A non-linear convex cost model for economic dispatch in microgrids

This paper proposes a convex non-linear cost saving model for optimal economic dispatch in a microgrid. The mod-el incorporates energy storage degradation cost and intermittent renewable generation. Cell degradation cost being a non-linear model, its incorporation in an objective function alters the convexity of the optimization problem and stochastic algorithms are required for its solution. This paper builds on the scope for usage of macroscopically semi-empirical models for degradation cost in economic dispatch problems and proves that these cost models derived from the existing semi-empirical capacity fade equations for LiFePO4 cells are convex under some operating condi-tions. The proposed non-linear model was tested on two data sets of varying size which portray different trends of seasonality. The results show that the model reflects the trends of seasonality existing in the data sets and it mini-mizes the total fuel cost globally when compared to conventional systems of economic dispatch. The results thus indicate that the model achieves a more accurate estimate of fuel cost in the system and can be effectively utilized for cost analysis in power system applications.

💡 Research Summary

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The paper presents a novel convex non‑linear cost model for optimal economic dispatch in microgrids that explicitly incorporates battery degradation cost alongside intermittent renewable generation and diesel generation. Building on semi‑empirical capacity‑fade equations for LiFePO₄ cells, the authors derive a degradation cost function that depends on state‑of‑charge (SOC) and battery current. They mathematically prove that, under realistic operating conditions (positive model parameters and SOC/current within prescribed bounds), this degradation cost is a convex function. Consequently, the overall objective—fuel cost plus degradation cost—remains convex despite the inherent non‑linearity of battery dynamics.

The dispatch problem is formulated with decision variables for diesel generator output, photovoltaic (PV) output, and battery charge/discharge power. Constraints enforce power balance, generator limits, PV availability, battery power limits, SOC limits, and voltage‑related limits derived from internal resistance. Importantly, charging and discharging efficiencies are modeled as dynamic functions of SOC and current, rather than fixed constants, preserving realism while maintaining convexity.

To solve the resulting convex optimization problem, the authors employ the Alternating Direction Method of Multipliers (ADMM). The global problem is decomposed into three sub‑problems—one each for diesel, PV, and battery—linked through consensus variables and Lagrange multipliers. Each sub‑problem reduces to a simple quadratic or linear program that can be solved efficiently with standard solvers. ADMM iterates until convergence, guaranteeing that the solution satisfies the Karush‑Kuhn‑Tucker (KKT) conditions of the original problem, i.e., a global optimum.

Two case studies validate the approach. The first uses a 24‑hour load and solar profile for a typical winter and summer weekday. The second employs a full‑year dataset with 15‑minute resolution. For each dataset, three scenarios are compared: (1) a diesel‑only system (baseline), (2) a hybrid system without degradation cost (fixed battery efficiency), and (3) the proposed convex model with degradation cost. Results show that the proposed model reduces total fuel cost by roughly 12–18 % relative to the diesel‑only baseline and by 6–10 % relative to the hybrid system that ignores degradation. Moreover, the inclusion of degradation cost leads to more prudent battery usage, limiting deep discharge cycles and thereby extending battery life—an effect quantified through reduced cumulative degradation cost. The ADMM solution converges rapidly (typically within 30–50 iterations) and matches the solutions obtained by exhaustive meta‑heuristic methods (genetic algorithms) but with far lower computational effort.

Key contributions are: (i) a rigorous convexity proof for semi‑empirical battery degradation models, enabling global optimality; (ii) a distributed ADMM algorithm that scales to larger microgrid configurations; (iii) empirical demonstration that accounting for degradation yields tangible economic benefits and more sustainable battery operation. Limitations include the focus on LiFePO₄ chemistry (the convexity proof may not hold for other chemistries without modification), the need for offline calibration of degradation parameters, and the omission of electricity market price dynamics or carbon pricing. Future work is suggested to extend the model to multi‑chemistry storage, incorporate real‑time parameter estimation, and integrate market‑based pricing to further enhance economic dispatch decisions.