Problem Formulation

Here, we explain the mathematical formulation of two V2G scheduling optimization problems that SH-V2G-Simulator aims to solve.

System model

We consider a situation where $N_\text{c}$ chargers participate in a charging scheduling program. Our goal is to determine the charging powers of the chargers over $T$ time slots, each of which is $\Delta$ minutes long.

For each $n=1,\dots,N_\text{c}$ and $t=1,\dots,T$, we denote by $u_n[t]$ the charging power of charger $n$ at time slot $t$, with units given in kW. A negative charging power implies that the charger is in a discharging mode.

We index the EVs plugged into the chargers using the symbol $i=1,\dots,N_\text{ev}$. The following symbols represent the characteristics of EV $i$.

Table 1. Characteristics of EV $i$

Symbol	Description	Restriction	Unit
$t_i^\text{arr}$	arrival time slot	$\in\{1,\dots,T\}$
$t_i^\text{dep}$	departure time slot[1]	$\in\{t_i^\text{arr}+1,\dots\}$
$U_i^{\text{min}}$	minimum charging power[2]	$\leq0$	kW
$U_i^\text{max}$	maximum charging power	$\geq0$	kW
$E_i^\text{cap}$	battery capacity	$>0$	kWh
$E_i^\text{min}$	minimum allowable energy	$\in \big[0,E_i^\text{cap}\big]$	kWh
$E_i^\text{ini}$	energy level at arrival	$\in \big[E_i^\text{min},E_i^\text{cap}\big]$	kWh
$E_i^\text{ref}$	reference (desired) energy at departure	$\in \big[E_i^\text{min},E_i^\text{cap}\big]$	kWh

Note

Although a single EV may be connected to chargers multiple times, for simplicity, we assign a distinct index to each connection.

Constraints

In the following, we state the constraints that $u_n[t]$ must satisfy.

EV’s charging/discharging capacity

Suppose that EV $i$ is connected to charger $n$ from time slot $t_i^\text{arr}$ to $t_i^\text{dep}$. Then, $u_n[t]$ must satisfy that

\[U_i^\text{min} \leq u_n[t] \leq U_i^\text{max},\quad t=t_i^\text{arr},\dots,t_i^\text{dep}-1.\]

If charger $n$ is vacant at time slot $t$, then $u_n[t]=0$.

Battery capacity

Again, suppose that EV $i$ is connected to charger $n$. Let $e[t]$ denote the EV $i$’s stored energy at the beginning of time slot $t=t_i^\text{arr},\dots,t_i^\text{dep}$. Then, $e[t]$ must lie within the bounds

\[E_i^\text{min} \leq e[t] \leq E_i^\text{cap},\quad t=t_i^\text{arr},\dots,t_i^\text{dep}-1,\]

while also satisfying the relations[3]

\[\begin{split}\begin{aligned} e[t_i^\text{arr}] &= E_i^\text{ini},\\ e[t+1] &= e[t] + \frac{\Delta}{60} \cdot u_n[t],\quad t=t_i^\text{arr},\dots,t_i^\text{dep}-1. \end{aligned}\end{split}\]

Congestion constraint

We also consider a congestion constraint

\[ P^\text{min} \leq z \leq P^\text{max} \]

where $z$ is the aggregate charging power defined by

\[ z[t] = \sum_{n=1}^{N_\text{c}} u_n[t],\quad t=1,\dots,T. \]

This may arise due to the capacity limit of the infrastructure (e.g., transformers or bidirectional converters) or a contract with the distribution system operator.

Note

If there is no congestion constraint, one may set $P^\text{max}$ and $P^\text{min}$ sufficiently large (in absolute value) to inactivate the congestion constraint.

Load flattening problem

One problem adressed by SH-V2G-Simulator is the load flattening problem,[4] which aims to flatten the total load (i.e., base load + EV load) by controlling the charging powers of the EVs. By flattening the total load, peak load rises can be mitigated (enabling the deferral of infrastructure investment) and the generation costs can be reduced.

