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Storage Size Determination for Grid-Connected Photovoltaic Systems

Yu Ru, Jan Kleissl, Sonia Martinez

arXiv:1109.4102v2math.OCeess.SY

TL;DR

The paper addresses how to size batteries for grid-connected PV systems while balancing grid transaction costs, battery aging, load satisfaction, and peak-purchase reduction. It derives an economic-value criterion and capacity bounds, then uses an efficient algorithm to compute the critical size. Simulations validate the approach, including reported cost savings for residential and commercial loads.

  • Problem

    The paper studies battery sizing for grid-connected PV systems where storage affects electricity transaction costs, battery capacity loss, load satisfaction, and peak grid purchases.

  • Method

    The paper derives an economic-value criterion and lower and upper capacity bounds, then introduces an efficient algorithm based on those bounds to calculate the critical capacity.

  • Results

    16.10% cost can be saved for a residential load with a 18320(Wh) battery, while 25.79% can be saved for a commercial load with a 15747(Wh) battery over T = 96(h).

  • Takeaways & Limitations

    The results show benefits from batteries in grid-connected PV systems through peak shaving and cost reductions compared with systems without batteries.

Abstract

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In this paper, we study the problem of determining the size of battery storage used in grid-connected photovoltaic (PV) systems. In our setting, electricity is generated from PV and is used to supply the demand from loads. Excess electricity generated from the PV can be stored in a battery to be used later on, and electricity must be purchased from the electric grid if the PV generation and battery discharging cannot meet the demand. Due to the time-of-use electricity pricing, electricity can also be purchased from the grid when the price is low, and be sold back to the grid when the price is high. The objective is to minimize the cost associated with purchasing from (or selling back to) the electric grid and the battery capacity loss while at the same time satisfying the load and reducing the peak electricity purchase from the grid. Essentially, the objective function depends on the chosen battery size. We want to find a unique critical value (denoted as $C_{ref}^c$) of the battery size such that the total cost remains the same if the battery size is larger than or equal to $C_{ref}^c$, and the cost is strictly larger if the battery size is smaller than $C_{ref}^c$. We obtain a criterion for evaluating the economic value of batteries compared to purchasing electricity from the grid, propose lower and upper bounds on $C_{ref}^c$, and introduce an efficient algorithm for calculating its value; these results are validated via simulations.

I. INTRODUCTION

Grid-connected PV storage sizing differs from off-grid sizing because batteries target cost reduction, arbitrage, and peak shaving rather than autonomy. The paper derives a unique critical capacity, bounds it theoretically, and computes it with an efficient algorithm validated by simulations.

  • I. INTRODUCTION: Off-grid sizing requires sufficient discharge rate for peak load and capacity for the longest night-time use and cloudy-period autonomy.Grid-connected applications instead use storage for output smoothing, demand-charge reduction, and arbitrage.
  • I. INTRODUCTION: Grid-connected batteries serve behind-the-meter loads by storing excess PV, shifting purchases under time-of-use pricing, and selling electricity when prices are high.Unlike off-grid systems, autonomy is secondary; the objective emphasizes energy cost, aging-related investment loss, and peak grid purchases.
  • I. INTRODUCTION: A unique critical capacity C_ref^c leaves total cost unchanged at larger sizes, while smaller capacities incur strictly higher cost.The total includes electricity purchase and battery investment loss associated with capacity degradation.
  • I. INTRODUCTION: The paper derives an economic-value criterion, lower and upper bounds, and an efficient algorithm for calculating the critical battery capacity.The bounds depend on PV generation, loads, and the cost-minimization period.
  • A. Photovoltaic Generation: The PV-generation model uses global horizontal irradiation, panel area, and cell efficiency, while omitting panel-temperature effects.The capacity analysis relies on PV power rather than a detailed generation model, allowing more complex models to be incorporated.
  • C. Battery: The model represents battery charge, discharge, aging, stored energy, and power-rate constraints, with capacity loss occurring only during discharge.Usable capacity starts at C_ref, and cumulative loss is nonnegative and non-decreasing over time.

D. Load

The load is modeled as a piecewise continuous power-demand function without explicit general assumptions. Residential demand may follow fixed schedules or flexible schedules constrained by comfort.

  • D. Load: P_load(t) denotes the load at time t and is assumed to be piecewise continuous.The model does not impose further explicit assumptions on the load in the analyzed setting.
  • D. Load: Residential loads may be fixed, such as lights and televisions, or flexible, such as refrigerators and air conditioners.Flexible schedules are allowed when room temperature remains within a comfortable range.

E. Converters for PV and Battery

PV and batteries connect to the AC bus through DC/AC conversion stages modeled with constant efficiencies. The battery-side converter and battery are treated together when determining AC-bus power exchange.

