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

Y. Shi, H. D. Tuan, A. V. Savkin, T. Q. Duong, H. V. Poor

arXiv:1708.07626v1eess.SYcs.SY

TL;DR

Random PEV arrivals leave future charging demand unknown, complicating joint service of residences and vehicles. The paper develops an MPC-based approach to coordinate PEV charging and power control without assuming an arrival distribution. Its reported benchmark results show solutions equal or close to lower bounds, supporting its claim of global optimality.

Abstract

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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.

I. INTRODUCTION

PEV charging demand is uncertain until vehicles arrive, while large-scale integration can affect grid operation. The paper develops joint charging scheduling and power control for multiple stations under randomly arriving PEV demand.

I. INTRODUCTION

  • Random PEV arrivals make individual charging loads and deadlines unknown in advance, complicating scheduling for grids that serve residences and vehicles together.Massive PEV integration can also cause voltage deviation, increased load variations, and power loss.Source: §I · p. 1
  • Prior studies impose restrictive assumptions on arrivals, future demand, charging deadlines, or single-slot charging, while some use infinite rather than finite horizons.The paper notes that typical charging periods span 12 hours, with 30-minute or one-hour slots, and that PEVs are rarely fully charged in one slot.Source: §I · p. 1
  • The paper targets joint PEV charging scheduling and power control for many randomly arriving PEVs at different stations, without assuming an arrival distribution.It seeks to reduce PEV service and power-generation costs; conventional MPC is stated to be inapplicable without an arrival distribution.Source: §I · p. 1
  • The paper develops an MPC-based approach and compares its online solution with an offline solution to assess optimality.The paper outline assigns system modeling and computational challenges to Section II, the online method to Section III, and offline computation and comparison to Sections IV–V.Source: §I · p. 1

II. PROBLEM STATEMENT AND COMPUTATIONAL CHALLENGES

The paper models a grid with distributed-generator buses that also serve as charging stations, and formulates joint PEV charging and power control over a finite time horizon. The optimization combines generation and charging costs subject to charging and grid constraints.

II. PROBLEM STATEMENT AND COMPUTATIONAL CHALLENGES

  • Each bus is connected through flow lines, and buses with distributed generators also serve as charging stations for their arriving PEVs.The number of charging stations equals the number of generator-connected buses.Source: §II · p. 2
  • PEVs arrive and depart within a finite set of time slots, and each vehicle must receive enough charging by departure to reach full charge.The charging rate is a decision variable, while hardware capacity imposes an additional limit.Source: §II · p. 2
  • The network model balances bus supply and demand using generated power, residential demand, PEV charging, and electrical quantities determined by line admittances and bus voltages.Generated real and reactive power are bounded, and voltage amplitude and phase must satisfy stated constraints.Source: §II · p. 2
  • The objective minimizes distributed-generation energy cost and PEV charging cost, but the formulation is nonconvex and relies on arrivals, demands, and departure times that are unknown beforehand.The objective includes the generation-cost function and the known PEV charging price for each time interval.Source: §II · pp. 2–3

III. MODEL PREDICTIVE CONTROL (MPC)-BASED

Because future arrivals and charging needs are unpredictable, the paper constructs an MPC-based method using information available at the current time. It combines semidefinite relaxation with a nonsmooth optimization procedure to handle rank-one constraints.

III. MODEL PREDICTIVE CONTROL (MPC)-BASED

  • At each time, the method solves an OPF problem over a prediction horizon but uses only current-time decisions to update the online solution.The optimization is formulated using information known at the present time, unlike the unpredictable full-horizon problem.Source: §III · p. 3
  • The method rewrites voltage variables as positive semidefinite matrices with rank-one constraints, then applies semidefinite relaxation to the difficult nonconvex formulation.When the relaxed solution is rank one, it recovers the voltage solution to the original problem; otherwise, further optimization is used.Source: §III · p. 3
  • For non-rank-one solutions, a nonsmooth optimization algorithm penalizes deviation from the rank-one condition using the trace minus the largest eigenvalue.The penalty formulation is stated to be exact for a finite penalty parameter; the iterative procedure converges at least to a local minimizer of the penalized problem.Source: §III · pp. 3–4
  • The online computation coordinates current PEV charging through an SDP and updates voltage and generated power using the nonsmooth optimization procedure when needed.If the relaxed matrix is already rank one, the method directly recovers the voltage solution.Source: §III · p. 4

IV. LOWER BOUND BY OFF-LINE OPTIMIZATION

The off-line formulation supplies a lower bound for the online MPC solution. Its semidefinite relaxation drops rank-one constraints, and two methods address cases where the relaxation does not recover a rank-one solution.