For this purpose, SH-V2G-Simulator minimizes the objective function

\[ (\text{load flattening cost}) + (\text{battery degradation cost}) + (\text{reference energy penalty}), \]

where each term is defined as follows:

The load flattening cost function is defined by
\[ (\text{load flattening cost}) = \sum_{t=1}^{T}\frac{\Delta}{60} \Big(D[t]+\sum_{n=1}^{N_\text{c}}u_i[t]\Big)^2. \]
It penalizes deviations from the mean total load. Hence, given the total amount of electricity consumption, minimizing the load flattening cost function $J$ results in a flattened total load.
The battery degradation cost is introduced to mitigate excessive charge-discharge cycling that may reduce the battery lifespan. In particular, we define
\[ (\text{battery degradation cost}) = \sum_{i=1}^{N_\text{ev}} \frac{\Delta}{60} \cdot \sigma_i \Big\| u_n \big[t_i^\text{arr}:(t_i^\text{dep}-1) \big] \Big\|_2^2, \]
where $\sigma_i\geq0$ represents the battery degradation parameter selected by the user of EV $i$. A larger value of $\sigma_i$ implies that the user places greater concern on battery lifespan reduction. This term penalizes fluctuations in the charging power.
While flattening the total load, the desired reference energies of the EVs should also be considered. In this regard, we introduce a penalty function
\[ (\text{reference energy penalty}) = \sum_{i=1}^{N_\text{ev}} \rho \cdot \max\big(0,E_i^\text{ref}- e_i\big[t_i^\text{dep}\big]\big) \]
where $\rho>0$ is a penalty parameter. It penalizes the reference energy unfulfillment.

User-friendly V2G problem

The other problem that SH-V2G-Simulator deals with is the user-friendly V2G problem,[5] which focuses on fulfilling various utilities of EV users. Here, we define the utility of EV $i$ as

\[ (\text{utility of EV $i$}) =\frac{1}{w_{\text{cs},i}+w_{\text{pc},i}+w_{\text{bs},i}} \big({w_{\text{cs},i} f_{\text{cs},i} + w_{\text{pc},i} f_{\text{pc},i} + w_{\text{bs},i} f_{\text{bs},i}}\big), \]

which is a convex combination of three utility functions $f_{\text{cs},i}$, $f_{\text{pc},i}$, and $f_{\text{bs},i}$, weighted by $w_{\text{cs},i}$, $w_{\text{pc},i}$, and $w_{\text{bs},i}$, respectively. (It is assumed that the weights are not all zero.)

The utility functions, each of which is normalized to enable a fair comparison, are defined as follows:

The cost saving utility function is defined by
\[f_{\text{cs},i}=\sum_{t=t_i^\text{arr}}^{t_i^\text{dep}-1}\frac{C[t]}{\bar C}\cdot\frac{\Delta}{60}\cdot u_n[t],\]
where $C[t]$ is the electricity price at time slot $t$ and $\bar{C} = \max_{t=1,\dots,T} C[t]$.
The priority charging utility function is defined by
\[f_{\text{pc},i}=\sum_{t=t_i^{arr}}^{t_i^{dep}-1}\frac{\Delta}{60}\cdot \frac{t-t_i^{arr}}{t_i^{dep}-t_i^{arr}}\max\big(0,E_i^{ref}-e_i\big[t_i^\text{dep}\big]\big),\]
whose minimization yields a charging schedule that charges the EV as quickly as possible.
The battery saving utility function (for mitigating battery degradation) is defined by
\[f_{\text{bs},i}=\sum_{t=t_i^\text{arr}}^{t_i^\text{dep}-1}\frac{\Delta}{60} \cdot\frac{1}{\bar U_i}u_n^2[t],\]
where $\bar U_i=\max(-U_i^\text{min},U_i^\text{max})$.

The objective function of the user-friendly problem is given by

\[ \sum_{i=1}^{N_\text{ev}}(\text{utility of EV $i$}) + (\text{reference energy penalty}). \]

Here, the total utility of EVs is combined with the reference energy penalty term, which is the same as that defined in the load flattening problem, to enforce reference energy fulfillment.