# Residential electrical vehicle charging strategies: the good, the bad and the ugly

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## Abstract

In recent years, a wide variety of centralised and decentralised algorithms have been proposed for residential charging of electric vehicles (EVs). In this paper, we present a mathematical framework which casts the EV charging scenarios addressed by these algorithms as optimisation problems having either temporal or instantaneous optimisation objectives with respect to the different actors in the power system. Using this framework and a realistic distribution network simulation testbed, we provide a comparative evaluation of a range of different residential EV charging strategies, highlighting in each case positive and negative characteristics.

## Keywords

Electric vehicle Smart grid Decentralised control Centralised control Demand side management## 1 Introduction

Electric vehicles (EVs) avoid the use of petroleum, and so are seen as an efficient and effective replacement for traditional internal combustion engine based vehicles (ICEVs). They are expected to play a vital role worldwide in the near future in terms of addressing CO_{2} reduction targets, combating climate change and improving energy security [1, 2, 3]. However, as EV penetration increases, the extra demand due to EV charging will have a considerable impact on the design and operation of electrical power systems [4]. In [5, 6, 7, 8, 9, 10], it has been shown that the widespread adoption of EVs could negatively impact the distribution network if charging is not properly coordinated. Grid impacts of uncoordinated charging include, but are not limited to, increased voltage imbalance [7], increased grid losses [8], overloading [6], fluctuation of grid frequencies [11], and increased harmonic distortion [12]. In turn these effects result in a decrease in the operational efficiency of the grid and in the life span of electrical devices. It is also observed in these studies that in order to accommodate the extra EV charging loads, it will be necessary for utilities to invest in and reinforce grid infrastructures in heavily loaded areas, to accommodate both EV and household loads.

It has been shown in [4, 10] that by using suitable charging strategies, it is possible to mitigate some of the adverse impacts of charging, which could in turn reduce or postpone the need for infrastructure reinforcement. To date, there have been several different strategies proposed for charging groups of EVs connected to low-voltage distribution networks [9, 13, 14, 15]. These strategies can be classified from the perspective of the different actors in the power system, which consist of EV consumers, distribution system operators (DSOs) and transmission system operators (TSOs) [16].

Consumer oriented algorithms typically focus on maximising the amount of charge that can be allocated to a customer in a given time period. It is usually desired to achieve this in a fair manner, providing a satisfactory quality of service (QoS), without violating system constraints, and minimising the cost to the customer. Many algorithms have been proposed using a centralised framework, the aim of which is to maximise the amount of charge allocated to customers [13, 17]. Here centralised implies that all the information in the network is available to a centralised controller, which in turn processes the information and decides the charge each EV will receive. These algorithms are typically based on linear programming techniques. While centralised coordination gives the best performance possible [18], centralised algorithms require access to global system information, which might not always be accessible, and centralised algorithms typically do not scale well [18, 19, 20]. For these reasons, several decentralised strategies have been proposed recently for EV charging [21, 22, 23, 24, 25, 26]. In decentralised charging strategies individual EVs are given a certain level of decision making autonomy. Often individual EV chargers send a limited amount of information to a centralised unit which in turn provides some global coordination of their decisions, to a degree determined by the algorithm in use.

From the perspective of DSOs, charging strategies are usually designed to achieve a grid related objective such as the minimisation of power losses, while satisfying grid constraints, and providing satisfactory customer service. Several centralised coordination strategies of this nature have been developed in [15, 27, 28, 29, 30, 31]. Vehicle-to-grid (V2G) techniques for grid regulation have also been proposed. For example, in [28], V2G was used for grid regulation on a daily basis and for peak reduction at times of high demand. Some decentralised optimisation approaches have also been proposed to enhance grid regulation [32, 33].

Charging strategies aligned to TSO priorities include those focused on the scheduling of power supplies in an economic way, and those seeking to maximise the use of renewables on the grid. Centralised TSO based charging strategies include [34], where serial quadratic programming techniques were used to minimise the variance in the U.K. national demand profile, and [35] which examined the use of quadratic programming for load flattening under different penetrations of EVs. A more general study of adapting different centralised optimisation strategies for the coordination of EVs from the perspective of TSOs is presented in [36]. Several decentralised approaches developed from the TSO perspective are given in [37, 38]. In [39, 40, 41, 42], EVs are used for storage and control in order to maximise the utilisation of renewable energy. In [41], mixed-integer-linear-programming (MILP) was used to schedule EV charging loads in order to reduce charging costs and carbon emissions. In addition to this, EVs have been used to provide ancillary services, such as frequency control [43, 44].

