Chance Constrained Optimization for Energy Management in Electric Vehicles

E-powertrain of future electric vehicles could consist of energy generation units (e.g., fuel cells and photovoltaic modules), energy storage systems (e.g., batteries and supercapacitors), energy conversion units (e.g., bidirectional DC/DC converters and DC/AC inverters) and an electric machine, which can work in both generating and motoring modes [1-6]. An energy management system is responsible to operate the above-mentioned components in a way that the technical constraints are satisfied. This task should be accomplished by solving an optimization problem, which could aim at minimizing the total operation costs [5]. The optimization problem has been widely addressed by deterministic approaches [7], which take into account the forecasted values of active-reactive load profile. However, as shown in Figure 1 (a), it is impossible to accurately forecast the values, meaning that the solutions coming from deterministic approaches could lead to infeasible operations (i.e., constraint violations). Therefore, stochastic optimization approaches [8] should be utilized to find optimal solution strategies while considering uncertain parameters.

There are several mathematical approaches for optimization under uncertainty each of which could be suitable for a specific type of application. For instance, robust optimization and worst-case optimization is frequently used in many applications in which constraint violations are not tolerated [9]. However, in electric vehicles, there exist some types of constraints (e.g., upper and lower limit of state of the charge in batteries [10]), which are allowed to be violated to some degree and also for a limited time. For this, the method of chance constrained programming [11-13] (see Figure 1 (b)) can be used, by which optimal operations are obtained, and the satisfaction of constraints is ensured with a ‘predefined probability’ level. The deterministic equivalent of chance constraints can be determined easily if the model is linear and the uncertain values are ‘normally’ distributed. However, the e-powertrain model is mixed-integer nonlinear [14] and the uncertain variables are described by non-Gaussian probability density functions. In addition, differential equations of energy storage units introduce further complexities to the problem as stochastic ‘dynamic’ optimization problem has to be solved. For this, MicroFuzzy GmbH in collaboration with Ilmenau University of Technology formulates and solves this complex problem using the powerful methods [15,16] which have been recently developed in the Department of Process Optimization of that university. The solutions obtained by utilizing the chance constrained method could lead to significant reduction in total operation costs.

Figure 1: (a) Uncertain active-reactive load power of the electric machine. (b) General formulation of the chance constrained optimization problem.

1. Fathabadi H (2018) Novel fuel cell/battery/supercapacitor hybrid power source for fuel cell hybrid electric vehicles. Energy 143: 467-477. Link: https://bit.ly/2ECRx42
2. Fu Z, Li Z, Si P, Tao F (2019) A hierarchical energy management strategy for fuel cell/battery/supercapacitor hybrid electric vehicles. International Journal of Hydrogen Energy  44: 22146-22159. Link: https://bit.ly/3gCQ76T
3. García P, Torreglosa JP, Fernández LM, Jurado F (2013) Control strategies for high-power electric vehicles powered by hydrogen fuel cell, battery and supercapacitor. Expert Systems with Applications 40: 4791-4804. Link: https://bit.ly/2EFdhMC
4. Khaligh A, Li Z (2010) Battery, ultracapacitor, fuel cell, and hybrid energy storage systems for electric, hybrid electric, fuel cell, and plug-in hybrid electric vehicles: State of the art. IEEE transactions on Vehicular Technology 59: 2806-2814. Link: https://bit.ly/2DcxyZy
5. Li H, Zhou Y, Gualous H, Chaoui H, Boulon L (2020) Optimal Cost Minimization Strategy for Fuel Cell Hybrid Electric Vehicles Based on Decision Making Framework. IEEE Transactions on Industrial Informatics. Link: https://bit.ly/2Qz84IQ
6. Wu Y, Gao H (2006) Optimization of fuel cell and supercapacitor for fuel-cell electric vehicles. IEEE transactions on Vehicular Technology 55: 1748-1755. Link: https://bit.ly/2QwyL0T
7. Wegmann R, Döge V, Becker J, Sauer DU (2017) Optimized operation of hybrid battery systems for electric vehicles using deterministic and stochastic dynamic programming. Journal of Energy Storage 14: 22-38. Link: https://bit.ly/2QwyL0T
8. \Moura SJ, Fathy HK, Callaway DS, Stein JL (2010) A stochastic optimal control approach for power management in plug-in hybrid electric vehicles. IEEE Transactions on control systems technology 19: 545-555. Link: https://bit.ly/2G7yorx
9. Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press 28. Link: https://bit.ly/2YFUAPK
10. Mohagheghi E, Alramlawi M, Gabash A, Blaabjerg F, Li P (2020) Real-time active-reactive optimal power flow with flexible operation of battery storage systems. Energies 13: 1697. Link: https://bit.ly/32waoWI
11. Charnes A, Cooper WW (1959) Chance-constrained programming. Management science 6: 73-79. Link: https://bit.ly/3gJnYLu
12. Geletu A, Klöppel M, Zhang H, Li P (2013) Advances and applications of chance-constrained approaches to systems optimisation under uncertainty. International Journal of Systems Science 44: 1209-1232. Link: https://bit.ly/2QyIyUc
13. Geletu A, Klöppel M, Hoffmann A, Li P (2015) A tractable approximation of non-convex chance constrained optimization with non-Gaussian uncertainties. Engineering Optimization 47: 495-520. Link: https://bit.ly/32B3H5w
14. Murgovski N, Johannesson L, Sjöberg J, Egardt B (2012) Component sizing of a plug-in hybrid electric powertrain via convex optimization. Mechatronics 22: 106-120. Link: https://bit.ly/34GHpSD
15. Geletu A, Hoffmann A, Kloppel M, Li P (2017) An inner-outer approximation approach to chance constrained optimization. SIAM Journal on Optimization 27: 1834-1857. Link: https://bit.ly/32BBo6T
16. Geletu A, Hoffmann A, Li P (2019) Analytic approximation and differentiability of joint chance constraints. Optimization 68: 1985-2023. Link: https://bit.ly/31zVFdN