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Stochastic Optimal Control Problems for the Cost-Optimal Management of a Standalone Microgrid

Paul Honore Takam, Nathalie Fruiba

Abstract

In this paper, we consider a domestic standalone microgrid equipped with local renewable energy generation such as photovoltaic panels, consumption units, and battery storage to balance supply and demand and investigate the stochastic optimal control problem for its cost-optimal management. As a special feature, the manager does not have access to the power grid but has a local generator, making it possible to produce energy using fuel when needed. Such systems are very important for rural electrification, particularly in developing countries. However, these systems are very complex to control due to uncertainties in the weather and environmental conditions, which affect the energy generation and the energy demand. In addition, we assume that the battery and the fuel tank have limited capacities and that the fuel tank can only be filled once at the beginning of the planning period. This leads us to the so-called finite fuel problem. In addition, we allow the energy demand to not always be satisfied, and we impose penalties on unsatisfied demand, the so-called discomfort cost. The main goal is to minimize the expected aggregated cost of generating power using the generator and operating the system. This leads to a mathematical optimization problem. The problem is formulated as a discrete-time stochastic control problem and solved numerically using methods from the theory of Markov decision processes.

Stochastic Optimal Control Problems for the Cost-Optimal Management of a Standalone Microgrid

Abstract

In this paper, we consider a domestic standalone microgrid equipped with local renewable energy generation such as photovoltaic panels, consumption units, and battery storage to balance supply and demand and investigate the stochastic optimal control problem for its cost-optimal management. As a special feature, the manager does not have access to the power grid but has a local generator, making it possible to produce energy using fuel when needed. Such systems are very important for rural electrification, particularly in developing countries. However, these systems are very complex to control due to uncertainties in the weather and environmental conditions, which affect the energy generation and the energy demand. In addition, we assume that the battery and the fuel tank have limited capacities and that the fuel tank can only be filled once at the beginning of the planning period. This leads us to the so-called finite fuel problem. In addition, we allow the energy demand to not always be satisfied, and we impose penalties on unsatisfied demand, the so-called discomfort cost. The main goal is to minimize the expected aggregated cost of generating power using the generator and operating the system. This leads to a mathematical optimization problem. The problem is formulated as a discrete-time stochastic control problem and solved numerically using methods from the theory of Markov decision processes.

Paper Structure

This paper contains 72 sections, 14 theorems, 72 equations, 18 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review on microgrids
  3. Literature review on stochastic optimal control
  4. Literature review on Markov decision processes
  5. Literature review on Markov decision processes
  6. Our contribution
  7. Paper Organization
  8. Mathematical Model of a Microgrid
  9. Management of a Microgrid
  10. Residual demand
  11. Battery state of charge
  12. Fuel tank level
  13. Control System
  14. State process
  15. Control process
  16. ...and 57 more sections

Key Result

lemma thmcounterlemma

The closed-form solution of the SDE dyn-Z at time $t_{n+1}$ with initial condition $Z(t_n) = z$ is given by:

Figures (18)

  • Figure 1: Simplified model of a microgrid that use solar panels, a battery and fuel driven generator to satisfy the electricity demand of a building.
  • Figure 2: Residual demand with daily and yearly seasonal patterns over a period of one year (left) and a zoom into a week (right) with parameters $\beta_R = 0.2, \sigma_R = 0.45, \kappa_1^R = 1,\kappa_2^R =0.4, \mu_0^R = 0.1, \delta_1= 365, \text{ and } \delta_2 = 1$.
  • Figure 3: Left: Efficiency as a function of the state of charge $q$ with $\eta_E^C(q) = 0.8+1.32q(1-q)^2$ and $\eta_E^D(q)= 0.8+1.32q^2(1-q)$. Right: State of charge and residual demand as functions of time in response to control actions.
  • Figure 4: Set of feasible controls $\pazocal{U}(n,x)=\pazocal{U}_Q(n,x) \cup\pazocal{U}_G(n,x)$ for a positive residual demand $(r>0)$ and $R_{Q0}=R_{G0}$. Left: $r< R_{Q0}$. Right: $r\geq R_{Q0}$
  • Figure 5: Running cost as a function a function of a positive residual demand $r>0$.
  • ...and 13 more figures

Theorems & Definitions (38)

  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • ...and 28 more