Communication-Censored-ADMM for Electric Vehicle Charging in Unbalanced Distribution Grids

Fig. 1 illustrates the topology of a grid-connected Residential EV Energy System for a single customer, consisting of a meter, an EV and residential load situated behind the Point of Common Coupling (PCC). In what follows, we consider a set of $N$ customers $\mathcal{N}:=\left\{1,\dotsc,n,\dotsc,N\right\}$, where each customer $n$ charges or discharges $EV_{n}$. We assume that for each customer, $EV_{n}$ is enabled with both real and reactive power control by way of the EV battery inverter.

Let $\Delta$ be a time interval length ²²2Emerging grid sensors such as micro-phasor measurement units (micro-PMUs) provide voltage and current phasor data at a high temporal resolution (see [22, 23]). We anticipate customer metering technology to follow a similar trajectory, reporting high temporal voltage and current data, which would also improve the applicability of our approach., and $\mathcal{T}:=\left\{1,\dotsc,T\right\}$ be a time horizon partition, so that the planning horizon is $[0,\Delta T]$.

Let $p_{n}(t)\in\mathbb{R}$ denote the real power to or from $EV_{n}$ over the time period $[\Delta(t-1),\Delta t]$ (in kW), and similarly, let $q_{n}(t)\in\mathbb{R}$ denote the reactive power to or from $EV_{n}$ over the time period $[\Delta(t-1),\Delta t]$ (in kVAR).

The power profile of $EV_{n}$ over $[0,\Delta T]$ is defined as

Each $EV_{n}$ has associated with it the following set of design parameters; arrival time index $a_{n}\in\mathcal{T}$, departure time index $d_{n}\in\mathcal{T}$, battery capacity $b_{n}$, inverter capacity $\overline{s}_{n}$, initial state of charge (SoC) $\hat{\sigma}_{n}$, target SoC $\sigma^{*}_{n}$, minimum SoC $\underline{\sigma}_{n}$, maximum SoC $\overline{\sigma}_{n}$, maximum charge rate $\overline{p}_{n}$, and minimum charge rate $\underline{p}_{n}$. These design parameters are collected into the set

Let the apparent power to or from $EV_{n}$ be denoted by ${s}_{n}(t)$, defined (for all $\Delta t\in[0,\Delta T]$) by

The apparent power through the inverter of $EV_{n}$ is constrained as $s_{n}(t)\leq\overline{s}_{n}$, where $\overline{s}_{n}$ denotes the inverter capacity. It follows that, for all $\Delta t\in[0,\Delta T]$,

Each customer $n$ must specify a departure time and target SoC which is feasible within the departure time, i.e., it must hold that

The battery SoC of $EV_{n}$ at time $\Delta t\in[0\Delta T]$ is denoted by $\sigma_{n}(t)$, which can be defined as

and the SoC profile of $EV_{n}$ is then defined as

The battery of $EV_{n}$ is constrained to prevent over-(dis)charging, in the following way

which implies that the SoC profile must satisfy

Let $\underline{\alpha}_{n}=(\underline{\sigma}_{n}-\hat{\sigma}_{n})/\Delta$, and $\overline{\alpha}_{n}=(\overline{\sigma}_{n}-\hat{\sigma}_{n})/\Delta$. Then using (2) together with (3) gives

where

Due to the limited (dis)charging power of the battery, we have

which implies that the power profile vector $\mathbf{p}_{n}$ is constrained as

To incorporate the charging demand for $EV_{n}$, we let $e_{n}=(\sigma^{*}_{n}-\hat{\sigma}_{n})/\Delta$ and enforce $\sigma_{n}(T)\geq\sigma_{n}^{*}$. Combining this charging demand with (2) gives

To limit the duration of (dis)charging for $EV_{n}$ to the period $EV_{n}$ is grid connected ($[a_{n},d_{n}]$), we define the availability matrix $L_{n}$ as follows

Accordingly, we further constrain $\mathbf{p}_{n}$ and $\mathbf{q}_{n}$ by

where $I\in\mathbb{R}^{T\times T}$ is the identity matrix, and $\mathbf{0}\in\mathbb{R}^{T}$ is the zero vector. Let us define

We now combine all of the battery constraints and the apparent power constraints, to write the feasibility set of power profile vectors

The following proposition can be easily proven.

We assume that each customer is compensated for delivering power to the grid at the same rate they are billed for consuming power from the grid. For our time horizon $\mathcal{T}$, we let the price profile be denoted by

where $\eta(t)$ is the price of electricity (in $/kWh) over the time period $[\Delta(t-1),\Delta t]$. The operational cost for $EV_{n}$ is then defined by

where $\kappa_{n}$ is a (small) regularization parameter which reduces to occurrence of unnecessary EV battery charge-discharge cycles. Specifically, we include the second term in (7) to maintain a smoother battery profile and avoiding unnecessary charge-discharge cycles [24], and in this way, we include the cost of battery degradation within our operational costs for $EV_{n}$. Positive values of $\Omega_{n}(\mathbf{p}_{n})$ represent financial costs whereas negative values of $\Omega_{n}(\mathbf{p}_{n})$ represent financial gains.