With such a wide variety of algorithms available for EV charging it is desirable to have a common framework under which their performance can be compared. The objective of this paper is to present such a framework for residential charging of EVs, and to provide a comparative evaluation of a range of residential EV charging strategies within this framework using a realistic distribution network simulation testbed and representative charging scenarios. Through this comparison the positive and negative characteristics of each approach are identified, and the good, the bad and the ugly, so to speak, are highlighted.

The remainder of the paper is organised as follows. The proposed framework is introduced in Section 2. The EV charging algorithms considered are summarised in Section 3 and the testbed and simulation results are presented in Section 4. The results are discussed in Section 5 and finally conclusions are presented in Section 6.

## 2 EV charging problem formulation

*S*is defined as the number of distribution transformers. Let \(\underline{S}\) denote the set {0, 1,…,

*S*}. These transformers are then connected to the medium-voltage (MV) substation bus, SubBus. The substation bus is powered by a transformer called TR(0), which connects to an external bulk power system.

A number of simplifications are used in the system model. The load power consumption in the network is discretised into *M* discrete time slots, each of length Δ*T*. For indexing purposes, let \(\underline{M}\) denote the set {1, 2,…, *M*}. The loads are classified as non-EV loads, and EV loads, in the low-voltage (LV) areas. The number of houses across all LV areas is given by *N.* Let \(\underline{N}\) denote the set {1, 2,…, *N*}. The index set of all houses connected to the transformer *TR*(*i*) is given by \(\phi_{h}^{i}\), and similarly the index set of all EVs connected to the transformer *TR*(*i*) is given by \(\phi_{c}^{i}\).

*j*

^{th}house at time slot

*k*is given by

*h*

_{ j }(

*k*), and

*c*

_{ i }(

*k*) denotes the charge rate of the

*i*

^{th}active EV charge point at time slot

*k*, for all \(k \in \underline{M}\). The electricity price signal at sample time

*k*, denoted

*E*(

*k*), can represent either time-of-use (TOU) [45] or real-time pricing. The corresponding electricity price signal vector is given by \({\user2{E}}^{\text{T}} : = \left[ {E(1),\;E(2), \ldots ,E(M)} \right]\). The charge rate vector for all EVs is given by \({\mathbf{c}}(k)^{\text{T}} : = \left[ {c_{1} (k),c_{2} (k), \ldots ,c_{N} (k)} \right]\) for all \(k \in \underline{M}\). The charge rate profile for the

*i*

^{th}EV is specified by \(\mathbf{c}_{i}^{T} : = [c_{i} (1),c_{i} (2), \ldots ,c_{i} (M)]\). A charge rate matrix is also defined as \(\varvec{C}: = \left[ {\mathbf{c}(1),\mathbf{c}(2), \ldots ,\mathbf{c}(M)} \right]\). The plug-in time and plug-out time of the

*i*

^{th}EV are given by \(\tau_{\text{in}}^{i}\) and \(\tau_{\text{out}}^{i}\), respectively. Therefore, the

*i*

^{th}EV must be charged within \(\left[ {\tau_{\text{in}}^{i} ,\;\tau_{\text{out}}^{i} } \right]\). Let

*P*

_{av}(

*k*) denote the maximum available power that can be drawn from the external grid at time

*k*Due to the battery specification, each EV may have a different battery size (kWh), and this parameter is denoted as

*B*

_{i}for the

*i*

^{th}vehicle. The state-of-charge (SOC) for the

*i*

^{th}EV at time

*k*,

*SOC*

_{ i }(

*k*), within \(\left[ {\tau_{\text{in}}^{i} ,\;\tau_{\text{out}}^{i} } \right]\) is given by:

*i*

^{th}EV when it plugs in. The maximum achievable SOC for the

*i*

^{th}EV is given by:

*i*

^{th}EV charge point. A feasible charging profile is a charging profile which satisfies both plug-in constraints and the state of charge condition, i.e. \(SOC_{i} (M) = SOC_{ \rm{max} }^{i}\) [46].

