Multiple Initial Point Approach to Solving Power Flows for Monte Carlo Studies

Abstract

Power flow solvers typically start from an initial point of power injection. This paper constructs a system of multiple initial points (SMIP) to enable selection of an appropriate initial point, with the objective to achieve a balanced improvement in the solution speed and accuracy, for problems with a large number of power flows. The intent is to recover time cost of forming the SMIP through the improvements to each power flow. The SMIP is tested on a time series based Monte Carlo study of Electric Vehicle (EV) hosting capacity in a low voltage distribution network, which has 5.4 million power flows. SMIP is applied to two power flow solvers: a Taylor series approximation and a Z-bus method. The accuracy of the quadratic Taylor series approximation was improved by a factor of 30 with a 27% increase in the solve time when compared against a single no-load initial point. A Z-bus solver with SMIP, limited to two iterations, gave the best performance for the EV hosting capacity case study.

1. Introduction

Probabilistic methods, including the time series based Monte Carlo technique, assess uncertainties such as the impact of Loads, Distributed Energy Resources (DER), Energy Storage (ES), or Electric Vehicles (EV) on electricity networks. Sampling of time at sufficient resolution, length of the time period, and the sampling of the probability space requires a large number of power flows to be solved. This paper analyses two types of power flow methods for the simulation of networks to optimise both the computational time and the solution accuracy for a study requiring many power flows. A new system of multiple initial points is introduced to control the trade-off between computational time and accuracy, and reduce the simulation time of Monte-Carlo studies. This system can be applied to numerous power flow solving techniques. In this paper, it is demonstrated with Taylor series approximation and the fixed-point iterative method/Z-bus method. These two methods are chosen for their robustness in solving power flows in distribution networks.

The Taylor series approximations can be applied to solve power flow equations up to any order [1,2]. Recent attention to these methods has been in the complex variable forms, where [3] expresses the Jacobian in a complex form through Wirtinger calculus, and [4] applies the formulation to Taylor series approximations. For this paper both linear and quadratic approximations are constructed for three phase, four wire distribution networks. Other than the Taylor series method, there are two other linearization techniques that are not considered as they cannot include quadratic terms: (1) DC equivalent power flows linearises each branch equation before the formulation of an equivalent admittance matrix. These approaches differ in the choice of independent variables and the method of linearisation, which are reviewed in [5] and the error is analysed. (2) Perturbation methods recognise that small changes in voltage create small linear perturbations in power injection. A quadratic term, $\Delta P\approx \Delta V^2$, is present, but since the perturbation is squared this term is small, and is removed from the equations for it to be solved [6]. The characteristic equation shows close similarity to the Jacobian in Taylor series methods.

The Z-bus approach has seen significant attention recently. It was determined that each iteration step of the solver can be formulated as a contraction mapping. The Banach Fixed-Point Theorem is applied to the contraction mapping to ensure that a unique solution exists in a space of power injections bound by an inequality constraint. Ref. [7] first applied the Banach Fixed-Point Theorem to the power flow equations. Later research has extended these ideas to distribution networks in the positive sequence component [8] and with three phase representation, wye and delta connections, and constant impedance, current, and power (ZIP) models for loads [9,10].

The novel contribution of this paper is the development of a System of Multiple Initial Points (SMIP). The idea is to implement an initial point reflective of the overall loading of the network for each power flow, instead of the same initial point. The desired outcome is that the number of iterations required to solve each power flow can be reduced by one, which is a significant improvement in computational speed for a Monte Carlo study, if only three to four iterations were previously required. A methodology of creating the system of initial points and an efficient mechanism for selecting the best initial point for each power flow is developed. The SMIP adds calculations to the initial setup of the power flow solver, but reduces the number of steps for each power flow. To compensate for the initial computational time of formulating the SMIP, the number of power flows has to be at least an order of magnitude greater than the number of initial points. Accuracy can be adjusted by changing the spacing between initial points, and more coarsely by the number of terms evaluated in the Taylor series expansion or the number of iterations in the Z-bus method. The application of SMIP could be extended to other solvers such as the Backward/Forward Sweep, where the first Backward pass in [11] is evaluated based on a selected initial voltage; Newton Raphson [12], where the Jacobian matrix is calculated for each initial point, but remains constant for each iteration; or Holomorphic embedding methods, where the embedding solves the power flow equations of the initial point at $s=0$, which could be achieved upon modification to a standardized form analysed in [13].

The SMIP is a method of improving initialisation to reduce power flow simulation time, and can be employed with other improvement methods, which reduce the number of iterations and time required to perform each iteration. For this paper, the largest computation demand in each iteration, for both the Taylor series approximations and Z-bus solvers, is a matrix multiplication requiring an identical number of floating point operations. The inverse Jacobian and Z-matrix are explicitly calculated and fully stored in the computer, which allows for an easier and more predictable comparison of initialisation and solver techniques. In practice, the Jacobian and admittance matrices would be factorised into a LU decomposition, and internally stored in a sparse form. This method has been generalised by [14] for distribution networks through modified augmented nodal analysis (MANA). Mature techniques of minimising the number of non-zero components in the LU factorization by row and column reordering should also be employed. These speed-up methods are actively researched in the area of high performance computing, such as parallel computing with GPUs [15,16]. Further parallel operation through Multi-Area Thévenin Equivalents (MATE) with reordering to blocked-bordered-diagonal form (BBDF) [17] is unnecessary for Monte-Carlo simulations, where individual scenarios can be divided amongst computer cores.

The SMIP is applied to an Electric Vehicle (EV) hosting capacity study as an example to demonstrate its performance. There are three general approaches for network analysis under increasing penetrations of EV uptake: deterministic, time-series Monte Carlo simulation, and probabilistic.

(1) The deterministic method considers one operating point of the network which typically would be when it is most stressed and inserts additional EV charging load according to the penetration level. This idea can be extended to include multiple scenarios and time snapshots of a network, or multiple characteristic time series.

(2) Time series simulation captures a period of interest, either including a period when the network is most stressed, or a full range of operation, simulating the network at regular time steps. EV charging profiles are created by sampling EV battery size, charger size, battery state of charge, consumer behaviour, driving behaviour, etc. Usually, most of these parameters are uncertain, so a Monte Carlo simulation is formed to consider a wide range of scenarios. Each Monte Carlo trial samples the underlying distribution for each uncertain parameter. This approach has the benefit of estimating the duration and occurrence of high load peaks, and their likelihood, but comes with a high computational burden. The number of scenarios can be dynamically adjusted to achieve the required accuracy for statistical outputs, which reduces computational requirements and is demonstrated in [18] for EV hosting capacity study. Furthermore, parallel computing has been employed for probabilistic studies in [19] on distribution networks.

(3) Probabilistic methods represent household and EV loads by probability density functions (PDF). These PDFs of input and output power are converted via the network equations to PDFs of voltage and current. Because of the general inability of finding closed form solutions to the power flow equations, approximations are applied in the conversion, or point estimations (these are representative power flows). This approach reduces the computational burden in comparison to the time series Monte Carlo approach, and provides a statistical aspect not seen in the deterministic approach. In [20], the PDFs for power injection are represented by moments and cumulants, which are transformed according to the Jacobian matrix of the network to obtain cumulants for nodal voltage. Then the cumulative distribution functions for nodal voltages are constructed by a Gram–Charlier expansion. A linearisation process specific to radial networks is employed by [21] to demonstrate that voltage magnitude can be approximated by a Nakagami distribution, provided the covariance matrix of the random variables is known. A point estimation method is employed by [22] to estimate moments of the power flow equation outputs, where a t-copula and a Kronecker product is used to select point estimates. These methods can require many power flow simulations depending on how the point estimation process is implemented. These methods cannot easily represent the complex nature of EV charging behaviour with a single random variable, and are less flexible than Monte Carlo methods.

An EV hosting capacity study was conducted for a local New Zealand distribution network operator (DNO) as an example. In New Zealand 92% of vehicles are parked at home overnight [23] making residential EV charging the convenient option. In the transition to EV, the charging load can appear on the distribution network without notification to the DNO. Recent scenarios reported by the International Energy Agency (IEA) show that private-chargers make up 90% of all chargers globally, with the majority of these based at home [24]. When undertaking congestion studies, the key is to have region specific and realistic representations of EV charging profiles. Synthetic EV charging profiles can achieve the desired representation by forming distributions of the EV charging parameters to align with the regional characteristics.

The main contributions of the paper are as follows:

• The formulation of a SMIP to speed up the solution for large quantities of power flows in a Monte Carlo study.
• The development of Taylor series approximations for three phase four wire distribution networks.
• Application of SMIP to the analysis of the impact of EV charging on distribution network congestion for a New Zealand distributor according to local EV fleet composition and behaviour.
• Comparison of baseline results with data recorded by network monitoring equipment, highlighting the importance of network data.