IV. LOWER BOUND BY OFF-LINE OPTIMIZATION

  • The off-line computation reformulates the problem to provide a lower bound on the optimal value of the online computation.The reformulation uses W = {W(t′)} over the full time horizon.Source: §IV · p. 4
  • The semidefinite relaxation drops the rank-one constraints; when its solution is rank one it yields a global solution, but this condition is rarely met.The paper therefore proposes two methods for handling the rank-one constraints.Source: §IV · p. 4
  • Under condition (13g), the rank-one constraints are expressed as a spectral constraint and added to the objective with penalty parameter μ > 0.The iterative SDP is initialized with the relaxed solution.Source: §IV · p. 4
  • The alternative scalable algorithm fixes Pkn(t) to the value from (22), leaving an optimization over W(t) and R(t) at each time.This procedure is identified as NOA 2 and iterates until its stopping tolerance is met.Source: §IV · pp. 4–5
  • The distributed NOA 3 procedure initializes W(t) from (22), iteratively updates W(t) and R(t), and accepts the resulting solution to (25).The algorithm is described as computing the alternative optimization in W(t) and R(t) only.Source: §IV · pp. 4–5

A. Simulation setup

The simulations use four Matpower networks, Tesla Model S vehicles, and four residential profiles. Across the networks, MPC solutions match or nearly match the off-line lower bound, while voltage bounds are satisfied.

A. Simulation setup

  • The simulations use four Matpower networks, Sedumi through CVX, and a stopping tolerance of 10^-4.Computations ran on a Core i5-3470 processor.Source: A. Simulation setup · p. 5
  • PEVs arrive from 6 pm to midnight during a 6 pm–6 am charging horizon divided into 24 half-hour slots, and must be fully charged.Arrival times are independent and generated from a truncated normal distribution (20, 1.52).Source: A. Simulation setup · p. 5, Figure 2 · p. 5
  • The modeled PEVs are Tesla Model S vehicles with 100 KWh batteries and 20% initial state of charge, connected at generator buses serving as charging stations.Network limits and generation cost functions come from Matpower.Source: A. Simulation setup · p. 5
  • Four residential load and energy-price profiles use evening-to-morning data from February, March, April, and May 2017.Residential load combines the Matpower-specified demand with a residential load profile.Source: A. Simulation setup · p. 5, Figure 3 · p. 5, Figure 4 · p. 5
  • MPC values match the lower bound for Case14 and Case30 and are almost the same for Case9 and Case118mod; average runtime is much shorter than a 30-minute slot.The comparison is reported for the proposed MPC computation using NOA 1.Source: Table II · p. 6, 1) Four network simulation · p. 6
  • Voltage bounds are satisfied in all four networks, and voltage behavior is stable and smooth during charging.The voltage profiles are shown for the four networks over the charging period.Source: 1) Four network simulation · p. 6

2) Four residential profile simulation:

For Case30, the residential profile affects whether the initial relaxation is rank one, but the proposed computation yields values equal or close to the lower bound. The reported active-power and voltage profiles remain aligned with residential demand and stable.

2) Four residential profile simulation:

C. Off-line computation and comparison with MPC-based online computation

The MPC-based online computation achieves objective values very close to the offline lower bounds, showing global optimality. The comparisons also show how residential profiles and energy prices relate to generation and charging schedules.

C. Off-line computation and comparison with MPC-based online computation

  • For Case9, Case14, and Case30, both offline algorithms match the lower bounds exactly; for Case118mod, their values are almost the same as the lower bound.The lower bounds come from SDR (22), and both algorithms produce the same values.Source: C. Off-line computation and comparison… · p. 7
  • With residential profile 2, the tested cases were Case9, Case14, Case30, and Case118mod; Algorithm 2 required less running time than Algorithm 3 on small-scale Case9 and Case14.Source: C. Off-line computation and comparison… · p. 7
  • The MPC-based computation achieves objective values very close to offline lower bounds, showing global optimality.Table V compares the online and offline computations for Case9 and Case30 across four residential profiles.Source: C. Off-line computation and comparison… · p. 7
  • Figures 8 and 9 plot online and offline power generation and PEV charging schedules, respectively, for Case30 across four residential profiles.The cited prose identifies the plotted quantities and case; the figures are identified by their captions.Source: C. Off-line computation and comparison… · p. 7, Figure 8 · p. 7, Figure 9 · p. 7
  • In profile 3, charging load drops significantly when energy prices rise at 11:30 pm and midnight; after midnight, online and offline charging loads match.The passage attributes the post-midnight match to there being no new PEV arrivals after that time.Source: C. Off-line computation and comparison… · p. 7, Figure 9 · p. 7

VI. CONCLUSIONS

Jointly scheduling PEV charging and controlling power generation to serve PEV and residential demand at competitive costs is difficult because PEV arrivals and demands are random. The paper proposes an easily implemented MPC-based algorithm that can achieve a globally optimal solution.

VI. CONCLUSIONS

  • Joint PEV charging scheduling and power control aim to serve PEVs and residential demand at competitive costs.Source: §VI · pp. 7–8
  • Random PEV arrivals and demands make this joint scheduling and control problem very difficult.Source: §VI · pp. 7–8
  • The proposed easily implemented MPC-based computational algorithm can achieve a globally optimal solution.Source: §VI · p. 8
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