Scalable Electric Vehicle Charging Protocols
Although electric vehicles are considered a viable solution to reduce greenhouse gas emissions, their uncoordinated charging could have adverse effects on power system operation. Nevertheless, the task of optimal electric vehicle charging scales unfavorably with the fleet size and the number of control periods, especially when distribution grid limitations are enforced. To this end, vehicle charging is first tackled using the recently revived Frank-Wolfe method. The novel decentralized charging protocol has minimal computational requirements from vehicle controllers, enjoys provable acceleration over existing alternatives, enhances the security of the pricing mechanism against data attacks, and protects user privacy. To comply with voltage limits, a network-constrained EV charging problem is subsequently formulated. Leveraging a linearized model for unbalanced distribution grids, the goal is to minimize the power supply cost while respecting critical voltage regulation and substation capacity limitations. Optimizing variables across grid nodes is accomplished by exchanging information only between neighboring buses via the alternating direction method of multipliers. Numerical tests corroborate the optimality and efficiency of the novel schemes.
Electric vehicles (EVs) have received significant attention from the automotive industry and the government due to their capacity to reduce greenhouse gas emissions and mitigate oil dependency. Nevertheless, the overall load profile will be greatly affected with increasing numbers of EVs. Uncoordinated charging of even a 10% penetration of EV loads will notably affect power system operation, giving rise to voltage magnitude fluctuations and unacceptable load peaks . On the other hand, with proper coordination scheme, EV loads can be controlled to minimize charging costs or perform valley-filling tasks relying on advanced power electronics.
Different charging control schemes have been proposed. A centralized scheduling scheme to minimize total charging costs based on the time-of-use price has been devised in . However, new load peaks may arise during low-price (also termed valley) periods. In , vehicle plug-in times are decided using random numbers, hence neglecting the specific charging requests of individual EV users. Charging rates have been also optimized in a centralized manner to facilitate voltage regulation , yet the number of control variables scales unfavorably with the number of vehicles.
Decentralized control strategies not only offer computational and communication savings, but they oftentimes enhance the privacy of vehicle users since they do not require the charging requests of EVs to be uploaded to the control center. Decentralized charging protocols are available based on congestion pricing schemes similar to those used in Internet Protocol (IP) networks; nevertheless, their optimality is not guaranteed . Presuming identical plug-in/-out times and energy requests for all vehicles, a game-theoretic charging scheme attaining a Nash equilibrium has been developed in . Iterative schemes based on Lagrangian relaxation are suggested in , while  builds on the alternating direction method of multipliers (ADMM). Distribution locational marginal prices are leveraged to coordinate vehicle charging in . Reference  proves a feasible valley-filling charging profile to be optimal for any convex cost, and it develops a decentralized protocol. A multi-agent based EV charging scheme is proposed in .
Vehicle charging under distribution grid limitations has been studied too. Centralized EV scheduling is studied under different linear models for multiphase networks in  and . The objective function is confined to be linear and the optimal solution is found using generic commercial solvers without exploiting the problem structure. A method for heuristically checking network constraint violations after vehicles have been scheduled is reported in . Presuming at most one EV per bus, management under balanced network constraints has been tackled using a water-filling algorithm .
The optimal vehicle charging problem considered here can be rigorously stated as follows. Given charging requests from EVs across time, a utility company schedules their charging to minimize certain cost function, e.g., the power supply cost or the load variance. The latter is equivalent to the so termed the valley-filling task. Depending on whether grid specifications are taken into account, two charging scenarios can be recognized. The first scenario ignores any grid-related constraints. Such a scenario arises for example when the EV load is relatively low and is not expected to incur voltage or feeder violations; see the valley-filling task in . In this first scenario, vehicle charging may be alternatively performed by a charging station or a load aggregator to minimize its power supply cost. Under the second scenario, EV penetration is high, and thus, voltage regulation and feeder limitations must be enforced by the utility. Apparently, the first scenario constitutes a relaxation of the second scenario of network-constrained vehicle charging. Thus, protocols for the first scenario will be used as building modules for the second one.
Our contribution is two-fold. First, a decentralized charging method based on the Frank-Wolf algorithm is developed (Section II). Different from existing schemes, the novel protocol requires minimal requirements from the vehicle charging controllers and involves privacy-preserving updates. Numerical tests demonstrate that the closed-form low-complexity updates yield significant convergence improvement over existing alternatives (Section V-A). Secondly, building on an approximate distribution grid model, network-constrained EV charging is formulated as a convex quadratic program (Section III), and tackled using a decentralized scheme based on ADMM (Section IV). Compared to existing centralized schemes, the decentralized protocol requires communication only between neighbors and preserves the privacy of EV owners. Numerical tests on the unbalanced IEEE 123-bus feeder corroborate the optimality of the proposed charging protocol (Section V-B).