The novel SMIP technique is established and its performance is demonstrated for a multi-variable Monte Carlo EV hosting study through the following steps: Section 2 presents the Taylor series approximation and Z-bus method of calculating power flows. SMIP is introduced in Section 3 as a means of improving convergence time with sufficient accuracy. The different methods are compared in Section 4. An EV case study is defined in Section 5. Section 6 presents voltage and current results for an example network. This network is validated with low voltage (LV) monitoring data. Section 7 discusses the SMIP in light of the challenges of modelling the network and the limitations of Monte Carlo simulation.

2. Power Flow Solvers

2.1. Taylor Series Approximations

A quadratic approximation is created for node voltages, $V\in {\mathbb{C}}^N$, from its Taylor Series expansion about an initial point, $S^{\left(0\right)}\in {\mathbb{C}}^N$, where the relationship between voltage and power is defined implicitly by the power flow equations:

$$S=\mathrm{diag}\left(V\right)\overline{I}$$

(1)

$$I=YV$$

(2)

The nodal current injection, $I\in {\mathbb{C}}^N$, is conjugated in (1), as shown by the overbar, and voltage is transformed from a column vector to a diagonal matrix as indicated by the operation, $\mathrm{diag}(·)$. The nodal admittance matrix, $Y\in {\mathbb{C}}^{N\times N}$, is formed from the primitive admittance matrices, in this instance, include three phases and neutral of each circuit in the distribution network. Approximations of circuit currents are calculated by their respective primitive admittance matrix. N is the number of nodes in the network. A node is distinguished from a bus, where, if all phases are present at a bus, it consists of co-located phase a, b, c nodes and a neutral node. It is assumed that every circuit has a neutral conductor, but not necessarily one or two of the phase conductors.

The quadratic approximation is first derived without adaption for a voltage source or neutral nodes. The approximation solves for a power injection $S\in {\mathbb{C}}^N$. A function $S\left(\lambda \right)$ is created to form a line between the initial point and the power injection:

$$S\left(\lambda \right)=(1-\lambda )S^{\left(0\right)}+\lambda S$$

(3)

where $\lambda \in [0,1]$. Consequently, node voltages become a function of $\lambda$, $V\left(\lambda \right)$, where the solution to the power flow is $V\left(1\right)$, and the Taylor Series expansion, $\tilde{V}\left(\lambda \right)$, is evaluated about $V\left(0\right)$:

$$\tilde{V}\left(\lambda \right)=V^{\left(0\right)}+\frac{dV}{d\lambda }{|}_{\lambda =0}\lambda +\frac{1}{2}\frac{d^{2}V}{d{\lambda }^2}{|}_{\lambda =0}{\lambda }^2$$

(4)

where $V^{\left(0\right)}=V\left(0\right)$ and later $I^{\left(0\right)}$ are solutions for voltage and current of the initial point power flow. The first order derivative in (4) is solved by first substituting (2) into (1) to create a direct relationship between power injection and node voltage:

$$S\left(\lambda \right)=\mathrm{diag}\left(V\left(\lambda \right)\right)\overline{Y}\overline{V}\left(\lambda \right)$$

(5)

Differentiating (5) with respect to $\lambda$ and evaluating at $\lambda =0$:

$$\Delta S=\mathrm{diag}\left({\overline{I}}^{\left(0\right)}\right)\frac{dV}{d\lambda }+\mathrm{diag}\left(V^{\left(0\right)}\right)\overline{Y}\frac{d\overline{V}}{d\lambda }$$

(6)

where $\Delta S=S-S^{\left(0\right)}$. A Jacobian, J, is formed from (6) by separating out the real and imaginary, components and equations:

$$\left[\begin{array}{c}\Delta P\\ \Delta Q\end{array}\right]=J\left[\begin{array}{c}d{V}_{Re}/d\lambda \\ d{V}_{Im}/d\lambda \end{array}\right]$$

(7)

where

$$J=\left[\begin{array}{cc}{C}_{Re}+{\mathsf{\Pi }}_{Re}& C_{Im}+{\mathsf{\Pi }}_{Im}\\ -C_{Im}+{\mathsf{\Pi }}_{Im}& C_{Re}-{\mathsf{\Pi }}_{Re}\end{array}\right]$$

(8)

and the real and imaginary components are identified by its appropriate subscripts, and $C=\mathrm{diag}\left(I^{\left(0\right)}\right)$, $\mathsf{\Pi }=\mathrm{diag}\left(V^{\left(0\right)}\right)\overline{Y}$. The first order derivative of (4) is calculated by solving the matrix Equation (7). The second derivative term of (4) is calculated by twice differentiating (5) with respect to $\lambda$ and evaluating at $\lambda =0$:

$$\mathbf{0}=J\left[\begin{array}{c}{d}^2{V}_{Re}/d{\lambda }^2\\ d^2{V}_{Im}/d{\lambda }^2\end{array}\right]+2\left[\begin{array}{c}{D}_{Re}\\ D_{Im}\end{array}\right]$$

(9)

where

$$D=\mathrm{diag}\left(\frac{dV}{d\lambda }\right)\frac{d\overline{I}}{d\lambda }$$

(10)

$$\frac{dI}{d\lambda }=Y\frac{dV}{d\lambda }$$

(11)

Once (9) has been solved by the matrix inversion of J, which has already been inverted in solving (7), (4) is evaluated at $\lambda =1$ to give the approximate solution. The linear approximation is (4) without the quadratic term.

To consider the voltage source and neutral nodes, specific rows or elements of $\Delta S$, C, $\mathsf{\Pi }$, and D are modified, while the form of (4) and (7)–(11) remains unchanged. A specific row is indicated by the first index in the subscript, e.g., ${\mathsf{\Pi }}_i$ is the i’th row vector of $\mathsf{\Pi }$. If a specific element is required, a second index determines the column.

An AC voltage source is modelled with an impedance, which is converted to a Norton equivalent current source. Set A is a collection of all nodes connected to the Norton equivalent current source(s). For $i\in A$, $\Delta S_i$ is an unknown quantity and (7) cannot be solved without modification. Fortunately, $d{I}_i/d\lambda =0$, assuming that there is always at least one circuit between the source nodes and loads. Therefore in differentiating (5) for the source nodes:

$$\Delta S_i={\overline{I}}_i^{\left(0\right)}\frac{d{V}_i}{d\lambda }$$

(12)

After subtracting (12) from (6), the following modifications are made for all $i\in A$:

$$\Delta S_i\leftarrow 0C_{i,i}\leftarrow 0D_i\leftarrow 0$$

All loads are assumed to be connected in Wye formation between phase and neutral. There is also an earthing resistance between neutral and earth for every bus with a load. The power injection between phase and neutral is known, as it is an input from the study. However, $\Delta S$ is the power injection with voltage referenced from earth and is unknown prior to solving the power flow. Therefore, the equations are modified to change the voltage reference from earth to neutral to allow for the known power injections to be the input. Let $\mathsf{\Phi }$ be the set of all phase nodes with loads attached. Let $\Delta S^{\prime }\in {\mathbb{C}}^N$ be the power injections with voltage referenced from the neutral. For all $i\notin \mathsf{\Phi }$, $\Delta S_i^{\prime }=0$ for completeness. The function, $n(·)$, determines the neutral node index for a phase node index on the same bus. The following modifications are made for all $i\in \mathsf{\Phi }$:

$$\Delta S_i\leftarrow \Delta S_i^{\prime }{C}_{i,n\left(i\right)}\leftarrow -C_{i,i}{\mathsf{\Pi }}_i\leftarrow (V_i^{\left(0\right)}-V_{n\left(i\right)}^{\left(0\right)}){\overline{Y}}_i$$

$$D_i\leftarrow \left(\frac{d{V}_i}{d\lambda }-\frac{d{V}_{n\left(i\right)}}{d\lambda }\right)\frac{d{\overline{I}}_i}{d\lambda }$$

Let U be the set of all neutral nodes with loads connected to them, i.e., for $i\in U$ there exists a $j\in \mathsf{\Phi }$ such that $i=n\left(j\right)$. At present, for $i\in U$, $\Delta S_i$ remains undetermined prior to solving the load flow. Therefore, to determine a value of 0 for $\Delta S_i$, replace it with a known condition that the total current injected into a bus is zero:

$$0=\left(Y_i+\sum _{k\in {\mathsf{\Phi }}_i}{Y}_k\right)V$$

(13)

where for each $i\in U$ create the largest set ${\mathsf{\Phi }}_i$ such that for all $j\in {\mathsf{\Phi }}_i$, $n\left(j\right)=i$. Therefore the following modifications are made after differentiating (13) with respect to $\lambda$ (twice in the case of $D_i$):

$$\Delta S_i\leftarrow 0C_{i,i}\leftarrow 0{\mathsf{\Pi }}_i\leftarrow Y_i+\sum _{k\in {\mathsf{\Phi }}_i}{Y}_k{D}_i\leftarrow 0$$

