Storage Size Determination for Grid-Connected Photovoltaic Systems

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.

💡 Research Summary

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The paper addresses the problem of sizing battery storage for grid‑connected photovoltaic (PV) systems, where the goal is to minimize the total cost consisting of electricity purchase (or revenue from selling back) and the degradation cost of the battery while satisfying load demand and a peak‑shaving constraint. The authors model PV generation as a function of global horizontal irradiance, panel area, and conversion efficiency, and they represent the battery dynamics with a simple energy balance equation. Battery aging is captured by a capacity‑loss term that grows only during discharge, with a degradation rate constant Z. The cost function includes the integral of time‑of‑use electricity prices multiplied by net grid exchange power, plus a term K ΔC that penalizes the cumulative capacity loss after a planning horizon T.

A key contribution is the identification of a unique critical battery capacity (C_{ref}^{c}). If the installed battery capacity (C_{ref}) is greater than or equal to this critical value, the optimal control policy will not increase total cost; any additional capacity beyond (C_{ref}^{c}) is redundant. Conversely, if (C_{ref}<C_{ref}^{c}), the total cost is strictly higher because the system cannot fully exploit low‑price periods or meet the peak‑shaving limit (D). The authors prove this property by establishing monotonicity of the optimal cost with respect to battery size under two reasonable assumptions: (i) constant round‑trip efficiency (\eta_B) and a fixed minimum charge/discharge time (T_c); (ii) non‑negative, time‑varying electricity prices and a finite peak‑shaving bound.

To make the critical capacity computable, the paper derives analytical lower and upper bounds. The lower bound (C_{low}) is the smallest capacity that yields a net electricity‑price saving larger than the degradation penalty, essentially the point where marginal savings from arbitrage outweigh marginal degradation cost. The upper bound (C_{up}) is the minimum storage needed to satisfy the peak‑shaving constraint given the worst‑case net load (load minus PV) and the charging/discharging rate limits. Between these bounds, the authors propose a binary‑search algorithm. At each iteration, a convex optimization problem is solved for the control variable (u(t)=P_{BC}(t)) (the power exchanged between battery and AC bus). The solution provides the total cost for the current trial capacity; the algorithm then adjusts the search interval based on whether the cost is decreasing or increasing, converging rapidly to (C_{ref}^{c}).

Simulation studies use real irradiance data and time‑of‑use tariffs from San Diego Gas & Electric, covering both residential and commercial load profiles. Results demonstrate that for capacities below (C_{ref}^{c}) the system heavily discharges the battery, incurring large ΔC and thus a high degradation cost that outweighs the electricity‑price savings. When the capacity reaches (C_{ref}^{c}), the total cost plateaus: additional storage does not further reduce the electricity bill, but it does guarantee that the peak‑shaving limit (D) is never violated. The simulations also illustrate the dual benefit of batteries: reducing peak grid draw (which can lower demand charges) and exploiting price differentials (buy low, sell high).

Compared with prior work that largely relies on exhaustive simulations or heuristic sizing rules, this study provides a rigorous analytical framework that yields explicit bounds and a provably optimal capacity. The inclusion of battery degradation as a direct monetary term makes the model suitable for investment‑return analysis and for comparing different battery chemistries (through the parameters Z and K). The authors suggest future extensions to incorporate more detailed PV temperature effects, stochastic load and generation forecasts, and multi‑energy micro‑grid configurations. Overall, the paper offers a valuable tool for engineers and policymakers aiming to design cost‑effective, grid‑interactive PV‑battery systems.