block diagram illustrating PEV integration into the grid.

The serving time period of the grid is divided into time slots of length , which usually varies from minutes to an hour. Under the definition , PEV arrives at and needs to depart at . The constraint

expresses the PEV ’s time demand. Suppose that and are the battery capacity and initial state of charge (SOC) of PEV . It must be fully charged by the departure time , i.e.

where is the charging efficiency of the battery and is a decision variable representing the power charging rate of PEV at time .

Due to the limited capacity of the hardware, the following constraint must be imposed:

for a given . For ease of presentation, we set

Let be the admittance of line . The current at node is , where is the complex voltage at bus during the time slot . For , the total supply and demand energy is balanced as , where and are the real and reactive powers generated by DG , and and are respectively known real and reactive price-inelastic demands at bus to express the residential power demand. By using the last two equations, we obtain

Similarly,

The next constraints relate to the acceptable range of generated power by the DGs:

where , and , are the the lower limit and upper limit of the real generated and reactive generated powers, respectively.

The constraints of voltage are

where and are the lower limit and upper limit of the voltage amplitude, while are given to express the voltage phase balance.

The problem of interest is to minimize both the energy cost to DGs and charging cost for PEVs. Thus, by defining , , and , , , , and , , the objective function is given by , where is the cost function of real power generation by DGs, which is linear or quadratic in , and is the known PEV charging price during the time interval .

The joint PEV charging scheduling and power control is then mathematically formulated as

The above problem (Equation 9) is very computationally challenging because the quadratic equality constraints (Equation 5) and (Equation 6) and nonlinear inequality constraints (Equation 8) and ( ?) constitute nonconvex constraints. Moreover, the arrival time of each individual PEV , its charging demand and its departure time are unknown.

## 3Model predictive control (MPC)-based computational solution

Considering and as the plant state and control, respectively, equations ( ?), (Equation 6), and (Equation 7) provide state behavioral equations [18] with the end constraint (Equation 2) while equations (Equation 8) and ( ?) provide control constraints. On the surface, (Equation 9) appears to be a control problem over the finite horizon . However, all equations in (Equation 9) are unpredictable beforehand, preventing the application of conventional model predictive control [16]. We now follow the idea of [19] to address (Equation 9).

At each time denote by the set of PEVs that need to be charged. For each , let be its remaining demand for charging by the departure time . Define

At time we solve the following optimal power flow (OPF) problem over the prediction horizon but then take only for online updating solution of (Equation 9):

with . One can notice that ( ?) includes only what is known at the present time . Of course, ( ?) is a still difficult nonconvex optimization and in the end we need only its solution at , so we propose the following approach in tackling its solution at .

Define the Hermitian symmetric matrix , which must satisfy and . By replacing , , in ( ?)-( ?), we reformulate ( ?) to the following optimization problem in matrices , :

Instead of ( ?), which is difficult due to multiple nonconvex matrix rank-one constraints in ( ?), we solve its semi-definite relaxation (SDR)

Suppose that and , are the optimal solution of (Equation 11). If , , then such that together with and constitute the optimal solution of the nonconvex optimization problem ( ?). Otherwise, we consider the following problem:

Note that in contrast to ( ?) involving matrix variables , and also variables , and , there is only single matrix variable in ( ?). The power generation variable in ( ?) is latent as it is inferred from in equation ( ?).

Following our previous works [20], a nonsmooth optimization algorithm (NOA) is proposed to deal with the discontinuous matrix rank-one constraint ( ?) in the optimization problem ( ?). Under condition ( ?) in ( ?),

where stands for the maximal eigenvalue of . The discontinuous matrix rank-one constraint ( ?) is then equivalently expressed by the following continuous spectral constraint:

because it means that has only one nonzero eigenvalue. Thus the quantity expresses the degree of the matrix rank-one constraint satisfaction (Equation 12), which is incorporated into the objective ( ?), leading to the following penalized optimization problem:

where is a penalty parameter. The above penalized optimization is exact because the constraint ( ?) can be satisfied by a minimizer of (Equation 13) with a finite value of . On the other hand, any feasible for (Equation 13) is also feasible for ( ?), implying that the optimal value of (Equation 13) for any is upper bounded by the optimal value of ( ?).