The referencing of power injection from phase to earth to phase to neutral is completed. One more minor modification remains. This is to ensure J is non-singular during no-load (${S^{\prime }}^{\left(0\right)}=\mathbf{0}$). Let $U^{★}$ be a set of all neutral nodes that do not have loads at its bus. During no-load, $C_i={\mathbf{0}}^T$ and ${\mathsf{\Pi }}_i={\mathbf{0}}^T$ for all $i\in U^{★}$; the former equation occurs for all conditions because ${\overline{I}}_i^{\left(0\right)}=0$, but the later equation is only valid when $V_i^{\left(0\right)}=0$. Complete rows of J are therefore zero and the matrix is not invertible. To overcome this, ${\mathsf{\Pi }}_i\leftarrow Y_i$, for all $i\in U^{★}$, i.e., the derivative of the neutral current injection is zero for all these nodes. Note, it has not been proven that J is non-singular for every load condition, but it is helpful to recognise that $\mathsf{\Pi }$ retains a similar diagonal structure to Y, and only under contrived conditions could J be singular.

2.2. Z-Bus Method

The Z-bus method is an iterative approach that starts with an initial voltage, $V^{\left(0\right)}$, and calculates the current injection by rearrangement of (1):

$$I^{\left(1\right)}=\mathrm{diag}{\left({\overline{V}}^{\left(0\right)}\right)}^{-1}\overline{S}$$

(14)

Next, a new estimate for voltage is created by the inversion of (2):

$$V^{\left(1\right)}=Y^{-1}{I}^{\left(1\right)}$$

(15)

This process is usually repeated until the solution converges to within the desired precision. Here the process is limited to a predetermined number of iterations. Starting with (15), this two step process is refined. The nodal admittance matrix is inverted once to give $Z=Y^{-1}$, the nodal impedance matrix. Current injection from the Norton equivalent current source is constant, and it is computationally efficient to divide Z into two parts $Z=\left[Z_S\right|Z_L]$ and evaluate the voltage component from the current source once, $V_S=Z_S{I}_S$, where $Z_S\in {\mathbb{C}}^{N\times \left|A\right|}$ and $\left|A\right|$ is the number of source nodes. The second part of Z is padded, so that $Z^{\prime }=\left[{\mathbf{0}}^{N\times \left|A\right|}\right|Z_L]$ and the same indexing can be retained. The presence of ${\mathbf{0}}^{N\times \left|A\right|}$ would indicate an unnecessary multiplication of zeros; this is removed in implementation. Therefore (15) has been modified, and with an iteration number, m, it becomes:

$$V^{\left(m\right)}=V_S+Z^{\prime }{I}^{\left(m\right)}$$

(16)

where $I^{\left(m\right)}\in {\mathbb{C}}^N$. Because the total current injection into a bus across all nodes equals zero and power injections are referenced from the neutral voltage, it is possible to remove current injections for neutral nodes from $I^{\left(m\right)}$ and minimise the number of calculations required. This requires a modification to $Z^{\prime }$ initially, where specific columns of $Z^{\prime }$ are combined. For all $i\in \mathsf{\Phi }$:

$$Z_{|i}^{\prime }\leftarrow Z_{|i}^{\prime }-Z_{\left|n\right(i)}^{\prime }{Z}_{\left|n\right(i)}^{\prime }\leftarrow \mathbf{0}$$

(17)

strictly in that order, where | indicates the column vectors of $Z^{\prime }$ are being modified. Therefore, the current injection is modified from (14) to account for the neutral voltage to become:

$$I_i^{(m+1)}=\left\{\begin{array}{cc}{\overline{S}}_i^{\prime }/({\overline{V}}_i^{\left(m\right)}-{\overline{V}}_{n\left(i\right)}^{\left(m\right)})\hfill & \mathrm{if}i\in \mathsf{\Phi }\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(18)

where $S^{\prime }$ is the power injection when referenced from the neutral voltage and is used to calculate $\Delta S^{\prime }$ in Section 2.1.

3. System of Multiple Initial Points

Both the Taylor Series approximations and the Z-bus method of solving power flows are dependent on an initial point. It is possible to provide a better initial point than the no-load scenario to start each solver, so that solver speed and solution accuracy can be improved. This section develops a System of Multiple Initial Points (SMIP) for this purpose. The challenge is the space of potential power flows has $2\left|\mathsf{\Phi }\right|$ dimensions, one each for real and reactive power of each load. If each dimension were just sampled twice, the number of initial points is $2^{2\left|\mathsf{\Phi }\right|}$. For example, a network with 12 loads will require 17 million initial points, which is greater than the 5.4 million power flows in the time series Monte Carlo study described later in the paper. Therefore, the curse of dimensionality has to be avoided, as most networks have more than 12 loads.

The solution is to reduce the number of dimensions down to three: the mean real power per load on each of the three phases of the network. There are four reasons to justify this choice. (1) Reactive power is ideally small to minimise losses, therefore more variation is expected in real power. (2) Node voltage, circuit current, and circuit powers flow are typically roughly linearly dependent on the total downstream load from the point of measurement, i.e., small changes in load do not cause major differences in the results. (3) When a network is most stressed there can be a significant unbalance in the total load between each phase. Therefore necessitating a minimum of three dimensions, one for each phase. (4) The next major division in a network is the difference in the load between feeders. However, the number of feeders varies between networks, therefore the number of dimensions cannot be controlled.

3.1. Formulation of SMIP

The SMIP is a finite set of evenly spaced points in ${\mathbb{R}}^3$, expressing the real power per load for each phase, $\mathbf{p}\left[k\right]={[p_a\left[k\right],p_b\left[k\right],p_c\left[k\right]]}^T$. The integer, k, is a position marker for initial points. For each initial point, $\mathbf{p}\left[k\right]$ is expanded according to translation rules to a vector, $S^{\prime }\left[k\right]\in {\mathbb{C}}^N$, of power injections. A power flow solver then determines $V^{\left(0\right)}$ for each $S^{\prime }\left[k\right]$, which is required for initialising Taylor Series and Z-bus solvers. The SMIP is separated into two sets: firstly, those along the main diagonal, $\mathbf{p}\left[k\right]$, and those radiating out from the diagonal in $\mathbf{p}[k,l,m]$. The first set is constructed according to the first basis vector:

$$\mathbf{p}\left[k\right]=k{P}_{\Delta }{\mathbf{e}}_0$$

(19)

where $P_{\Delta }$ is the interval size, and ${\mathbf{e}}_0={[1,1,1]}^T$, where each phase is weighted equally. $\mathbf{p}\left[k\right]$ is then translated to $S^{\prime }$:

$$S_i^{\prime }\left[k\right]=\left\{\begin{array}{cc}-p_a\left[k\right](1+j\varphi )\hfill & i\in \mathsf{\Phi }\mathrm{and}\mathrm{if}\mathrm{load}i\mathrm{is}\mathrm{on}\mathrm{phase}\mathrm{a}\hfill \\ -p_b\left[k\right](1+j\varphi )\hfill & i\in \mathsf{\Phi }\mathrm{and}\mathrm{if}\mathrm{load}i\mathrm{is}\mathrm{on}\mathrm{phase}\mathrm{b}\hfill \\ -p_c\left[k\right](1+j\varphi )\hfill & i\in \mathsf{\Phi }\mathrm{and}\mathrm{if}\mathrm{load}i\mathrm{is}\mathrm{on}\mathrm{phase}\mathrm{c}\hfill \\ 0\hfill & i\notin \mathsf{\Phi }\hfill \end{array}\right.$$

(20)

where j is the imaginary unit, and $\varphi$ is the expected ratio between real and reactive power on a distribution network. The power flow, $S_i^{\prime }\left[k\right]$, is solved to a low tolerance on the node voltage. The solver used for this step of the study was OpenDSS [25].

The first integer to solve is $k=0$. k is increased (positive power export into residential dwelling), solving for each value, until one node voltage magnitude drops below 80% or above 120% of the nominal value, and the solution to that power flow is not kept. Alternative voltage magnitude ranges that are sufficiently broader than the statutory voltage limits can be chosen. The process is repeated, now decreasing k from 0 to account for distributed generation.

The second set of initial points is created from 12 basis vectors and incorporates phase imbalance. These are listed in

Table 1. List of basis vectors for the creation of phase imbalance in the initial points.

