PaperlayerView exact revision on arXiv

Source-linked AI summary

Model Predictive Control for Distributed Microgrid Battery Energy Storage Systems

Thomas Morstyn, Branislav Hredzak, Ricardo P. Aguilera, Vassilios G. Agelidis

arXiv:1702.04699v1eess.SYcs.CE

TL;DR

The paper tackles dynamic optimal power flow for distributed battery storage in AC microgrids, where battery state and network constraints must be managed over time. It proposes a convex MPC formulation using linearised d–q models and variable-efficiency updates, and evaluates it in real-time simulations. The method approaches non-convex optimization performance while reducing computation time by a factor of 1000.

  • Problem

    Dynamic optimal power flow must account for network constraints, battery state of charge, and state-dependent battery efficiency across a time horizon.

  • Method

    The paper formulates convex MPC using a linear d–q voltage-current model, linearised power-flow approximations, and sampling-interval updates of battery charge and discharge efficiencies.

  • Results

    A factor of 1000 reduction in computation time allowed the proposed strategy to approach non-convex optimization performance in the reported comparison.

  • Takeaways & Limitations

    The proposed formulation is suitable for real-time MPC implementation while accounting for line losses and voltage drops in the microgrid.

Abstract

from arXiv · show

This paper proposes a new convex model predictive control strategy for dynamic optimal power flow between battery energy storage systems distributed in an AC microgrid. The proposed control strategy uses a new problem formulation, based on a linear d-q reference frame voltage-current model and linearised power flow approximations. This allows the optimal power flows to be solved as a convex optimisation problem, for which fast and robust solvers exist. The proposed method does not assume real and reactive power flows are decoupled, allowing line losses, voltage constraints and converter current constraints to be addressed. In addition, non-linear variations in the charge and discharge efficiencies of lithium ion batteries are analysed and included in the control strategy. Real-time digital simulations were carried out for an islanded microgrid based on the IEEE 13 bus prototypical feeder, with distributed battery energy storage systems and intermittent photovoltaic generation. It is shown that the proposed control strategy approaches the performance of a strategy based on non-convex optimisation, while reducing the required computation time by a factor of 1000, making it suitable for a real-time model predictive control implementation.

I. INTRODUCTION

The paper addresses dynamic optimal power flow for distributed battery storage in islanded AC microgrids, where storage state, network constraints, and renewable variability complicate optimization. It proposes a convex MPC formulation intended to retain relevant power-flow constraints while enabling real-time computation.

  • Motivation: Dynamic optimal power flow must account for network topology, operating constraints, battery state of charge, and efficiency-dependent objectives.Battery overcharge or undercharge can cause lifetime deterioration, while very low state of charge can limit power delivery.
  • Related approaches: Existing distributed-storage MPC strategies simplify the problem by aggregating storage, assuming ideal transfers, or neglecting losses and line-flow limits.DC power-flow approaches can be convex but omit line losses and bus-voltage limits, which is unsuitable for microgrids with relatively low line X/R ratios.
  • Related approaches: Non-convex MPC can model unbalanced microgrids, but scalability is limited and solvers are guaranteed only to find locally optimal solutions.Recursive dynamic programming can obtain a globally optimal solution, but its numerical complexity grows as the number of energy-storage systems increases.
  • Proposed strategy: The proposed strategy formulates dynamic optimal power flow using a linear d–q voltage-current model and linearised power-flow approximations, producing a convex QCQP.The formulation does not decouple real and reactive power, allowing line losses, voltage constraints, and converter-current constraints to be included.
  • Proposed strategy: Battery charge and discharge efficiencies are approximated by second-order polynomials and updated at each sampling interval.The simulations use an islanded IEEE 13 bus prototypical feeder with distributed battery storage and intermittent photovoltaic generation.
  • Implementation: The MPC supplies converter-voltage references to local controllers and operates on a slower time scale than the network dynamics.A 1 minute sampling period is chosen to adjust power flows in response to photovoltaic changes while accounting for battery state of charge.

III. MICROGRID MODELLING

The microgrid model represents passive network elements and regulated converter buses in a synchronous d–q reference frame. It uses VSC output voltages as inputs and calculates steady-state currents with a static model suited to the slower MPC time scale.

  • Network model: The network model includes coupling filters, RL lines, and loads between buses actively regulated by voltage source converters.The full microgrid contains sets of loads, lines, and VSCs, with line elements modeled as RL circuits.
  • Network model: State equations combine VSC output currents, load currents, and line currents into a state-space model driven by regulated VSC output voltages.The current vectors are expressed through their d–q components for VSC outputs, loads, and lines.
  • Steady-state approximation: The MPC uses a static steady-state network model to calculate network currents from VSC output voltages.This simplification follows from the MPC operating much more slowly than the network dynamics.
  • Model assumption: The d–q voltage-current model assumes a balanced three-phase system.If voltage unbalance is a concern, the paper states that it can be addressed with lower-level virtual-impedance compensation.
  • Model requirements: Microgrid impedances are required to develop the voltage-current model and may be obtained using online identification methods when unavailable.