For any feasible for the convex constraints ( ?)-( ?), let be the normalized eigenvector corresponding to the eigenvalue . Then

i.e. the function is lower bounded by the linear function . Accordingly, the following semi-definite program (SDP) provides an upper bound for the nonconvex optimization problem (Equation 13):

because according to (Equation 14).

Suppose that is the optimal solution of SDP (Equation 15). Since is also feasible for (Equation 15), it is true that , so is a better feasible point of (Equation 13) than .

In Nonsmooth Optimization Algorithm (NOA) ? we propose an iterative procedure, which is initialized by the solution of SDR (Equation 11) and generates a feasible point at the -th iteration for , as the optimal solution of SDP (Equation 13). As proved in [13], this algorithm converges at least to a local minimizer of (Equation 13). Note that the procedure terminates at , so the spectral constraint (Equation 12) for the matrix rank-one is satisfied with the computational tolerance .

In summary, our proposed MPC-based computation for (Equation 9) is based on solving SDP (Equation 11) for online coordinating PEV charge and solving (Equation 13) by NOA ? for online updating the generated voltage for the generated power by

whenever the solution of SDR (Equation 11) is not of rank-one. If , it is obvious that with the normalized eigenvector corresponding to is the optimal solution of ( ?), which is what we need.

## 4Lower bound by off-line optimization

To investigate the optimality of the MPC-based online computation proposed in the previous section, in this section we address an off-line computation for (Equation 9), which provides a lower bound for the optimal value of its online computation. Under the additional definition , we reformulate (Equation 9) as

First, we solve its SDR by dropping the matrix rank-one constraints in ( ?):

Suppose that and are the optimal solution of SDP (Equation 17). If , then a global solution of the original problem (Equation 9) is found as , and and with , . However, such a matrix rank-one condition is rarely achieved. In what follows we propose two methods to address the matrix rank-one constraints in ( ?).

Again, under condition ( ?) for in ( ?), the rank-one constraints in ( ?) are equivalently expressed by the single spectral constraint , which is incorporated into the objective function in ( ?) for the following penalized function optimization:

with a penalty parameter . Initialized by , the following SDP is solved at -th iteration to generate and :

This computational procedure is summarized in NOA ?.

Alternatively, we propose the following scalable algorithm for computing ( ?). By replacing by , which was found by solving from (Equation 17), in ( ?) at every , we obtain the following optimization problem in and only:

which is computed by the distributed NOA Algorithm (DNOA) ?.

## 5Simulation results

### 5.1Simulation setup

The SDPs (Equation 11), (Equation 15), (Equation 17) and (Equation 19) are computed using Sedumi[24] interfaced by CVX [25] on a Core i5-3470 processor. Four power networks from Matpower [26] are chosen. The tolerance is set for the stop criterions.

Generally, PEVs are charged after their owners’ working hours. We focus on the charging period from 6:00 pm to 6:00 am of the next day, which is then uniformly divided into time slots of minute length [27]. Accordingly, the charging time horizon is . It is also reasonable to assume that the PEVs arrive during the time period from 6:00 pm to midnight. The PEVs must be fully charged after being plugged into the grid. The arrival times of PEVs are assumed to be independent and are generated by a truncated normal distribution , which is depicted by Figure 2.

We assume that the PEVs are Tesla Model S’s, which have a battery capacity of 100 KWh [28]. The SOC of all PEVs is set as 20%. The structure and physical limits of the considered grids are given in the Matpower library [26] together with the specific cost functions .

Without loss of generality, PEV loads are connected at the generator buses, which means each generator bus will serve as a charging station.