${\mathbf{e}}_1={[1,-1/2,-1/2]}^T$	${\mathbf{e}}_7={[-1,1/2,1/2]}^T$
${\mathbf{e}}_2=\sqrt{3}{[1/2,0,-1/2]}^T$	${\mathbf{e}}_8=\sqrt{3}{[-1/2,0,1/2]}^T$
${\mathbf{e}}_3={[1/2,1/2,-1]}^T$	${\mathbf{e}}_9={[-1/2,-1/2,1]}^T$
${\mathbf{e}}_4=\sqrt{3}{[0,1/2,-1/2]}^T$	${\mathbf{e}}_{10}=\sqrt{3}{[0,-1/2,1/2]}^T$
${\mathbf{e}}_5={[-1/2,1,-1/2]}^T$	${\mathbf{e}}_{11}={[1/2,-1,1/2]}^T$
${\mathbf{e}}_6=\sqrt{3}{[-1/2,1/2,0]}^T$	${\mathbf{e}}_{12}=\sqrt{3}{[1/2,-1/2,0]}^T$

, and forms a twelve pointed star. Each of these basis vectors are orthogonal to ${\mathbf{e}}_0$, but not necessarily orthogonal to each other. The calculation of initial points is extended:

$$\mathbf{p}[k,l,m]=k{P}_{\Delta }{\mathbf{e}}_0+m{P}_{\Delta }{\mathbf{e}}_l$$

(21)

where $l=1,2,\dots 12$ and m is a positive non-zero integer. For each l, power flows are solved first for $m=1$, m is then increased until the voltage magnitude falls outside the 80 to 120% range.

The SMIP has so far assumed that all loads are single phase and the total number of loads per phase is identical. Multi-phased loads and different number of loads per phase are incorporated by improving the translation of $\mathbf{p}\left[k\right]$ to $S^{\prime }\left[k\right]$ and $\mathbf{p}[k,l,m]$ to $S^{\prime }[k,l,m]$. In addition, further improvement is made by weighting each load according its size in translation. For example, if a commercial or industrial customer is connected to the network it is given a higher proportion of the load in the initial point than the residential loads. The exact formulation of these translations is not explained.

3.2. Selecting Initial Point for a Power Flow

The closest initial point, $S^{\prime }\left[k\right]$ or $S^{\prime }[k,l,m]$, is selected for each power flow calculation. The orthogonality between ${\mathbf{e}}_0$ and ${\mathbf{e}}_l$ is exploited for this process. Calculate the average real power per phase, $\mathbf{p}$, for the power injection, $S^{\prime }$. Each component of $\mathbf{p}$ represents a phase and the mean real power is calculated according to the standard formula:

$$p_{\phi }=-\frac{1}{N_P}\sum _{i\in {\mathsf{\Phi }}_{\phi }}Re\left\{S_i^{\prime }\right\}$$

(22)

where ${\mathsf{\Phi }}_{\phi }$ is a set of all loads on phase $\phi \in \{a,b,c\}$. $N_P$ is the total number of loads on a phase. The overall mean real power is $p_{\mathrm{mean}}=(p_a+p_b+p_c)/3$. Therefore, the choice for k, rounded to the nearest integer, is:

$$k^{\ast }=\mathrm{round}\left\{\frac{p_{\mathrm{mean}}}{P_{\Delta }}\right\}$$

(23)

Situations may occur where $k^{\ast }$ exceeds either the smallest or largest value for an initial point, because for an initial point with $k^{\ast }$ node voltage falls outside the 80–120% voltage range. In these cases the minimum or maximum value of k is selected for $k^{\ast }$. The selection of l is determined by the direction, ${\mathbf{e}}_l$, that most closely matches $\mathbf{p}$:

$$l^{\ast }=\underset{l=1\dots 12}{arg\; max}\left(\mathbf{p}•{\mathbf{e}}_l\right)$$

(24)

The dot-product in (24) is only evaluated for the first six basis vectors in Table 1. The remaining six have opposite sign to the first six. The selection of m is by projecting $\mathbf{p}$ onto ${\mathbf{e}}_{l^{\ast }}$ and determining the size of the projection:

$$m^{\ast }=\mathrm{round}\left\{\frac{\mathbf{p}•{\mathbf{e}}_{l^{\ast }}}{P_{\Delta }\sqrt{3/2}}\right\}$$

(25)

If $m^{\ast }=0$ then the initial point is $S^{\prime }\left[k^{\ast }\right]$, otherwise the initial point is $S^{\prime }[k^{\ast },l^{\ast },m^{\ast }]$. There is the possibility that $m^{\ast }$ may be too large, as a valid initial point does not exist. In such instances reduce $m^{\ast }$ to the largest value for m given $k^{\ast }$ and $l^{\ast }$.

4. Comparison of Computational Complexity for Solver Techniques

A total of 10 solver techniques are compared to analyse the benefits of using a SMIP against other techniques. These techniques are analysed on one network with source nodes represented as constant current injections, and loads by constant power injections. These solver techniques are listed as follows:

• LINSINGLE and QUADSINGLE—A linear or quadratic approximation (4) is applied about a single initial point, $S^{\prime }\left[0\right]$.
• LINSMIP($P_{\Delta }$) and QUADSMIP($P_{\Delta }$)—A linear or quadratic approximation (4) is applied using the SMIP.
• ZBUS1SMIP($P_{\Delta }$), ZBUS2SMIP($P_{\Delta }$) and ZBUS3SMIP($P_{\Delta }$)—The Z-bus method is applied with either 1, 2 or 3 iterations of (16) and (18) using the SMIP.
• ZBUS1TIME, ZBUS2TIME and ZBUS3TIME—The Z-bus method is applied with either 1, 2 or 3 iterations of (16) and (18), but instead of using the SMIP, the solution from the previous time step is used as the initial point.

These solvers execute a definite number of iterations, and do not compute the residual error, hence better known as approximations. In this Section, it is demonstrated that the several of the identified solvers offer sufficient accuracy for the purposes of conducting a Monte Carlo study through an example study. Section 7 discusses the solver error in comparison to external sources of error from network modelling and Monte Carlo simulation.

To critically analyse the solvers, the computational time and space requirements for each method is first predicted. The largest computational time demand for each method centres around the matrix multiplication of J⁻¹ for the Taylor Series approximations and $Z^{\prime }$ for the Z-bus methods. Both matrix multiplications require $8N\times N_L$ floating-point operations, where $N_L$ is the number of single phase loads, $N_L=\left|\mathsf{\Phi }\right|$. There are many zero components in ${[\Delta P^T\Delta Q^T]}^T$ from (7), D (because of $dI/d\lambda$) from (9), and $I^{\left(m\right)}$ in (16), which results in the same columns in J⁻¹ and $Z^{\prime }$ that can be excluded from the calculation. The $N_L$ columns corresponding to phase nodes with loads are retained. In order to compare the different methods, a time score is given to each method in

Table 2. Relative score of computational time and space requirements for each power flow method.

Solver	Time Score	Space Requirements
LINSINGLE	1	Low
QUADSINGLE	2.1	Low
LINSMIP($P_{\Delta }$)	1.2	High
QUADSMIP($P_{\Delta }$)	2.3	High
ZBUS1SMIP($P_{\Delta }$)	1.1	Medium
ZBUS2SMIP($P_{\Delta }$)	2.1	Medium
ZBUS3SMIP($P_{\Delta }$)	3.1	Medium
ZBUS1TIME	1	Low
ZBUS2TIME	2	Low
ZBUS3TIME	3	Low

, where the unit of measurement is the time required for 1 million multiplications of either J⁻¹ or $Z^{\prime }$.

For a study with 1 million power flows and 1000 initial points, Table 2 predicts the time score and space requirements for the different solvers. The time to form the SMIP with its approximations is considered secondary to their application, as the number of initial points is orders of magnitude fewer than the number of power flows. Solving the initial point power flows to form the SMIP includes forming the matrix J and inverting it; this gives the score of 0.2 to LINSMIP($P_{\Delta }$) and QUADSMIP($P_{\Delta }$). For ZBUS1SMIP($P_{\Delta }$), ZBUS2SMIP($P_{\Delta }$) and ZBUS3SMIP($P_{\Delta }$) a score of 0.1 is given for the formation of the SMIP, as only power flows at the initial point are performed to find the initial $V^{\left(0\right)}$.

A time score of 0.1 is added for QUADSINGLE and QUADSMIP($P_{\Delta }$) because of the matrix multiplication of (11). Although Y has the same size as $Z^{\prime }$ and implying that a time score of 1.0 should be added, the multiplication of Y is achieved with the faster sparse matrix operation.

Space requirements are large for time-series simulations. The additional space required for the initial points can have a significant contribution, especially if the inverse Jacobian is stored for numerous initial points (LINSMIP($P_{\Delta }$) and QUADSMIP($P_{\Delta }$)). Table 2 also compares the space to store the initial point data.