  • E. Converters for PV and Battery: PV generation reaches the AC bus through a DC-to-AC converter with constant efficiency η_pv.PV, battery, grid, and loads are connected to the AC bus.
  • E. Converters for PV and Battery: Battery charging uses AC-to-DC conversion, while discharging uses DC-to-AC conversion, with both converter efficiencies assumed constant and equal to η_B.These stages are shown in the system configuration and determine the battery-side power representation.
  • E. Converters for PV and Battery: P_BC(t) denotes power exchanged with the AC bus when the converters and battery are treated as one entity.The combined representation incorporates converter and battery behavior into the AC-bus exchange.

F. Cost Minimization

The optimization balances net grid electricity cost against battery capacity-loss cost while meeting load and peak-purchase constraints. Time-of-use pricing encourages discharge during expensive periods, creating a tradeoff between savings and aging.

  • F. Cost Minimization: The cost-minimization problem combines net power-purchase cost with battery capacity-loss cost while satisfying demand and peak-shaving requirements.Grid purchases contribute positive cost, whereas electricity sold back to the grid contributes negative cost.
  • F. Cost Minimization: The formulation treats PV-generated electricity as free and enforces power balance throughout the cost-minimization interval.The peak-shaving requirement is represented by the constraint P_g(t) ≤ D.
  • F. Cost Minimization: Frequent battery use lowers net power-purchase cost but raises capacity-loss cost, whereas infrequent use has the opposite tradeoff.The optimal control policy determines battery charging/discharging and grid exchange under this tradeoff.
  • F. Cost Minimization: Dynamic pricing encourages battery discharge and reduced grid purchases during peak-price periods, while P_g(t) ≤ D limits purchases when peak load occurs off-peak.The pricing schedule and peak constraint jointly support peak shaving.

G. Storage Size Determination

The paper formulates battery sizing for grid-connected PV systems as selecting a critical capacity based on an objective that varies with battery size. It seeks non-trivial critical capacities and develops bounds for determining them.

  • The optimization uses battery control u(t) as its independent variable while enforcing battery-state, capacity, and load-related constraints.The feasible-control set consists of controls satisfying all constraints in the reformulated optimization problem.
  • The objective J(Cref) combines time-of-use grid electricity costs with battery capacity-loss costs.With other parameters fixed, the objective is treated as a function of the reference battery capacity.
  • Increasing Cref cannot increase J because additional storage can store PV electricity, enable low-price grid purchases, and later serve loads or support sales.The supplied passage states that the decrease may not be strict.
  • The storage-size determination problem asks for a critical Cref whose cost behavior changes at the threshold.The paper notes that direct analytic solution is difficult because of nonlinear constraints and unavailable analytic load and PV profiles.
  • The paper addresses this difficulty by identifying conditions for positive finite critical capacities and proposing lower and upper bounds.

III. BOUNDS ON Cc

This section characterizes how the optimal cost changes with storage capacity and establishes when the critical capacity is meaningful, unique, and economically justified. It then derives bounds under feasibility and cost assumptions.

  • Feasibility requires initial conditions and charging opportunities that prevent the battery from needing discharge before it can charge.Assumption 1 excludes cases with no feasible control or only the unused control u(t)=0.
  • J(Cref) is non-increasing in battery capacity, with J(0) having the largest value when the battery cannot be used.
  • The critical capacity Cref^c is unique under the fixed-parameter optimization problem.
  • If battery capacity-loss cost exceeds the price-spread benefit adjusted by conversion efficiency and aging, using storage is not economically desirable and Cref^c can be zero.
  • Under Assumptions 1 and 2, the paper derives lower and upper bounds on the critical battery capacity.The upper-bound discussion also covers the case where charging opportunities are absent, implying a zero critical capacity.

IV. ALGORITHMS FOR CALCULATING Cc

The paper presents two algorithms for calculating the critical battery capacity: a grid-search procedure and a more efficient bisection method based on cost monotonicity.

  • Algorithm 1 evaluates discretized candidate capacities from the calculated upper bound using repeated optimization solves.Its approximation accuracy is controlled by δt, τcap, and τcost.
  • The critical capacity can be implemented with fixed-capacity batteries connected in parallel when the minimum charging time is fixed.
  • The continuous optimization is discretized into a mixed integer program with indicator constraints and solved using CPLEX.
  • Algorithm 1 stops after at most L+1 optimization problems and can approach arbitrary prescribed precision by decreasing τcap.
  • Algorithm 2 uses bisection because J(Cref) is non-increasing, maintaining an interval containing the critical capacity.
  • Each Algorithm 2 iteration halves the search interval, yielding a logarithmic bound on the number of optimization problems.