We consider a three phase, unbalanced, radial distribution feeder that includes two phase and single phase laterals, represented by a graph $\mathcal{G}=\left\{\mathcal{V}_{0},\mathcal{E}\right\}$. The set of vertices $\mathcal{V}_{0}=\left\{0,\dotsc,i,\dotsc,k\right\}$ represent nodes along the feeder and the edges $\mathcal{E}\subseteq\mathcal{V}_{0}\times\mathcal{V}_{0}$ represent line segments along the feeder with cardinality $\lvert\mathcal{E}\rvert=k$.

Node $0$ is considered the root node (feeder head) which is taken as an infinite bus, decoupling the interaction between the feeder and the wider power grid. An edge $(ij)\in\mathcal{E}$ exists if there is a line segment between node $i$ and node $j$, with node $i$ being closest to node $0$.

We assume that the graph $\mathcal{G}$ is simple with no repeated edges or self loops for any $i\in\mathcal{V}_{0}$. Let $\mathcal{V}=\mathcal{V}_{0}\backslash\left\{0\right\}$, and for each vertex $i\in\mathcal{V}$ let $\mathcal{E}^{i}\subseteq\mathcal{E}$ be the set of edges on the unique path from node $0$ to node $i$.

Each edge $(ij)\in\mathcal{E}$ is either a single phase, two phase, or three phase edge, and each vertex $i\in\mathcal{V}_{0}$ is either a single phase, two phase, or three phase node, where phases are labelled $\mathfrak{a},\mathfrak{b}$ and $\mathfrak{c}$. The set of all phases at node $i\in\mathcal{V}_{0}$ are denoted $\Phi^{i}$, e.g., for a three phase node we would have $\Phi^{i}=\left\{\mathfrak{a},\mathfrak{b},\mathfrak{c}\right\}$. We assume node $0$ is a three phase node.

The set of phases along edge $(ij)\in\mathcal{E}$ is denoted as $\Phi^{ij}$, e.g., $\Phi^{ij}=\left\{\mathfrak{a},\mathfrak{b},\mathfrak{c}\right\}$ for a three phase line segment. For any edge $(ij)\in\mathcal{E}$ we have

Next, we denote phases as $\phi$ and $\psi$, where $\phi,\psi\in\left\{\mathfrak{a},\mathfrak{b},\mathfrak{c}\right\}$.

Let $\left\{(i:\phi)\right\}_{\phi\in\Phi^{i}}$ represent the tuple of phases at node $i\in\mathcal{V}$, e.g., if $i$ is a three phase node, then we have $\left\{(i:\phi)\right\}_{\phi\in\Phi^{i}}=\left\{(i:\mathfrak{a}),(i:\mathfrak{b}),(i:\mathfrak{c})\right\}$. Now let

with $\lvert\mathcal{K}\rvert=K$ giving the sum of the total number of phases across all nodes in $\mathcal{V}$. Each element $(i:\phi)\in\mathcal{K}$ represents a supply point which serves $N^{i:\phi}\geq 0$ customers, such that

We label the customers by initializing the customer index at $n=1$ and increment the index through all customers at a supply point, before continuing the index at each following supply point, and across all supply points in $\mathcal{K}$ in ascending order, respectively.

To identify the location of each customer with reference to a supply point $(i:\phi)\in\mathcal{K}$, let $\Theta$ be a binary matrix with rows indexed by elements of $\mathcal{K}$, and columns indexed by elements of $\mathcal{N}$, and define the entries of $\Theta$ as

Let $r^{ij,\phi\psi}$ and $x^{ij,\phi\psi}$ denote the resistance and reactance of line segment $(ij)\in\mathcal{E}$ respectively, where $\phi$ and $\psi$ are the phases along the line segment. Then $z^{ij,\phi\psi}:=r^{ij,\phi\psi}+\mathfrak{i}x^{ij,\phi\psi}$ (where $\mathfrak{i}^{2}=-1)$, so that each line segment has a complex impedance $\mathbf{z}^{ij}$, e.g., for a three phase line segment

Let $Z^{ij,\phi\psi}:=\sum_{(uv)\in\left(\mathcal{E}^{i}\cap\mathcal{E}^{j}\right)}z^{uv,\phi\psi}\in\mathbb{C}$ denote the sum of the impedances along the unique path which is common to the paths from node $0$ to node $i$, and node $0$ to node $j$, such that both paths incorporate phases $\phi$ and $\psi$.