4.1. Computational Time and Accuracy Benchmarking

Each of the 10 power flow solvers are tested to assess computational time and accuracy by application to an EV hosting capacity case study. This study creates power flows on a network with 265 loads. The formulation of the case study is described in Section 5. A total of 5,376,000 power flows are solved for each test. These cover 8 EV penetration levels from 0 to 100%, 336 half-hour periods over the study week, and 2000 Monte Carlo scenarios per penetration level. For each penetration level, 2000 power flows are randomly selected (identical selection for all power flow solvers) to assess solver accuracy. These 16,000 power flows covering the 8 EV penetration levels, are solved in OpenDSS [25] to determine their base line solution from which the error is determined.

Each solver is implemented in MATLAB, and runs on a computer with an Intel i7-9700 CPU with 96 GB of RAM. The solve times for each solver is listed in

Table 3. Solve time for each power flow method averaged across three trials, $P_{\Delta }=$ 200 W.

Solver	Trial 1 (s)	Trial 2 (s)	Trial 3 (s)	Average (s)
LINSINGLE	1290	1281	1291	1290
QUADSINGLE	2587	2598	2548	2580
LINSMIP(200)	3182	3178	3199	3190
QUADSMIP(200)	5994	6067	6093	6050
ZBUS1SMIP(200)	1863	1878	1871	1870
ZBUS2SMIP(200)	2324	2299	2324	2320
ZBUS3SMIP(200)	2771	2777	2760	2770
ZBUS1TIME	1703	1639	1652	1660
ZBUS2TIME	2102	2091	2076	2090
ZBUS3TIME	2536	2529	2513	2530

for three simulations runs. The overall solver time is averaged to the closest 10 s. Comparing Table 3 with the expected relative performance in Table 2, the solve time for the Taylor series approximations with (Linear and Quadratic) is longer than expected. There are two reasons for the longer execution time in practice: (1) the time required to form the approximation for each initial point is not trivial, (2) upon closer inspection of the algorithm using MATLAB Profiler, bringing a different J⁻¹ for each power flow (depending on selected initial point) into the CPU is slowing the calculation.

Next, the impact initial point separation has on the solve time and accuracy is assessed. The space of feasible power flows defined by the 80 to 120% voltage limits is fixed. Therefore, increasing the separation distance means fewer initial points can fit in the feasible space. Increasing the separation distance from $P_{\Delta }=$ 200 W to 500 W, the number of initial points decreases from 3577 to 541, which has an appreciable reduction in solve time as shown in

Table 4. Comparison of solve times for SMIP methods with different initial point separation distances, $P_{\Delta }$.

Solver	Average (s)	Solver	Average (s)
LINSMIP(200)	3190	LINSMIP(500)	1780
QUADSMIP(200)	6050	QUADSMIP(500)	4050
ZBUS1SMIP(200)	1870	ZBUS1SMIP(500)	1790
ZBUS2SMIP(200)	2320	ZBUS2SMIP(500)	2220
ZBUS3SMIP(200)	2770	ZBUS3SMIP(500)	2640

. Figure 1 plots the error standard deviation as $P_{\Delta }$ is varied for both QUADSMIP and ZBUS2SMIP, as expected there is a trade-off between accuracy and the number of initial points. Noticeably, a data point is missing for QUADSMIP(100) in Figure 1 because the available computer memory was exceeded.

The accuracy of each solver is tabulated for node voltages and branch currents in

Table 5. Overall solver accuracy as measured by the mean bias ($\mu$) and standard deviation ($\sigma$) of the error.

Solver	Node Voltage (V)		Branch Current (A)
	$\mathbf{\mu }$	$\mathbf{\sigma }$	$\mathbf{\mu }$	$\mathbf{\sigma }$
LINSINGLE	1.2 × 100	8.3 × 10−1	−4.6 × 100	4.4 × 100
QUADSINGLE	1.9 × 10−1	2.0 × 10−1	−7.2 × 10−1	1.1 × 100
LINSMIP(200)	4.5 × 10−2	5.7 × 10−2	−2.1 × 10−1	3.6 × 10−1
QUADSMIP(200)	−4.8 × 10−4	6.1 × 10−3	2.1 × 10−3	3.7 × 10−2
ZBUS1SMIP(200)	−1.2 × 10−3	1.8 × 10−1	−3.9 × 10−3	1.1 × 100
ZBUS2SMIP(200)	1.5 × 10−3	2.2 × 10−2	−6.7 × 10−3	1.5 × 10−1
ZBUS3SMIP(200)	−2.4 × 10−5	3.7 × 10−3	2.5 × 10−4	2.1 × 10−2
ZBUS1TIME	1.5 × 10−2	2.1 × 10−1	−6.7 × 10−2	1.2 × 100
ZBUS2TIME	1.7 × 10−3	2.4 × 10−2	−8.4 × 10−3	1.6 × 10−1
ZBUS3TIME	−7.5 × 10−5	3.9 × 10−3	6.5 × 10−4	2.1 × 10−2

for addressing network wide performance.

Table 6. Solver accuracy as measured by the mean bias ($\mu$) and standard deviation ($\sigma$) of the error for the node voltage and branch current with the largest absolute bias.

Solver	Node Voltage (V)		Branch Current (A)
	$\mathbf{\mu }$	$\mathbf{\sigma }$	$\mathbf{\mu }$	$\mathbf{\sigma }$
LINSINGLE	2.8 × 100	1.7 × 100	−2.5 × 101	1.5 × 101
QUADSINGLE	5.2 × 10−1	4.9 × 10−1	−4.5 × 100	4.1 × 100
LINSMIP(200)	1.1 × 10−1	1.5 × 10−1	−9.4 × 10−1	8.9 × 10−1
QUADSMIP(200)	−2.1 × 10−3	8.7 × 10−3	1.6 × 10−2	6.4 × 10−2
ZBUS1SMIP(200)	−7.3 × 10−2	2.3 × 10−1	6.3 × 10−1	1.8 × 100
ZBUS2SMIP(200)	8.3 × 10−3	5.5 × 10−2	−6.7 × 10−2	5.1 × 10−1
ZBUS3SMIP(200)	−6.9 × 10−4	3.5 × 10−3	6.5 × 10−3	3.0 × 10−2
ZBUS1TIME	3.4 × 10−2	4.4 × 10−1	−3.2 × 10−1	3.9 × 100
ZBUS2TIME	4.2 × 10−3	5.0 × 10−2	−4.0 × 10−2	5.8 × 10−1
ZBUS3TIME	−8.8 × 10−4	9.9 × 10−3	1.0 × 10−2	8.5 × 10−2

focuses on just one node and branch with the worst absolute mean bias. Branch currents are calculated from node voltages and primitive matrices. The SMIP provides a significant improvement to the application of Taylor Series approximations, improving the accuracy of quadratic approximations by 30 times. The SMIP also removes bias from the error when node voltage drops under higher loading as shown in Figure 2. This minimises distortion when assessing simulation results.

The Taylor Series approximations struggle to match the accuracy of the Z-bus approaches with comparable execution time. Both ZBUS3SMIP(200) and ZBUS3TIME have better accuracy and faster solve times than QUADSMIP(200). Overall, the SMIP has better accuracy, but is slower than the time sequential approaches as shown in Table 5 and Table 6.

The time performance of the Taylor series approximations is improved by not calculating the inverse Jacobian, J⁻¹, for every initial point, as not every initial point is used. Figure 3 shows how often each initial point is selected from the 16,000 test power flows for LINSMIP(500) and QUADSMIP(500). Only 54 out of the 541 possible initial points are selected, none with $m\ge 2$. Therefore, the Jacobian and its inverse are calculated when the corresponding initial point is first selected (Section 3.2).

The reason why initial points with significant phase imbalance are never selected is because a consistent difference in demand between loads on opposite phases is required, which is statistically unlikely. Residential loads share a similar daily demand profile and the total demand on each phase follows a similar pattern. A significant phase imbalance is possible if there are more residential loads on one phase than on another, or if a large single load is placed on one phase. However, it is the asset owner’s incentive to avoid these situations to maximise network capacity.

Further improvements to LINSMIP($P_{\Delta }$) and QUADSMIP($P_{\Delta }$) can be made by organising how data is moved within the computer. The problem is that each time a new J⁻¹ is required for different initial points it requires additional time to bring the data into the CPU. If J⁻¹ remains in CPU’s cache for more than one power flow, then faster solve times can be expected, as the cache is faster than the main memory. Therefore, if all power flows are grouped by their initial point, then each group can be executed all at once. The improved solve times for LINSMIP and QUADSMIP are shown in

Table 7. Improved solve times for LINSMIP($P_{\Delta }$) and QUADSMIP($P_{\Delta }$).