Regarding notation, column vectors (matrices) are denoted by lower- (upper-) case boldface letters, except for power flow vectors . Sets are represented using calligraphic symbols, and is the cardinality of . Symbol stands for transposition; while , , and , denote respectively the all-zeros, all-ones, and the -th canonical vectors. Operator defines a diagonal matrix having on its diagonal, and returns the real part of complex number .
Ii Optimal Vehicle Charging
This section studies EV charging without network constraints. Under this scenario, the utility company, a load aggregator, or a charging station would like to coordinate EVs to minimize the power supply cost or for valley-filling purposes. Upon formulating the problem, an optimal charging scheme is developed and contrasted to state-of-the-art alternatives.
Ii-a Electric Vehicle Charging Model
An EV scheduler coordinates the charging of EVs over a period of consecutive time slots comprising the set . The time slot duration can range from minutes to an hour, depending on charging specifications, the granularity of load forecasts, as well as communication and computation capabilities. Let denote the energy charge for vehicle at time with , and . Given that operational slots have equal duration, the terms power and energy will be used interchangeably. The charge can range from zero to its maximum value . Apparently, a vehicle can be charged only when it is connected to the grid. If is the set of time slots that vehicle is connected to the grid (not necessarily consecutive), then for all
where is the maximum charging rate determined by the battery of vehicle . Let be the charging profile for EV . Profile should belong to the compact and convex set
where is the total energy needed by EV . The latter depends on the initial state of charge, the desired state of charge, and the efficiency of the battery.
Through coordinated charging of electric vehicles, various objectives can be achieved, such as minimizing charging costs or valley-filling. Optimal EV charging can be posed as the optimization problem 
where the energy costs are assumed convex and differentiable. For charging cost minimization, can be linear or quadratic ; while for all when it comes to the valley-filling task. Parameters capture the based load for the EV scheduler, which is assumed inelastic and known in advance. The network constrained EV charging is postponed for Section III, wherein problem (2) turns out to be a building module. To facilitate scheduling, each electric vehicle controller is capable of two-way communication and of executing simple computation tasks. Before the beginning of the charging horizon , vehicle controller submit their charging requests to the charging station controller. Protocols for efficiently solving (2) are presented next.
Ii-B Scalable Charging Protocol
Observe that the total number of variables involved in (2) is . Therefore, although (2) is a convex problem, solving it is a non-trivial task, particularly for large EV fleets and/or decreasing control intervals . To derive a scalable solver, the Frank-Wolfe method is deployed next . Also known as conditional gradient algorithm, the Frank-Wolfe method aims at solving the generic problem
where is a differentiable convex function, and is a compact convex set. The method selects an initial , and iterates between the updates for as
with . Step (4a) finds such that is a feasible descent direction for the first-order approximation of the cost in (3). Step (4b) updates towards that direction after scaling it with the diminishing step size . The updated is always feasible, since it is computed as the convex combination of and .
Granted that (2) entails a differentiable cost and a compact feasible set; it is amenable to Frank-Wolfe iterations. In the first Frank-Wolfe step, the gradient of the cost in (2) with respect to must be obtained. Critically, due to the problem structure, the per-vehicle partial gradients of the cost are all identical to
It can be readily checked that the -th entry of the common partial gradient evaluated at is
Applying (4a) to the problem at hand requires solving
which is separable across vehicles. Thus, given , vehicle needs to solve the linear program
Problem (7) involves a linear cost minimized over a weighted budget and box constraints. The key observation here is that due to the aforementioned structure, problem (7) can be solved by a simple sorting algorithm [16, Chap. 4]: The entries of are first sorted in increasing order as
Since the problems in (7) share vector for all , the sorting operation is performed only once by the charging station. Then, for vehicle , we need to find the index for which
Subsequently, the entries of the minimizer of (7) can be computed per vehicle as
The solution in (10) simply selects the maximum possible charge during the cheapest time slots in a greedy fashion. Interestingly, finding from (10) requires knowing solely the rank order (smallest to largest) rather than the actual entries of the gradient vector .
The second Frank-Wolfe step updates the charging profiles via the convex combinations
for all vehicles .