Let $\omega=e^{\frac{-2\pi\operatorname{i}}{3}}$ and let $[[\phi]]$ denote phase $\phi$ as a numeral, where we set $[[\mathfrak{a}]]=0,[[\mathfrak{b}]]=1,[[\mathfrak{c}]]=2$. Let $R\in\mathbb{R}_{\geq 0}^{K\times K}$ be a square matrix with rows and columns indexed by elements of $\mathcal{K}$, with the entries defined as

where $i,j\in\mathcal{V},\phi\in\Phi^{i}$, and $\psi\in\Phi^{j}$. We similarly define the matrix $X$ as

where $i,j\in\mathcal{V},\phi\in\Phi^{i}$, and $\psi\in\Phi^{j}$. $\Re$ and $\Im$ denote the real and imaginary parts respectively, and ^∗ is the complex conjugate.

Let $v^{i:\phi}(t)$ denote the squared voltage magnitude of phase $\phi\in\Phi^{i}$ at node $i\in\mathcal{V}$, at time $t$. For a node $i\in\mathcal{V}$ and time $t\in\mathcal{T}$, let

for example, at a three phase node we have

Collecting across all nodes in $\mathcal{V}$ we define

and let $v^{0}$ be the squared voltage magnitude of each phase at the root node, which we set to the nominal voltage. We also define $\mathbf{V}^{0}=v^{0}\mathds{1}\in\mathbb{R}^{K}$.

Let ${p}^{i:\phi}(t)\in\mathbb{R}$ denote the net real power (in kW) to or from node $i\in\mathcal{V}$ on phase $\phi\in\Phi^{i}$ over the time period $[\Delta(t-1),\Delta t]$. In particular, ${p}^{i:\phi}(t)$ is the net real power to or from $N^{i:\phi}$ customers connected to phase $\phi\in\Phi^{i}$ at node $i\in\mathcal{V}$. Given a node $i\in\mathcal{V}$ we let

and collecting over all nodes in $\mathcal{V}$ we define

Similarly, let $q^{i:\phi}{\color[rgb]{0,0,0}(t)}\in\mathbb{R}$ denote the net reactive power to or from node $i\in\mathcal{V}$ on phase $\phi\in\Phi^{i}$ over the time period $[\Delta(t-1),\Delta t]$. And analogously define

and

Since each supply point $(i:\phi)\in\mathcal{K}$ serves $N^{i:\phi}$ customers, $p^{i:\phi}(t)$ and $q^{i:\phi}(t)$ have contributions from the EV and non-EV load. Accordingly, let us denote by $\tilde{p}^{i:\phi}(t)$ and $\tilde{q}^{i:\phi}(t)$, respectively, the net real and reactive power to or from the non-EV loads at node $i\in\mathcal{V}$ on phase $\phi\in\Phi^{i}$ over the time period $[\Delta(t-1),\Delta t]$. Likewise, we denote by $\hat{p}^{i:\phi}(t)$ and $\hat{q}^{i:\phi}(t)$, respectively, the real and reactive power to or from the EV loads at supply point $(i:\phi)\in\mathcal{K}$ over the time period $[\Delta(t-1),\Delta t]$. Ignoring losses, we define

Similarly, we decompose terms $\mathfrak{p}^{i}(t)$, $\mathfrak{q}^{i}(t)$, $\mathbf{P}(t)$ and $\mathbf{Q}(t)$, such that for all $t\in\mathcal{T}$, we define

Given an optimization problem

ADMM proceeds by first defining the augmented Lagrangian

with dual variable $\mathbf{y}$, and regularization parameter $\alpha$ ($\alpha$ can also be thought of as a stepsize, which balances convergence with constraints satisfaction). The decision variables $\mathbf{x,y,z}$ are then updated sequentially on each iteration $k$ as,

The ADMM algorithm has many desirable properties, such as strong duality, linear convergence rates, etc. Importantly, the ADMM algorithm can be implemented in a distributed way, and can be applied to a variety of problems, as there are very few restrictions on the function $g$ in (13). For a more detailed explanation, the reader should consult [13].

We consider a communication network represented by a graph $\mathbb{G}:=\left\{\mathcal{N},\mathbb{E}\right\}$, where the vertex set $\mathcal{N}$ is the set of EV customers (as defined previously), and the edges in $\mathbb{E}\subseteq\mathcal{N}\times\mathcal{N}$ represent communication links between EV customers (an example is presented in Fig. 2). Specifically, an edge $(uv)\in\mathbb{E}$ allows bidirectional communication between customer $u$ and customer $v$. Due to symmetry in the communication network, if edge $(uv)\in\mathbb{E}$, then we must also have $(vu)\in\mathbb{E}$. We denote the set of neighbors of customer $u$ as $\mathcal{N}_{u}:=\left\{v\in\mathcal{N}\;\middle|\;(uv)\in\mathbb{E}\right\}$. We make the following assumption about $\mathbb{G}$.

Let us introduce a slack variable $\mathbf{s}_{n}\in\mathbb{R}^{2KT}$ for each customer $n\in\mathcal{N}$, and define the following

Then, we write the inequality constraint (12c) as

Define the sets

Denote the indicator function by

and let