Solver	Average (s)	Solver	Average (s)
LINSMIP(200)	1840	LINSMIP(500)	1640
QUADSMIP(200)	3290	QUADSMIP(500)	2920

. For QUADSMIP(200), the solve time has almost halved from 6050 to 3290 s and is respectable in comparison to the Z-bus methods. Potentially Taylor series approximations can be a better option than the Z-bus method, e.g., when a network with distributed energy resources is modelled with constant Power-Voltage (PV) bus, as the Z-bus method cannot make full use of the constant voltage magnitude in step (16), which could have a sub-optimal impact on the accuracy after each iteration [26].

In summation, the SMIP provides the desired improvement to the Taylor series approximations. Table 5 and Table 6 show that LINSMIP(200) has greater accuracy than QUADSINGLE, while being computationally faster. LINSMIP(200) has a solve time of 1840 s (Table 7), while QUADSINGLE is significantly longer and requiring 2580 s (Table 3). This shows that the SMIP can reduce the number of iterations required for each power flow, and retain a similar or better accuracy.

5. EV Hosting Capacity Case Study

An EV hosting capacity study was conducted for 236 potentially vulnerable networks owned by a local New Zealand Distribution Network Operator (DNO). Based on the test network analysis in Section 4, ZBUS2SMIP(200) was selected as the solver for the EV hosting capacity study. It demonstrates a good balance between solver time and accuracy for networks dominated by PQ buses and a constant source current injection.

The study covers a worst case winter week (period of highest loading) at a half hour resolution. The 2000 separate scenarios ($N_{MC}$) specified for each network and each penetration level allow for placements of EVs at different connection points in the network, variations in EV battery capacity, charger ratings, car journeys undertaken, charging behaviour, and capturing the diversity of base loads. This section describes how the network and EV charging behaviour are represented, and how the EV scenarios are defined.

The literature presents a number of approaches to modelling EV charging in order to characterise their impact. Daina reviews EV modelling methods [27], classifying approaches in terms of time resolution, from annual to sub-hourly analysis, and the level of spatial aggregation. The modelling time-scale plays an important role in determining what insights studies can offer. When considering network congestion impacts [18,28,29,30] or capacity requirements of public EV charging infrastructure [31], sub-hourly time-scales are beneficial, and essential if strategies to shift EV charging demand are to be evaluated. One of the challenges of EV impact studies is the lack of suitable and publicly available EV charging data. A small number of studies have utilised real-world EV charging data, such as the My Electric Avenue study [32], which accessed the onboard charging data of over 200 Nissan Leaf’s to extract charging times, initial State of Charge (SOC), and final SOC data. Other studies have utilised a single EV charging profile to model EV impacts [30]. Even with available real charging data, extrapolating to future scenarios is difficult, as EVs with increased battery capacities become more prevalent with higher powered EV chargers in the home. An alternative approach is to synthesise EV loads. This is a complex multi-dimensional problem that needs to consider charging start time, duration of charging, SOC on arrival, and user behaviour aspects. National travel surveys, predominately of internal combustion engine vehicles, have been used [29,33,34] either directly, or to extract trip data from which to establish statistical distributions. Markov models are also becoming more commonly employed to generate random trip profiles [34,35,36] but still require justifiable inputs to ensure these models reflect reality. This study synthesises EV charging profiles by directly sampling from the NZ national travel survey to determine the timing of energy demand.

Equally important to predicting EV electricity load is predicting EV diffusion in a population, which due to socio-economic factors amongst others can be patchy. Models require additional demographic data, such as income distributions, pricing information for fuels (internal combustion engines (ICE) versus EVs), and incentives [37,38]. This paper considers fixed EV penetration levels to gauge network impacts, and recognises that different localities will achieve these penetrations at different times.

5.1. Distribution Network Modelling

The network is modelled with three phase four wire circuits. If a circuit does not include a phase, then it is omitted from its primitive matrix. Per unit length resistance is calculated according to the conductor’s cross-sectional area and a mid-range temperature of 40 °C. The positive sequence series reactance and shunt capacitance are provided by a database of overhead line and cable parameters. Each parameter is added to the diagonal components of the primitive matrix. Therefore, off diagonal components of the primitive matrices are zero, meaning the mutual coupling between phases is modelled in the positive sequence component alone. This was justified from analysing a test case, which included all mutual coupling components. These components were found to minimally affect the results.

The connection of the LV network to the medium voltage (MV) network is represented by a Thévenin equivalent voltage source. This models a 11 kV/415 V transformer with its short circuit impedance, and the estimated strength of the 11 kV network at the point of connection of 100 MVA three-phase short circuit power.

A network consists of numerous branches. A branch is a series of overhead line or cable sections with multiple residential or commercial loads connected. Each load connection represents a node on the branch. Voltage results are referenced from the neutral node, and only one voltage result is presented for each branch and power flow. Since there are multiple phases and multiple nodes within a branch, the minimum voltage is retained for the results. Similarly, there are many possible currents within a branch; only the maximum current is retained. The length of each branch is known, however, the distance between loads on a branch is unknown. An assumption is made that the loads are equally spaced along the branch. For each bus with at least one load between phase and neutral, there is a 1.0 Ω earthing resistance between neutral and earth.

For the 236 networks analysed, there was uncertainty regarding the the number of phases for 6% of branches, and 19% of branches have unknown composition. A large proportion of these branches are short tee-offs to customer premises. Furthermore, the particular phase(s) that a load is connected is unknown across all networks. The number of phases for each load is known. The assignment of phases to loads is explained in Section 5.2.11 to align the loads across phases according to the transformer’s Maximum Demand Indicator (MDI).

5.2. Ev Scenario Definition

A time-frame of a worst-case winter week was used for the study; this encompasses the highest network load due to increased residential space-heating, and allows for the evaluation of diurnal load patterns. A time resolution of half an hour (HH) was used, consistent with the available smart-meter load data. This study built up a pattern of EV charging profiles with NZ centric data, such as EV models registered in NZ at present and NZ travel survey data. The State of Charge ($SOC$) of each EV is evaluated at each time step assuming a simplified uniform charging power and a fixed distance-kWh relationship for all trips and vehicles.

5.2.1. EV Placement

To account for the impact of EV location on the network, EVs are randomly allocated to residential connections with equal probability. Allowance is made for a maximum of a single EV at each residential connection.

5.2.2. EV Penetration Level

The number of EVs in a network is expressed by an EV penetration level Y:

$$\mathsf{Y}=\frac{N_{EV}}{N_R}$$

(26)

where $N_{EV}$ is the number of households with an EV associated with it and $N_R$ ($\le N_L$) is the total number of residential households. Note that as the study’s focus is on residential charging, commercial and industrial connections are not assigned EVs. Where possible, eight penetration levels are modelled: 0, 10, 20, 30, 40, 50, 75, and 100%; or the penetrations levels closest to these values. The increased penetration level step-size after 50% reflects a greater interest in lower EV penetrations, which are more relevant for assessing hosting capacity in the short to medium term.

5.2.3. EV Charging Profiles

The key to the evaluation of EV charging impacts is the creation of EV charging profiles. Multiple random variables are used to construct each profile, where each variable is sampled according to its distribution; these include:

• EV battery capacity ($E_{\mathrm{batt}}$);
• EV charger rating ($S_{\mathrm{charger}}$);
• An index to a set of journeys ($\mathsf{\Gamma }$);
• Owner EV charging behaviour ($CB$);
• Owner Anxiety Factor ($AF$);
• Initial battery State of Charge ($SO{C}_{\mathrm{init}}$).

These random variables are described in the sections below.

5.2.4. EV Battery Capacity

A probability distribution function discretized to eight battery capacities was created on the basis of EVs registered in New Zealand at present [39]. Battery degradation was not generally incorporated into the model; an exception to this was the treatment of the Nissan Leaf. The NZ EV fleet has a high proportion of older second-hand Leafs imported from Japan and as they do not have active battery cooling, battery degradation is pronounced. To account for capacity reduction, Leaf batteries are degraded at a rate of 3% per year. The final distribution of battery capacities is illustrated in Figure 4. Plug-in hybrid EV models are not included in the EV analysis, due to their smaller battery capacities and less well-defined charging behaviour.

5.2.5. Charger Rating

The types of EV chargers employed for home charging are not well recorded. Local surveys undertaken by the DNO found that the majority of charging was Level 1 trickle charging. Probability distributions were established from surveys as a function of the battery capacity ($E_{batt}$), the maximum onboard charging capability of different models, and the number of available phase connections to a household. These probability distributions are listed in

Table 8. Probability distributions of charger rating as a function of battery capacity for single phase residences.