V. SIMULATIONS

The simulations calculate critical battery capacity using the proposed algorithm under residential and commercial settings. They consider PV profiles with relatively large and small output variations.

  • The simulation parameters are chosen based on typical residential home settings and commercial buildings.
  • Figure 2 reports PV output based on GHI measurements at La Jolla, California, with tick marks indicating local-standard-time noon each day.

A. Setting

The setting uses measured California PV data, representative residential and commercial loads, time-of-use grid prices, and hourly discretization to simulate battery operation. Two four-day periods represent relatively large or small PV-output variations.

  • PV and scenarios: Simulations use measured GHI from July 2010 in La Jolla, California, with PV efficiency η = 0.15 and panel area S = 10m2.These parameters give Ppv(t) = 1.5 × GHI(t)(W).
  • PV and scenarios: The study compares July 8 and July 13 four-day periods, representing relatively large and relatively small PV-output variations, respectively.Both periods begin at 0000 h local standard time.
  • Electricity pricing: Grid purchase prices are 16.5¢/kWh on-peak, 7.8¢/kWh semipeak, and 6.1¢/kWh off-peak.The rates apply from 11AM–6PM, 6AM–11AM and 6PM–10PM, and all other hours, respectively.
  • Battery model: Battery dynamics and aging are discretized with a 1-hour sampling interval, using the discretized forms of the continuous battery equations.The simulations assume lead-acid batteries with Z = 3 × 10−4, K = 0.15$/Wh, Tc = 12h, and converter efficiencies ηpv = ηB = 0.9.
  • Load profiles: Residential and commercial load profiles differ in peak timing: residential peaks occur early morning and late evening, while commercial peaks occur together during daytime.The profiles are used as representative load scenarios.
  • Peak shaving: The parameter D is set to 800(W) against load maxima of about 1000(W) to illustrate battery peak-shaving capability.The stated parameter choice satisfies Assumption 1.

B. Results

The simulations demonstrate how the critical battery capacity and operating strategy vary with horizon, load profile, photovoltaic variation, and starting date. Batteries reduce costs and shave peaks, with savings increasing over longer optimization durations in the reported residential and commercial cases.

  • Basic setting: 2.2 Wh of capacity loss occurs in the 16089 Wh residential case, consistent with aging only during battery discharge.The optimization objective for this case is J(16089) = −0.3222.
  • Residential load: 16089 Wh and 16096 Wh are the critical capacities for residential cases with 24-hour and 48-hour horizons, respectively.For the 48-hour case, the battery charges during the first half of each day and discharges during the second half until empty.
  • Commercial load: 13352 Wh is the commercial-load critical capacity; discharge during peak prices shaves load peaks, while surplus PV is sold to the grid.Commercial peak loads coincide with high prices, producing negative net grid purchases in the illustrated case.
  • Load and PV effects: Commercial loads tend to have higher cost because their peaks coincide with high prices, despite lower average cost than residential loads.Commercial loads tend to require larger optimum capacity under relatively large PV variation, whereas residential loads tend to require larger capacity under relatively small variation.
  • Cost comparison: 16.10% and 25.79% cost savings are reported for 96-hour residential and commercial cases using batteries of 18320 Wh and 15747 Wh, respectively.Relative battery savings increase as the cost optimization duration increases; Table II reports absolute savings because some costs are negative.

VI. CONCLUSIONS

The paper addresses battery sizing for grid-connected PV systems by proposing bounds and an efficient algorithm for calculating storage size. Its analysis assumes constant converter efficiency and identifies extensions to nonlinear converter behavior, distributed storage, and stochastic PV generation.

  • The paper proposes lower and upper bounds on battery storage size for grid-connected PV systems.
  • It introduces an efficient algorithm for calculating the storage size.
  • The analysis assumes constant efficiency for the PV and battery power converters.The paper notes that practical converter efficiency depends nonlinearly on input power.
  • Future work includes modeling nonlinear converter efficiency, distributed renewable energy storage, and stochastic PV generation.

APPENDIX

The appendix simplifies a nonlinear battery capacity-loss model into a linear ODE under short-horizon and discharge-specific assumptions. Capacity loss is modeled during battery discharge and excluded during charging or non-discharging operation.

  • The appendix combines prior battery capacity-loss equations and rewrites them using the paper’s notation.
  • The nonlinear capacity-loss equation is rewritten using ΔC(t) = Cref − C(t).
  • For time periods much shorter than battery lifetime, the capacity-loss percentage supports a simplifying approximation.
  • The approximation yields a linear ordinary differential equation for battery capacity loss during discharge.
  • When PB(t) ≥ 0, the model specifies no capacity loss; loss occurs only when the battery is discharging.
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