B. Voltage Source Converters

The converter model links VSC dynamics and local control to the MPC’s slower voltage-reference updates. It uses synchronous d–q variables and a static VSC representation to support power-flow optimization.

  • Converter control: The MPC controls microgrid power flows by providing voltage references to VSCs connected to renewable sources and energy-storage systems.Standard decoupled d–q inner current controllers and outer voltage controllers regulate each VSC output voltage to its reference.
  • Dynamic model: VSC inductor-current and output-voltage state equations are expressed in the synchronous d–q reference frame.The model defines the DC-link voltage, modulation factor, inductor-current d–q vector, and PWM control-signal d–q vector.
  • Static approximation: The local VSC controllers operate much faster than the MPC sampling period, allowing a static VSC model in the optimization.
  • Power interface: The VSC model defines the input-voltage d–q vector and uses it to formulate the real power supplied from the converter’s DC side.

C. Battery Energy Storage System

The battery model represents SoC dynamics and charge/discharge efficiencies while accounting for their nonlinear dependence on SoC and output power. Second-order polynomial approximations model efficiency across the operating ranges considered.

  • Battery model: Battery SoC dynamics use charging and discharging efficiencies together with the sampling period and maximum battery energy.The model defines separate efficiencies for charge and discharge.
  • Efficiency model: Battery efficiency varies nonlinearly with both SoC and output power, contrary to the constant-efficiency assumption commonly used in MPC.The underlying cell model represents nonlinear SoC dependence in open-circuit voltage and terminal resistance.
  • Battery model: Battery output current and voltage are obtained from cell-level quantities scaled by the numbers of parallel strings and series-connected cells.The cell current is determined by solving a quadratic equation for a specified SoC and output power.
  • Battery implementation: The simulated batteries are 100kWh, 900V lithium ion systems composed of 130 parallel strings of 215 series-connected 4.2V, 860mAh cells.The cell model parameters are taken from the cited reference.
  • Efficiency approximation: Second-order polynomials provide good efficiency approximations over output powers from -100kW to 100kW and SoC from 20% to 100%.The approximations cover both charging and discharging efficiency relationships.

IV. MODEL PREDICTIVE CONTROL FORMULATION

The MPC formulation uses a finite horizon with battery SoC states and VSC voltage and power decisions as optimization variables. Separate charging and discharging powers preserve distinct efficiencies while maintaining a convex formulation.

  • MPC formulation: The convex QCQP is implemented with a receding horizon for model predictive control.The optimization horizon is defined over discrete time indices.
  • Optimization variables: Battery SoC levels are state variables, while VSC d–q voltage references and charging and discharging powers are manipulated variables.These variables are defined across the MPC time horizon.
  • Optimization variables: Separate charge and discharge power variables allow different charging and discharging efficiencies within a convex QP formulation.The formulation distinguishes the two power directions rather than imposing one shared efficiency.

A. Objective Function

The islanded-microgrid objective minimizes real power losses over the MPC horizon. It includes losses from network lines, VSC filters, and battery charging or discharging, while updating efficiencies between optimization intervals.

  • Objective function: The objective minimizes real power losses over the time horizon.The objective includes line, VSC LCL-filter, and battery losses.
  • Network losses: Network losses associated with load, renewable-generation, and battery locations are incorporated through time-varying network matrices.The matrices vary with predicted microgrid loads over the horizon.
  • Efficiency handling: Efficiencies are updated before each optimization using current SoC estimates and previous-interval output powers, then treated as constant over the horizon.This avoids directly placing SoC- and power-dependent efficiencies in the objective, which would make the problem non-convex.

B. Constraints

The constraints enforce consistency between VSC power variables and voltage references while using linearization to replace non-convex quadratic equalities with approximate affine constraints. Additional limits represent battery, generation, load, and renewable-power operation.

  • Constraint structure: Power-quality requirements and device operating limits are represented by affine equalities and convex inequalities.This constraint structure supports the convex QCQP formulation.
  • VSC output power: VSC charging and discharging power variables must match the power flows implied by the output voltage variables for feasibility.This consistency links manipulated power variables to voltage-reference decisions.
  • VSC output power: Quadratic VSC power equalities are replaced by affine approximations based on nominal inductor currents and input voltages to obtain a convex formulation.The original quadratic equality is the source of non-convexity in this part of the optimization.
  • Battery constraints: Battery real output powers are constrained to each system’s permitted interval from negative maximum charging power to maximum discharging power.The bounds are expressed as [−P̄_chi, P̄_dis_i] for battery system i.
  • Source and load constraints: Constant-power loads and renewable sources are constrained to their predicted operating levels over the MPC horizon.Renewable predictions specify available generation, while load predictions specify required constant-power operation.
  • Source and load constraints: An integral local voltage-control loop corrects renewable-generation mismatch caused by the power-flow linearization.The loop adjusts the voltage reference until VSC power matches the predicted renewable generation.