Battery Capacity	EV Charger Rating Probabilities
(kWh)	1.9 kW	3.6 kW	7 kW
16	0.75	0.25	0
20	0.75	0.25	0
24	0.75	0.25	0
30	0.55	0.3	0.15
40	0.35	0.45	0.2
60	0.35	0.45	0.2
75	0	0	1
100	0	0	1

for a single phase residence. Probabilities were also determined for residences with three phase connections.

5.2.6. Journey Data

An EV driver’s charging profile is closely tied to their journey data, including how far they travelled, where and when they travelled. Traffic conditions, ambient temperature, road gradients and style of driving will also have an impact, but these are not considered and are simplified to a linear distance to battery discharge relationship. This study used a NZ domestic travel survey for light transport vehicles between 2015 and 2018 [40]. The survey listed individual journeys by vehicle code (to identify an individual vehicle), mode of transport, the purpose of the trip, distance and time. It includes household trips over a full week, which was valuable for identifying weekend/weekday temporal patterns.

The UK’s Electric Avenue trial observed that 70% of charging days used a single charge event, while 30% of charging days had a second charging event or more [32]. To enable the analysis to cater for a maximum of two charging events in a day, the raw travel survey data has been pre-processed to condense multiple trips in a day, down to a maximum of two trips per day. Trips are combined in a manner to preserve the largest time duration that the EV is parked at home, assuming that this is the most likely time for charging. The processed travel survey data, referred to as ‘Journey Data’. is then directly sampled to build up the EV scenarios.

5.2.7. Charging Behaviour

Each EV is assigned with one of three charging behaviours: (1) Charge on Arrival (COA), (2) Charge by Morning (CBM), and (3) Charge in Time of Use (CTOU). For COA, the EV immediately starts charging upon arrival at home, if the SOC is below a threshold. CBM attempts to fully charge the vehicle before the next departure, again if the SOC is below a threshold. Lastly CTOU preferentially charges the EV in time periods with lower per unit electricity tariffs. For this study, the preferred Time of Use period is the local network operator’s cheap night rate between 9 pm to 7 am. This study assumes 50% of people charge upon arrival home, 25% of people charge by morning, and 25% preferentially charge in the Time of Use periods.

5.2.8. Anxiety Factor

Each EV user has different comfort levels around when to recharge their vehicle. This study incorporates an Anxiety Factor ($AF$), which reflects an EV users comfort level around not having a fully charged EV [41]. A decision about whether to charge is made based on the current $SOC$, how much energy is required for the next trip, and a users $AF$. The $AF$ has a value between 0 and 1 and is randomly selected from a normal distribution with a mean of 0.75 and a standard deviation of 0.15. Any values that arise above one or below zero are constrained to the limits. The $AF$ determines a $SOC$ threshold ($SO{C}_{\mathrm{th}}$) for charging as given in the equation below.

$$SO{C}_{\mathrm{th}}=0.2+AF\times 0.8$$

(27)

5.2.9. Initial Battery State of Charge

The initial state of charge of each EV, ($SO{C}_{\mathrm{init}}$), is assigned from a normal distribution with a mean of 75% and a standard deviation of 15% and rounded to a minimum/maximum SOC of 20% and 100% respectively.

5.2.10. Base Loads

Base load data was assigned randomly to residential connections from a set of half-hour smart-meter data. The dataset comprises of ∼2000 households in the network’s region for the highest loaded winter week in 2015. Note that in 2015, fewer than 1000 EVs were registered in the whole of NZ, thereby providing a suitable base case scenario with little EV-charging [39]. Separate commercial and industrial load profiles were included to provide representative base case scenarios. Where available, the historic smart-meter data was obtained for specific business connections. Alternatively, half-hourly loads were synthesized based on smart-meter data for businesses with similar Australian and New Zealand Standard Industrial Classifications (ANZSIC) [42] and scaled according to monthly load figures. A default load power factor of 0.98 is assumed for base loads and EV charging loads.

5.2.11. Load Matching and Phase Balancing Using Transformer Maximum Demand Indicators

As the network’s asset database does not explicitly state phase connection information for residences and businesses, phase connections must be allocated. An algorithm was developed to assign connections according to the following criteria, either (1) to balance load across phases as best as possible, or (2) if a transformer’s MDI is available, assign phases in order to best match reported MDIs. For the second case, an algorithm selects a subset of allowable base load profiles based on peak load and correlation results with other load profiles, to enable best matching of MDI values. A minimum subset size of 500 load profiles was set to ensure sufficient load diversity. This approach assumes any phase unbalance measured at the transformer is uniformly representative across all the circuits.

5.3. Interpreting Network Congestion Limits

The Monte Carlo simulation consists of 2000 scenarios for each EV penetration level of interest and for each network to provide a representative sample. The results are summarised into a number of key percentiles. The statistical values are presented in Figure 5. For branch currents and transformer powers, the percentiles to the right of the median are relevant to understand asset overloading, i.e., P85.7, P99.4, and P99.97. For voltage, the percentiles to the left of the median are relevant to characterise the voltage drop along circuits, i.e., P14.3, P0.6, and P0.03. These percentile results were chosen to be interpreted in terms of the number of half-hours (HH) spent above or below this threshold in a week.

Network congestion is determined by the transformer power rating, maximum branch conductor current rating and voltage excursions outside of the NZ regulated range of ±6% of 230 V nominal voltage [43].

For the transformer power, the nominal transformer rating is considered the limit. In practice, it is generally acceptable for loading to exceed this limit intermittently for short durations, due to thermal cycling considerations. Accordingly, a lower percentile threshold between P85.7 and P99.4 may be deemed acceptable in determining hosting capacity. Ampacity limits for each branch were provided by the DNO, with the overhead lines current limits adjusted to a local average ambient winter temperature of 10 °C.

New Zealand regulations state that voltages should remain within ±6% of 230 V at the point of connection, i.e., between 216.2 and 243.8 V [43]. The EEA Power Quality guideline defines that this should be assessed based on 10 min r.m.s. readings using the 99th percentile and 1st percentile values as upper and lower bounds over a one-week period [44]. This corresponds closely to the 2 HH threshold (2 half hours out of a week) and equivalent to the 0.6 percentile (Figure 5).

6. Case Study Results

This section validates the model for an example network, #129 (Figure 6), and presents the aggregate results of EV hosting capability for 236 local DNO networks.

6.1. Example Network and Model Validation

Network #129 was selected as it has a high proportion of “known” circuits, i.e., circuits with confirmed asset information. The DNO has some LV monitoring data for this network as it was considered at risk of congestion. This provided an opportunity to validate simulation results (Section 6.1.2).

6.1.1. Transformer Maximum Current Distribution

Figure 7 shows a box and whisker plot of the distribution of maximum transformer phase currents for the 2000 scenarios modelled per EV penetration level. The 50th percentile is at the centre of each box with the edges representing the 25th and 75th percentiles. The whiskers extend to the 1st and 99th percentiles and the outliers are represented as dots with the three lowest and highest outliers presented for clarity. The recorded winter MDI values for the example network are shown as black dashes in Figure 7. These report significant unbalance with phase A’s MDI 20% higher than phase B or C. The results at 0% penetration (base load condition) show that the MDI phase matching algorithm was successful, as the median of the maximum phase currents are close to the MDI values. At 100% EV penetration, the median maximum current for phase A has increased by 26%.

6.1.2. Main Circuits and End of Line Monitoring

Simulated results for network #129 are validated with current and voltage measurements located at the transformers LV bus for circuits 1, 2, and 4 as identified in Figure 6. A temporary monitor was positioned at the end of Branch 36 (End of Line, EoL), which regrettably did not capture the highest load week (week in July).

Simulated and measured circuit currents near the transformer are compared for key percentiles in

Table 9. Comparison of modelled versus actual (measured) current percentiles for the peak winter week (July), for circuits 1, 2 and 4.

	Circuit 1 I(A)			Circuit 2 I(A)			Circuit 4 I(A)
	Model	Act.	Var.	Model	Act.	Var.	Model	Act.	Var.
Median	140	154	−9%	237	211	12%	18	14	29%
48 HH	181	194	−7%	313	306	2%	29	24	21%
2 HH	239	243	−2%	402	436	−8%	48	41	17%
0.1 HH	274	270	1%	451	470	−4%	60	46	30%

. Differences in the median current values are evident, where the model simulates higher median values on circuits 2 (Figure 8) and 4 compared to measured, while the reverse is true for circuit 1. Interestingly, the variances agree more closely at the higher current percentiles. This shows a good translation from transformer MDI values to ‘peak’ circuit loadings. The voltage measurements at the LV bus for July show good agreement with the simulations with variances of 0.7% or less as shown in

Table 10. Comparison of modelled versus actual (measured) voltage percentiles for the LV transformer bus and at the EoL of circuit 2 (branch 36).