To practically implement (5)–(11) during iteration , the charging control center evaluates the cost gradients defined in (5), and sorts them to determine the time slot ordering . This sorting operation can be performed using for example the Merge-Sort algorithm in operations . The price ordering of time slots is subsequently broadcast to all EV controllers as shown in Fig. (a)a. Based on its charging needs , the -th EV controller first finds from (9)–(10) in . It then updates its charging profile using (11) in . Note that operations (9)–(11) can be performed in parallel over the EV controllers. The updated charging profiles are communicated back to the charging center, where upon adding the base load , the center computes the updated cost gradient , and iterations proceed as tabulated in Algorithm 1. The developed solver converges to optimal charging profiles at the rate 
Algorithm 1 not only exhibits provable convergence and low computational cost (namely operations) per iteration. It further enjoys two additional advantages. First, the charging center does not require knowing the individual charging profiles , since their summation suffices for finding the gradient vector . In an effort to preserve the privacy of EV users, a simple communication protocol can be designed. Information flow can be arranged over a tree graph rooted at the charging center, and vehicle controllers constitute the remaining tree nodes. Each node receives aggregate charging profiles from its downstream nodes, adds them up to its own profile, and forwards the updated aggregate charging profile to its parent node. As a second feature, vehicle controllers do not need to know the precise value of the cost gradient vector , but only the ordering of its entries (current price ordering of time slots). This algorithmic feature lightens the communication load from the charging center to the vehicles, and enhances resiliency to price manipulations and data attacks to the solving scheme.
Ii-C Comparison with Previous Work
The optimal EV charging of (2) has been previously studied in , where a projected gradient descent (PGD) solver was developed. Interpreted here as a projected gradient algorithm applied to minimize the non-strongly convex cost in (2), the PGD method exhibits a convergence rate of . At iteration of the PGD method, controller solves in parallel
for a step size . In other words, every EV controller projects vector onto the simplex , which is a non-trivial task.
On the other hand, each iteration of Algorithm 1 involves closed-form updates, offering high computational efficiency and posing affordable hardware requirements on EV controllers. Although both Algorithm 1 and the PGD solver are decentralized schemes with convergence rate , the overall computation time for the former is significantly lower due to its simpler per-iteration updates: The numerical tests in Section V-A demonstrate that Alg. 1 provides a 100 times speed-up advantage over the PGD solver and the centralized solver SeDuMi. The SeDuMi solver would be a viable option for tackling (2) in a centralized manner after collecting all charging needs at the charging center.
Iii Network-Constrained EV Scheduling
The charging scheme of Section II applies to scenarios where EV charging can be transparently supported by the underlying grid. If higher levels of EV load incur voltage magnitude or feeder capacity violations, the underlying power distribution grid needs to be taken into account. In this context, upon reviewing an approximate model for unbalanced distribution grids, this section formulates a network-constrained vehicle charging task, while a decentralized solver scalable to the number of buses and EVs is developed in Section IV.
Iii-a Modeling Unbalanced Distribution Grids
Electric vehicles are connected to a distribution feeder comprising buses indicated by , and phases indexed by . Let represent the set of vehicles located on phase of bus , and . The distribution grid is assumed to be functionally radial with the substation bus numbered by . Every non-feeder bus with has a unique parent bus indexed by . The distribution line connecting bus with bus is denoted by . For bus , let also denote the set of its children buses, and the set of its phases.
To enforce distribution network and voltage regulation limitations, the underlying physical system is taken into account. For that purpose, the distribution grid can be captured either by the full AC power flow model or the linearized power flow model proposed in . The former becomes tractable under appropriate conditions using convex relaxations , . However, counterexamples indicate that convex relaxations are not always successful and they can increase computational requirements. On the other hand, several numerical tests indicate that the approximation error of the linearized model is within the order of in terms of calculating voltages , . Although the linearized grid model is adopted here to simplify calculations, extending our charging protocol to the full AC model is straightforward.
Let , , and be respectively the 3-dimensional vectors of squared voltage magnitudes and (re)active power injections for all phases of bus . For line , let be the related phase impedance matrix, and and be the vectors of (re)active power flows on all phases of line . If line losses are relatively small and voltages are roughly balanced, the linearized multi-phase power flow model reads , 
where ; ; ; and denotes complex conjugation. When not all phases are present, power injection and flow vectors and phase impedance matrices are zero-padded. For (14c) to hold, the entries of associated with non-existing phases are arbitrarily set to the corresponding entries of .
Iii-B Network-Constrained EV Scheduling
To facilitate network-constrained EV scheduling, the base active and reactive power loads for all and need to be predicted in advance. Active power loads consist of two parts: the base loads and the EV charging load. If and are respectively the total active load and the base load on phase of bus , it holds that . The cost of power supply from the main grid is convex and known in advance. Variables capture possible dispatchable generation distributed across the feeder, and is the associated convex quadratic cost for all and .