- 1)Aggregate non-EV base load at time
*k*is given by:$$b(k): = P_{\text{sub}}^{0} (k) - \mathop \sum \limits_{i = 1}^{N} c_{i} (k)$$(3) - 2)Aggregate non-EV base load profile is defined as:$$\varvec{b}^{\mathrm{T}}:=\left[b(1),b(2),\dots,b(M)\right]$$(4)

### 2.1 Plug-in constraints

*i*

^{th}EV, the maximum charge rate is denoted by \(c_{\rm{max} }^{i}\). Considering the charging rate over the course of the full

*M*time slots and noting that charging can only take place when the EV is plugged in and not already fully charged, the constraints are given as follows:

### 2.2 Power system constraints

*k*is inspected, and the voltage for each connected node

*v*

_{i}(

*k*) is used to evaluate the voltage level at each sample step. In this context, the power flow for the

*i*

^{th}transformer can be expressed as:

*i*

^{th}transformer, the power flow constraint can be expressed as:

*v*

_{min}, the voltage constraint is given by:

While the voltage level can be measured at each charge point, in general for monitoring purposes it only needs to be checked at the end of each phase since this will be the point where voltage violations will occur first.

### 2.3 Cost functions

Having established the model parameters and constraints, cost functions need to be defined to reflect the desired EV charging behaviour. Typically, cost functions for EV charging can be divided into two groups. The first are temporally based cost functions *J*(* C*) that evaluate actions taken over a period of time. The second are instantaneous cost functions

*J*(

*(*

**c***k*)) that only consider the current sample step.

*matrix can be determined, leading to sub-optimal solutions due to the uncertainty surrounding the predicted loads. These problems are typically computationally challenging and do not scale well. Typical temporal optimisation objectives include:*

**C**- 1)Minimising the total charging costs for all EVs over the course of all time slots [47]:$$J\left( \varvec{C} \right) = \mathop \sum \limits_{k = 1}^{M} \mathop \sum \limits_{i = 1}^{N} c_{i} (k) \cdot \Delta T \cdot E(k)$$(11)
- 2)Minimising the total energy losses on power transmission lines during EV charging period [48],where$$J\left( \varvec{C} \right) = \mathop \sum \limits_{k = 1}^{M} \mathop \sum \limits_{i = 1}^{{N_{l} }} I_{l,k}^{2} \cdot R_{l} \cdot \Delta T$$(12)
*N*_{ l }is the number of lines in a given area,*I*_{ l,k }the current on line*l*at time*k*, and*R*_{ l }the resistance for line*l*. - 3)Minimising the load variance (thereby flattening the load profile): [37, 46]where \(\overline{{P_{\text{sub}}^{0} }}\) is defined as the average power consumption measured at the main transformer over the course of$$J\left( \varvec{C} \right) = \mathop \sum \limits_{k = 1}^{M} \left( {P_{\text{sub}}^{0} (k) - \overline{{P_{\text{sub}}^{0} }} } \right)^{2}$$(13)
*M*time slots.