	Bus (V)					Circuit 2 EoL (V)
	Model	Act.	Var.	Act.	Var.	Model	Act.	Var.
		(Jul.)	(Jul.)	(Aug.)	(Aug.)		(Aug.)	(Aug.)
Median	234.4	235.5	−0.5%	236.0	−0.7%	217.7	225.7	−3.5%
48 HH	232.8	233.9	−0.5%	234.7	−0.8%	210.5	219.6	−4.1%
2 HH	231.0	232.4	−0.6%	233.3	−1.0%	201.2	210.6	−4.5%
0.1 HH	230.3	232.0	−0.7%	233.1	−1.2%	195.6	207.6	−5.8%

and Figure 9a.

Larger variances are found in the EoL voltages in Table 10. These are discussed in relation to the modelled no-load line-to-neutral voltage of 239.6 V, and the voltage drop from this value. The simulated and measured (August) 2HH percentile voltages are 201.2 and 210.6 V, which correspond to a load voltage drop of 38.4 and 29.0 V, respectively. The measured voltage drop is 24% less than that simulated. There are a number of factors that can contribute to this difference.

Firstly, EoL measurements are not available for July when the MDI was recorded. The measured July bus voltages are lower than August (Figure 9), which would imply the corresponding EoL would be lower. Extrapolating from the measured July bus voltage, the corresponding July EoL voltage is estimated to be 206.5 V. This calculation assumes that July voltages have the same proportions between no-load, LV bus, and EoL voltages as the August measurements. A 2 HH EoL of 206.5 V reduces the difference between simulated and measured voltage drop to 14%.

Secondly, the simulated LV bus voltage is out by 1.4 V from the measured 2 HH July value (Table 10). This error arises from the difference between the model of the MV network and the LV transformer and the network in reality. A 1.4 V difference would account for 4% of the voltage drop, bringing down the remaining difference to 10%.

Conservative modelling of the LV network could be the third factor. For example, the conductor resistance is based on a 40 °C core temperature. The MDI would be recorded on a winter evening when the outside air temperature is significantly below the winter average. It is conceivable that the core temperature could be lower than modelled. Network #129 has both copper and aluminium conductors, which have a temperature coefficient for resistivity of 0.004 K⁻¹. If the core temperature dropped by 10 °C, then each conductor’s resistance would reduce by 4%, accounting for a further 4% difference in the voltage drop.

The remaining 6% of the error could be a result of a number of factors that are hard to quantify. The phase connections of loads are unknown, the distribution of transformer MDIs are obtained from a single measurement, and the simulations assume that the MDI at the transformer is representative of the entire network.

6.2. Aggregated Results for Local DNO

Aggregated results are presented for 236 networks, which were preferentially selected as being at most risk of congestion. The voltage and current constraints are analysed to establish the hosting capacity. Hosting capacity is dependent on the choice of percentile and is determined by the penetration level that results in the percentile value exceeding the congestion limits (Section 5.3). Figure 10 presents the occurrence of violating voltage and current constraints for all 4189 branches in the 236 networks as a function of EV penetration. Focusing on the 2HH distribution results, the number of voltage constrained branches increases by 53% from 19.4% to 29.7% and the current constrained branches double from 3.8% to 7.9% as EV penetration changes from 0 to 100%. Figure 11 shows the distribution of hosting capacities based on voltage and current congestion. From Figure 11, it is clear that a significant number of networks have constraints before EV load is added. There are also a high number of networks that have sufficient hosting capacity to support EV charging between 90 and 100% or higher. The remaining networks have a strong sensitivity to the EV charging load.

7. Discussion

The accuracy of results from the example EV Hosting Capacity study is dependent on three factors: the accuracy of the electricity network model, the conditions of the Monte Carlo simulation, and the accuracy of the power flow solver. The error in the power flow solver should be smaller than the confidence interval for the Monte Carlo simulation results, which in turn should be smaller than the error from modelling the electricity network. This section discusses the choice of power flow solver in achieving the desired accuracy when considering the limitations in network modelling and Monte Carlo simulation. The network modelling challenges have been documented in Section 6.1.2, and are based on assumptions made in Section 5.

Two aspects of the Monte Carlo simulation are examined here: firstly, the way in which the results are presented, and secondly, how the number of scenarios affects the confidence in the results.

The raw results from the Monte Carlo simulation are the node voltages and branch currents for the 5.4 million power flows. These are too many to analyse individually. To meaningfully assess the impact of EV charging on the distribution network, the results can be condensed in at least two ways:

• Option 1—For each node voltage or branch current, determine percentiles from all power flows, as in Section 6.
• Option 2—For each node voltage or branch current, determine the percentiles for each scenario, then find the mean value for each percentile across all scenarios.

Option 1 is interpreted as if the network is indefinitely operating in a condition equivalent to the study week, then a percentile states the proportion of time a node voltage or branch current is below or above the given percentile value. Whereas for Option 2, the results are the expected values for the percentiles if an individual scenario was selected at random. Each option gives a slightly different result; for the 2 HH EoL voltage, Option 1 gives 201.2 V as shown in Table 10, and Option 2 provides a value of 202.7 V. The method of consolidating the results is important.

The second aspect is how the number of scenarios impacts the confidence in the results. An uncertainty in the results, however small, can never be excluded because it is not possible to solve every combination of random variable outcome.

This uncertainty is quantified by a confidence interval, which describes how close the unknown true value is to the result obtained. For results presented by both Options 1 and 2, the width of the confidence interval is proportional to $1/\sqrt{N_{MC}}$. Therefore, increasing the number of scenarios improves the confidence. For the EoL voltages in #129, a 95% confidence has an interval of ±0.07 V for Option 1, which is calculated according to asymptotic theory of order statistics [45]. Option 2 has a confidence interval of ±0.12 V, and is calculated according to Monte Carlo error analysis of percentiles [46].

The width of the confidence interval from the Monte Carlo process is better than the accuracy of the network model. For the EoL voltage in #129, the confidence interval is an order of magnitude smaller than the voltage difference as shown in Section 6.1.2. The network model could be improved with a load dependent model of the 11 kV bus voltage, or a thermal model of the branches to estimate their resistance based upon dynamic weather conditions.

For the error of the power flow solver to be smaller than the confidence interval, LINSINGLE, QUADSINGLE, LINSMIP(200), ZBUS1SMIP(200), and ZBUS1TIME are not viable options, based on their mean voltage bias as shown in Section 4.1. The accuracy of the power flow solver does not need to far exceed the confidence intervals either, as the overall accuracy is primarily dependent on network modelling. Therefore, the better accuracy of ZBUS3SMIP(200) and ZBUS3TIME would exceed the requirement. Since, ZBUS2SMIP(200) and ZBUS2TIME have better solve times than QUADSMIP(200), either of these two solvers would be suitable for such a study.

8. Future Potential

The SMIP can be extended to accommodate the complexities associated with networks containing a high level of converter interfaced technologies, such as DER, ES, and future large scale hybrid AC-DC networks. This will necessitate extending the application of SMIP to other power flow solvers and developing additional methods to speed up computation. These power flow solvers could include Newton’s Method to handle slack nodes and constant voltage magnitude nodes, which can represent resources with voltage control. To reduce solve times further, the SMIP could be implemented with parallel programming.

The SMIP is best suited for time series studies where power injections have noticeably changed between time step. The SMIP is not optimised for Quasi-Static Time Series (QSTS) simulation, where time series have time steps between one to five seconds, and control actions modelled between time steps. Solvers such as ZBUS2TIME that utilise results from previous time steps are better suited for QSTS.

9. Conclusions

Two power flows solvers, a Taylor series approximation and a Z-bus method, have been presented for distribution networks with three phase four wire representation. A SMIP was created to improve the accuracy of the power flow solvers with a minimal increase in solve time. The SMIP is a representative set of power flows, which are solved prior to a study. For each study power flow, the closest power flow in the SMIP is selected to initialise the solver. Therefore, starting the solver from a closer initial point requires fewer iterations to find the solution. The computation expense of forming the SMIP is recovered if the size of the study is large. The SMIP was demonstrated on a Monte Carlo based EV hosting capacity study with 5.4 million power flows, which improved the quadratic form of the Taylor series approximation by a factor of 30 for the example network (#129) with a 27% increase in computational time. This network was monitored in the field, which allowed for the modelling of 0% penetration level, i.e., no EVs, to be validated. After considering sources of error in the simulation results, it was determined that solvers ZBUS2SMIP(200) or ZBUS2TIME provided sufficient accuracy to assess a further 235 real-world networks from New Zealand. SMIP is a valuable tool for solvers to efficiently evaluate the impacts on future electricity networks.