2) Battery SoC:

The battery SoC model incorporates charge and discharge efficiencies while maintaining convex optimization through interval updates and constraints. Voltage constraints are convexified using a conservative affine lower bound around the nominal operating point.

  • SoC constraints are set at 20% and 100% for the battery energy storage systems.
  • SoC over the prediction horizon is expressed from the current SoC estimates and manipulated variables.
  • Charge and discharge efficiencies are updated at each interval and treated as constant over the prediction horizon.This preserves a linear system model for the convex formulation.
  • The objective discourages simultaneous charging and discharging because battery inefficiency makes one of those actions lower cost.
  • VSC Output Voltage: Standard AC microgrid voltage limits are VLL ± 10%, with the lower non-convex bound approximated conservatively by an affine d-axis constraint.The upper voltage bound remains a convex quadratic constraint, while the approximation is made around vodqi = (VLL, 0).

4) VSC Output Current:

The VSC current constraint limits converter RMS phase current and thereby restricts reactive power. The MPC problem combines a quadratic loss objective with affine approximations that produce a convex QCQP solvable by interior-point methods.

  • VSC Output Current: The maximum rated VSC switch current imposes an RMS phase-current constraint that also limits reactive output power.
  • Optimisation Formulations: The MPC loss-minimization objective is quadratic in the manipulated variables.
  • Optimisation Formulations: The convex QCQP replaces VSC output-power quadratic equalities and the quadratic voltage lower bound with affine approximations.
  • Optimisation Formulations: Interior-point methods can solve the resulting convex QCQP with hundreds of variables and constraints in several seconds on standard hardware.

D. Model Predictive Control Implementation

The proposed convex QCQP is implemented as receding-horizon MPC using updated battery, generation, load, and network information. Real-time simulations evaluate operation in an islanded IEEE 13-bus microgrid with distributed batteries and photovoltaic generation.

  • D. Model Predictive Control Implementation: At each interval, the implementation updates states and predictions, solves the convex dynamic optimal power-flow problem, applies voltage references, and advances the horizon.
  • A. Real-Time Digital Simulation: The real-time study controls four battery energy storage systems and photovoltaic generation in an islanded microgrid based on the IEEE 13-bus prototypical feeder.The batteries use a nonlinear model and the VSCs use switching converter models.
  • A. Real-Time Digital Simulation: The case study uses 100 kWh, 900 V lithium-ion batteries and a 100 kW photovoltaic source, with one-minute irradiance and temperature data.
  • A. Real-Time Digital Simulation: The MPC uses a one-minute sampling period and a 30-minute horizon with inaccurate photovoltaic and load predictions.
  • A. Real-Time Digital Simulation: The average QCQP solution time is 0.71 seconds on an Intel Core i7-4770 CPU using MATLAB, YALMIP, and IBM CPLEX.
  • A. Real-Time Digital Simulation: Battery SoC reaches 19.98% minimum and 100.05% maximum, while maximum average VSC phase current reaches 154.8 A against a 150 A limit.The SoC violations are attributed to the approximate battery model, and the current violation to photovoltaic prediction errors.
  • A. Real-Time Digital Simulation: Microgrid bus voltages remain within VLL ± 10%, with VSCs injecting or absorbing reactive power to regulate them.
  • A. Real-Time Digital Simulation: Average power loss is 12.638 kW, while relocating photovoltaic generation across three other buses produces losses from 12.634 kW to 12.654 kW.

B. Comparison with the Non-Convex QCQP

The convex QCQP achieves near-comparable power-loss performance to a non-convex formulation while reducing computation time enough for real-time implementation. Its variable-efficiency battery model also lowers average losses relative to a constant-efficiency formulation, although the approximation has limitations.

  • The convex QCQP produced 12.638kW average power loss versus 11.596kW for the non-convex QCQP, which was 8.25% lower.
  • The non-convex QCQP required 23 minutes on average, compared with 0.71 seconds for the convex QCQP, a 1000-fold reduction.This computation-time difference demonstrated the non-convex formulation’s unsuitability for real-time implementation.
  • The variable-efficiency battery model reduced average losses to 12.638kW, compared with 13.194kW for the constant-efficiency formulation, a 4.40% increase for the latter.
  • The variable-efficiency formulation is approximate, and fixed efficiency values may outperform it in some cases.Its stated advantage is generating reasonable approximate values from the microgrid operating state.
  • Overall, the convex strategy approaches non-convex performance while reducing computation time by a factor of 1000 for real-time MPC.
Loading 1702.04699v1…