The price-inelastic load is calculated as

where is the load demand specified by [26] and is the residential load demand taken from [29]. Four profiles are taken from different days in 2017. Profile 1 is the residential load and energy price from 6:00 pm on February 5th to 6:00 am on February 6th, Profile 2 is from 6:00 pm on March 5th to 6:00 am on March 6th, Profile 3 is from 6:00 pm on April 5th to 6:00 am on April 6th, and Profile 4 is from 6:00 pm on May 5th to 6:00 am on May 6th. Figure 3 and Figure 4 provide the residential load demand and energy price for these profiles.

### 5.2MPC-based online computational results

#### Four network simulation

We test MPC-based online computation for Case9, Case14, Case30 and Case118mod from [26] and profile 2 of the residential data. The information on these networks is given in Table. ?, where the first column is the name of network, the second column indicates the numbers of buses, generators and branches. The dimension of is given in the third column, while the total number of PEVs is shown in the last column.

Buses/Generators/Branches | Dim. of | PEVs | |
---|---|---|---|

Case9 | 9/3/9 | 291 | |

Case14 | 14/5/20 | 485 | |

Case30 | 30/6/24 | 582 | |

Case118mod | 118/54/186 | 5238 | |

The computational results are summarized in Table ?.

Rank | LB | Comp.value | Time(s) | ||
---|---|---|---|---|---|

Case9 | 9 | 10 | 27991.4 | 27992.3 | 7.4 |

Case14 | 1 | - | 40824.1 | 40824.1 | 8.5 |

Case30 | 1 | - | 4935.6 | 4935.6 | 8.7 |

Case118mod | 2 | 100 | 644245.9 | 644278.5 | 432.1 |

Again, the first column is the network name. The second column presents the initial rank of the optimal solution of SDR (Equation 11). It is observed that the rank of is the same for all . The value of the penalty parameter in (Equation 13) is given in the third column. As the initial rank of Case14 and Case30 are all rank-one, SDR (Equation 11) already outputs the optimal solution for ( ?). Comparing the lower bound (LB) in the fourth column by solving SDR (Equation 17) at each time and the value found by the proposed MPC-based computation with using NOA ? in computing (Equation 9) in each time reveals the capability of the MPC-based computation for (Equation 9). These values are either the same (for Case14 and Case30) or almost the same (for Case9 and Case118mod), so indeed the proposed MPC-based computation could exactly locate a globally optimal solution. The average running time for solving ( ?) to implement the proposed MPC-based computation is provided in the sixth column, which is very short compared with the minute time slot and thus is practical for this particular online application.

The voltage profile for the four networks during the charging period are shown in Figure 5. For all cases, the voltage bound constraints (Equation 8) are satisfied. The voltage behavior is stable and smooth.

#### Four residential profile simulation

We consider Case30 together with four different residential profiles. The computational results are provided in Table ?, whose format is similar to Table ?.

Rank | LB | Comp. value | Time(s) | ||
---|---|---|---|---|---|

Profile 1 | 2 | 10 | 31961.2 | 31963.5 | 10.9 |

Profile 2 | 1 | - | 4963.3 | 4963.3 | 8.7 |

Profile 3 | 2 | 10 | 10771.3 | 10774.7 | 8.7 |

Profile 4 | 1 | - | 8139.3 | 8139.3 | 8.1 |

It can be seen that, even for the same network, the rank of the optimal solution of SDR (Equation 11) may be different depending on the residential profiles. For profile 2 and profile 4, the initial rank is one and SDR (Equation 11) has located a globally optimal solution. However, for profile 1 and profile 3, NOA ? is needed for obtaining the rank-one solution. The convergence speed is fast, and the optimum values are all equal or close to the lower bound, which clearly shows the global efficiency of the proposed MPC-based computation.

The aggregated active powers generated at each time are shown in Figure 6, from which the trends of generated power are seen to be similar to the residential load demand in Figure 3.

The stable and smooth voltage profile for these 4 residential profiles during the charging period are shown in Figure 7.

