In mcanigueral/evflex: Tools to coordinate EV sessions based on user profiles and optimization objectives

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Overview

This article wants to define the optimization problem corresponding to a smart charging scenario for electric vehicles. The smart charging concept is widely used in different fields and applications. In this case, we define smart charging as a tool to coordinate individual EV charging sessions in order to obtain the optimal aggregated demand profile according to a certain objective. There are different practical strategies to coordinate each session, such as postponing sessions, reducing the charging power or the total energy charged, a combination of several strategies, etc. Currently, {evflex} package only supports postpone strategy.

Optimization problem

Objective function

The main goals of the EV demand optimization performed with {evflex}package are:

• Minimize grid dependency maximizing the use of renewable energy
• Minimize the peaks in the demand curve (i.e. peak shaving)
• Make the minimum changes as possible to the original schedule of charging sessions

These variables will be represented with the following nomenclature:

• $T$ : Number of time intervals within the optimization window
• $\Delta t$ : Time interval, in hours
• $w_1$ : Weight for grid balance optimization strategy
• $w_2$ : Weight for peak shaving optimization strategy
• $R_t$ : Renewable generation time series
• $L_t$ : Static EV load (BAU) time series
• $V_t$ : Flexible EV load (BAU) time series
• $O_t$ : Optimal flexible EV load time series
• $E$ : Total energy demand from flexible EV within the optimization window

The formulation of the problem is depicted by Equation \@ref(eq:opt-objective):

\begin{equation} min \sum_{t=1}^{T} w_1 (R_t-L_t-O_t)^2 + w_2 (V_t - O_t)^2 (#eq:opt-objective) \end{equation}

The first term represents the energy exchanged with the distribution grid since it is a subtraction between the renewable energy generation and the EV demand (both flexible and static). This subtraction is squared to highly peak values. Thus, minimizing this term corresponds to avoid the difference between renewable generation and EV demand on one hand, and peak shaving on the other hand.

The second term represents the difference between the original flexible EV demand and the final flexible EV demand modulated by the optimization algorithm. The difference is squared too in order to avoid negative values. Minimizing this term corresponds results in less changes to the original EV sessions schedule. It could be interesting for the charging point operator when every sessions postponed represents an economic cost.

Both terms have a corresponding weight parameter ($w_1$ and $w_2$) to customize the optimization result according to a defined interest. Increasing $w_1$ over $w_2$ would result to a more energy-focused optimization while probably higher values of $w_2$ over $w_1$ would correspond to a more practical approach.

If we develop the objective formula in Equation \@ref(eq:opt-objective) we obtain:

\begin{equation} min \sum_{t=1}^{T} w_1 (R_t² + L_t² + O_t² - 2 R_t L_t + 2 L_t O_t - 2 O_t R_t) + w_2 (V_t² - 2 V_t O_t + O_t²) (#eq:opt-objective-2) \end{equation}

\begin{equation} min \sum_{t=1}^{T} w_1 R_t L_t (R_t + L_t - 2) + O_t² (w_2 + w_1) + 2 O_t (w_1 L_t - w_1 R_t - w_2 V_t) + w_2 V_t² (#eq:opt-objective-3) \end{equation}

Now we can discard the constant terms of the equation since they don't produce any difference in the final result to minimize. Therefore, the final optimization problem equation is depicted in Equation \@ref(eq:opt-equation):

\begin{equation} min \sum_{t=1}^{T} O_t² (w_2 + w_1) + 2 O_t (w_1 L_t - w_1 R_t - w_2 V_t) (#eq:opt-equation) \end{equation}

Constraints

The optimization constraints depend of each study case and, in this case, it is a smart charging scenario considering postpone strategy. Therefore, our objective function in Equation \@ref(eq:opt-equation) will be constrained to Equations \@ref(eq:opt-constraint-energy), \@ref(eq:opt-constraint-foreshift) and \@ref(eq:opt-constraint-positive):

1. Total EV demand must remain the same:

\begin{equation} \sum_{t=1}^T O_t \Delta t = E (#eq:opt-constraint-energy) \end{equation}

1. Demand can only be shifted forwards, not backwards (Postpone strategy):

\begin{equation} \sum_{t=1}^U O_t \Delta t \le \sum_{t=1}^U V_t \Delta t \quad U = 1, 2, \dots, T (#eq:opt-constraint-foreshift) \end{equation}

1. Optimal EV demand can only have positive values:

\begin{equation} O_t \ge 0 \quad t \in T (#eq:opt-constraint-positive) \end{equation}

Quadratic programming formulation

The form of our optimization problem corresponds to Quadratic programming formulation, described in Equation \@ref(eq:cvx-formulation):

\begin{equation} \begin{array}{ll} \text{minimize} & \mathrm (1/2) x^\top \mathrm P \, \mathrm x + \mathrm q^{\top} \mathrm x \ \text{subject to} & \mathrm G \mathrm x \leq \mathrm h \ & \mathrm A \mathrm x = \mathrm b \end{array} (#eq:cvx-formulation) \end{equation}

To handle this type of optimization problem, the optimization engine used in {evflex} package is the CVXOPT Python package.

Simulation of smart charging

In this section, the demonstration of the smart charging algorithm has considered a set of sessions simulated by {evsim} package and the models built with package {evprof}. Figures \@ref(fig:opt1) - \@ref(fig:opt5) show the significant differences in the optimal setpoint definition taking into account different weights for the optimization objectives. Following the same nomenclature exposed in the previous sections, $w_1$ is the weight of Energy optimization (i.e. minimize the energy exchanged with the distribution grid) and $w_2$ is the weight of Cost optimization (i.e. minimize the difference between the original and optimal flexible EV demand).

In Figures \@ref(fig:opt1) - \@ref(fig:opt5) we see the following variables:

• Solar power generation r colorize("(Orange)", "orange")
• Business-as-usual EV demand simulated from our models r colorize("(Blue)", "blue")
• Optimal EV demand given by the optimization problem r colorize("(Dashed green)", "green")
• Final flexible EV demand obtained after coordinating EV sessions according to optimal setpoint r colorize("(Green)", "green")

knitr::include_graphics('figures/1-0.png')

knitr::include_graphics('figures/0.75-0.25.png')

knitr::include_graphics('figures/0.5-0.5.png')

knitr::include_graphics('figures/0.25-0.75.png')

knitr::include_graphics('figures/0-1.png')

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mcanigueral/evflex documentation built on Feb. 16, 2021, 8:01 p.m.