To capture operational constraints, the following limits are introduced. Let be the lower, and the upper limits for distributed generation at phase of bus . Define also as the limits of squared voltage magnitudes at phase of bus , as the apparent power flow limits on line , and as the rated capacity of the feeder transformer. The utility company aims to minimize the total operation cost by coordinating vehicle charging and generation dispatch, while respecting charging and operational limitations. The pertinent network-constrained EV scheduling task can be posed as:
Constraints (15b)–(15d) originate from the power flow model; constraints (15e)–(15f) enforce generation limits; voltage regulation is guaranteed via (15g); apparent power flows are upper bounded by (15h); the equalities in (15i) define demands across phases and buses; constraint (15j) is related to the per-vehicle charging profile; and (15k) results from the capacity limit of the feeder transformer.
The cost functions and all the constraints apart from the EV charging constraint in (15j) are separable across time. The capacity limit in (15k) couples flows across phases, while the voltage regulation constraints in (15d) and (15g) couple variables across buses and phases. For linear and convex quadratic costs, problem (15) can be reformulated as a standard quadratically-constrained quadratic program and tackled by standard solvers in a centralized manner. Nonetheless, for increasing grid sizes, longer time horizons , and/or shorter control periods, tackling (15) could be challenging. In addition, private information on a per-vehicle basis needs to be collected and processed by the utility. These considerations motivate well the privacy-preserving and scalable (both in space and time) scheme for solving (15) that is pursued next.
Iv Distributed Optimal Charging Protocol
This section delineates an ADMM-based method for decomposing (15) into smaller subproblems. Notably, each subproblem either enjoys a closed-form solution or it can be tackled efficiently by Alg. 1. As a brief review, ADMM solves problems of the form 
where and are convex functions; and are convex sets; and model the linear equality constraints coupling variables and . In its normalized form, ADMM assigns a Lagrange multiplier for the equality constraint and solves (16) by iterating over the following three recursions
for some . ADMM has been successfully applied to decentralize various power system tasks across buses , , . Related ideas are adopted here to decouple the spatially-coupled constraints (15b)–(15d).
To that end, each bus maintains a local copy of the variables associated with the squared voltage magnitude of its parent bus, and the power flows feeding its children buses. These auxiliary variables are marked with a hat as and . The duplicate variable stored at bus should agree with the original variable stored at bus . To decentralize the computations, we further introduce the consensus variable , and impose the constraints and for all non-leaf buses. By repeating this process for power flow variables and all , the physical grid model will be later decoupled across buses.
We also introduce duplicate variables for net loads to separate the tasks of EV charging and generation dispatch. As detailed later, imposing the constraints for all , enables isolating (15j) from the rest of the constraints in (15); resulting in localized EV charging subproblems that is a special case of (2).
For a compact representation define the aggregate variables:
for all and . With the newly introduced variables, problem (15) can be equivalently expressed as:
The equality constraints between duplicate variables in (18i)–(18l) are assigned Langrange multipliers according to Table I. Adopting the ADMM iterates of (17) to solve (18), variables and are updated in the first ADMM step, whereas variables are updated during the second ADMM step as detailed next.
Iv-a First Step of ADMM
Due to the form the generic update (17a) takes for the problem at hand, variables can be updated separately from the EV charging profiles . The updates for these two variable sets are studied next.
Heed that can be optimized independently across buses and time periods. Nevertheless, for fixed bus and time indices , variables and are coupled due to constraints (18b)–(18d). To simplify the presentation, we drop the time index and consider the canonical subproblems involved for all . Let for bus . Variables and are updated during the -th iteration as the minimizers of
For and due to the power supply cost from the main grid, variables are found as the minimizers of
We next focus on updating the vehicle charging profiles at iteration . Interestingly, the task of EV charging decouples over buses and phases. The charging profiles for vehicles can be updated as the minimizers of
In the first step of ADMM, each bus needs to collect from its parent and from all its children as depicted in Fig. (a)a. Meanwhile, each bus transfers to its EV scheduling center, where the charging profile of EVs are optimized using Alg. 1.
Iv-B Second Step of ADMM
Finding optimal can be performed independently across buses and time slots. Because of that, time indices are ignored. Every bus has to solve five sub-problems in parallel, each one associated with the variables , , , , and . Firstly, updating , , , and decouples over phases of bus too. It can be shown that per phase variables are updated as the minimizers of a univariate convex quadratic function possibly over box constraints. If the generation cost at bus is with , then is updated at iteration by solving
The minimizer of (IV-B) is expressed as
where . The entries of and are similarly found as
The entries of are obtained as the solutions of unconstrained univariate convex quadratic programs as
The optimizations involved in updating the consensus power flow variables decouple across phases. The consensus power flow variables are updated by solving the problems for all and :