### 5.3Off-line computation and comparison with MPC-based online computation

Firstly, Case9, Case14, Case30 and Case118mod are tested with the residential data of profile 2 to analyze the efficiency of off-line computation by using Algorithm ? and Algorithm ?. The computational results are summarized in Table ?.

Rank | Lower bound | Computed value | Opt. degree | Iterations | NOA time(s) | DNOA time(s) | ||
---|---|---|---|---|---|---|---|---|

Case9 | 9 | 1 | 27978.1 | 27978.1 | 100% | 2 | 11.2 | 23.2 |

Case14 | 1 | - | 40800.7 | 40800.7 | 100% | 1 | 8.9 | 8.9 |

Case30 | 1 | - | 4935.6 | 4935.6 | 100% | 1 | 24.5 | 36.3 |

Case118mod | 2 | 50 | 644225.3 | 644233.9 | 99.999% | 3 | 1094.8 | 363.5 |

The initial rank in the second column is the rank of the optimal solution of SDR (Equation 17), which is the same for all . The value of penalty parameter in (Equation 19) is in the third column. The fourth column provides the lower bound by computing SDR (Equation 17). The values found by solving (Equation 18) and ( ?) by Algorithm ? and Algorithm ? are in the fifth column as they are the same and are either exactly same as their lower bounds in the fourth column (Case9, Case14 and Case30) or almost the same (Case118mod). According to the seventh column both Algorithm ? and Algorithm ? converge in two and three iterations for Case9 and Case188mod, while for Case9 and Case30, SDR (Equation 17) already outputs the optimal rank-one solution. The running times of Algorithm ? and Algorithm ? are provided in the eighth and ninth column, respectively. Algorithm ? requires less running time for small-scale networks such as Case9, Case14 and Case118mod. However, its running time increases dramatically for large-scale networks such as Case118mod, for which the scalable Algorithm ? is clearly advantageous.

A performance comparison between MPC-based computation and off-line computation for Case9 and Case30 with the four mentioned residential profiles is provided in Table ?, which clearly shows the global optimality of the proposed MPC-based computation as it attains objective values very close to the lower bounds provided by the off-line computation.

Rank | MPC | Offline | Offline/MPC | MPC times(s) | Offline times(s) | ||
---|---|---|---|---|---|---|---|

Case9 |
Profile 1 | 9 | 31963.5 | 31963.1 | 99.99% | 161.1 | 15.2 |

Profile 2 | 9 | 27992.3 | 27978.1 | 99.94% | 177.4 | 15.1 | |

Profile 3 | 9 | 31102.9 | 30885.1 | 99.29% | 173.7 | 14.8 | |

Profile 4 | 9 | 29896.2 | 29870.9 | 99.91% | 178.4 | 15.0 | |

Case30 |
Profile 1 | 2 | 31963.5 | 31963.1 | 99.99% | 262.4 | 37.4 |

Profile 2 | 1 | 4963.3 | 4935.6 | 99.43% | 209.8 | 12.6 | |

Profile 3 | 2 | 10774.7 | 10330.8 | 95.70% | 208.2 | 24.5 | |

Profile 4 | 1 | 8139.3 | 8087.2 | 99.35% | 194.9 | 12.9 | |

Figure 8 plots online and offline power generations in Case30 with four residential profiles, while Figure 9 plots the corresponding PEV charging scheduling. The charging load drops dramatically after am, by which all PEVs have been integrated into the grid but some of them have already been fully charged. Obviously, the charging load is sensitive to the energy price. For example in profile 3, the increase of the energy price at pm and am leads to a significant drop of the charging load. The charging load under MPC-based and off-line simulation are the same after am because there are no new PEVs arriving after that time.

## 6Conclusions

Joint PEV charging scheduling and power control for power grids to serve both PEVs at a competitive cost and residential power demands at a competitive operating cost is very difficult due to the random nature of PEVs’ arrivals and demands. We have proposed a novel and easily-implemented MPC-based computational algorithm that can achieve a globally optimal solution.

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