*= 0.*

**C**- 1)Charge rate based fairness [33]: Here the objective is to minimise the difference between the charge rates of each EV while distributing the available power among EVs, that is:$$\begin{array}{*{20}c} {} & {\mathop {{\text{min}}}\limits_{{\user2{c}(k)}} \;\mathop \sum \limits_{{i = 1}}^{N} \left[ {c_{i} (k) - \frac{1}{N}\mathop \sum \limits_{{i = 1}}^{N} c_{i} (k)} \right]^{2} } \\ {{\text{subject}}\;{\text{to:}}} & {\left\{ {\begin{array}{*{20}c} {\mathop \sum \limits_{{i = 1}}^{N} c_{i} (k) = P_{{{\text{av}}}} (k),\;\forall k \in \underline{M} } \\ {{\text{Plug-in}}\;{\text{constraints}}} \\ {{\text{Power}}\;{\text{system}}\;{\text{constraints}}\;{\text{(optional)}}} \\ \end{array} } \right\}} \\ \end{array}$$(15)
- 2)Price based fairness [21]: In this case, charging fairness is defined from the perspective of the charging cost. In this problem, we expect the allocated charge rate for each EV to be proportional to the amount of money they would like to pay. A parameter
*ω*_{ i }is introduced for this purpose to reflect the level of willingness to pay of the*i*^{th}EV customer. The resulting optimisation problem can be expressed as:Comment: The algorithms proposed in [33] and [21] can be thought of as providing an approximate distributed solution to the optimisation problems defined in (15) and (16), respectively. The constraint \(\mathop \sum \limits_{i = 1}^{N} c_{i} (k) = P_{\text{av}} (k)\) in both (15) and (16) is considered to avoid trivial solutions. In addition, the power system constraint is not considered in [21, 33]. However, as a necessary option for the grid, we shall consider this constraint in the proposed enhanced algorithms given in Section 3.$$\begin{array}{*{20}c} {} & {\mathop {{\text{min}}}\limits_{{\user2{c}(k)}} \;\mathop \sum \limits_{{i = 1}}^{N} \left[ {\frac{{c_{i} (k)}}{{\omega _{i} }} - \frac{1}{N}\mathop \sum \limits_{{i = 1}}^{N} \omega _{i} \cdot c_{i} (k)} \right]^{2} } \\ {{\text{subject}}\;{\text{to:}}} & {\left\{ {\begin{array}{*{20}c} {\mathop \sum \limits_{{i = 1}}^{N} c_{i} (k) = P_{{{\text{av}}}} (k),\;\forall k \in \underline{M} } \\ {\frac{{c_{i} (k)}}{{c_{j} (k)}} = \frac{{\omega _{i} }}{{\omega _{j} }},\;\forall i \ne j \in \underline{N} ,\;\forall k \in \underline{M}} \\ {{\text{Plug-in}}\;{\text{constraints}}} \\ {{\text{Power}}\;{\text{system}}\;{\text{constraints}}\;{\text{(optional)}}} \\ \end{array} } \right\}} \\ \end{array}$$(16) - 3)Maximising available power utilisation [17]: Here, utility companies wish to maximise the power delivered to charging EVs. The mathematical formulation of this problem is given as follows:$$\begin{array}{*{20}c} {} & {\mathop { \hbox{max} }\limits_{{\varvec{c}(k)}} \;\mathop \sum \limits_{i = 1}^{N} c_{i} (k)} \\ {{\text{subject}}\;{\text{to:}}} & {\left\{ {\begin{array}{*{20}c} {{\text{Plug-in}}\;{\text{constraints}}} \\ {{\text{Power}}\;{\text{system}}\;{\text{constraints}}} \\ \end{array} } \right\}} \\ \end{array}$$(17)

*f*

_{ i }is defined as a utility function associated with the

*i*

^{th}EV, and can be used to influence charging behaviour. For instance, in [17], the functions were chosen such that EVs with lower SOC were given higher priority for charging.

## 3 Charging strategies

In this section, we introduce a number of different competing charging strategies that have been proposed in the literature, which we will evaluate on a realistic distribution network simulation testbed in Section 4.

### 3.1 Uncoordinated charging strategy

### 3.2 Decentralised AIMD algorithms

The additive increase multiplicative decrease (AIMD) algorithm was originally applied in the context of decentralised congestion control in communication networks [49]. Reference [33] proposed applying the AIMD algorithm to EV charging problems and demonstrated its effectiveness in a number of practical scenarios. Roughly speaking, the key characteristic of the AIMD algorithm is that it guarantees an equitable “average” distribution of the available power between active EV charge points if each charge point chooses the same parameters. The elegance of the approach is that it achieves this desirable property while requiring only a minimal communication infrastructure and limited computational power at each EV. In addition, the simple communication topology and minimal communication bandwidth make it a highly scalable and cost effective solution. Further details could be found in [33].

### 3.3 Enhanced distributed AIMD algorithms

It should be noted that in the basic decentralised AIMD algorithm proposed in [33], many practical power system constraints were not considered. As an extension, we introduced a number of enhancements to the basic decentralised AIMD in [50] so that power system constraints on voltage and loading are taken into account.

*P*

_{av}(

*k*) with TOU pricing information

*E*(

*k*) so that an artificial reduction in available power is created at times of high electricity prices, that is:

Here *δ* is a constant tuning parameter, *E* _{min} is the minimum TOU price during the day. Please refer to [50] for further details on this heuristic and the Enhanced AIMD (EnAIMD) implementation.

Comment: Other approaches to fair decentralised EV charging have also been proposed. In particular, [51] describe a decentralised methodology for achieving fair EV charging under transmission constraints. This is based on solving a centralised mixed-integer-nonlinear-program (MINLP) problem using decomposition techniques, where each EV determines its own charging schedule by iteratively solving a knapsack-type optimisation problem. However, this method has a large communication overhead arising from the need for transacting signals between the EVs and a central authority. More sophisticated algorithms are also possible using the AIMD based approach; for instance [52] develop a V2G implementation that provides reactive power compensation capabilities to the grid.

### 3.4 Distributed price feedback

In [21], Fan borrowed the concept of congestion pricing in internet traffic control and introduced a willingness to pay (WTP) parameter to model the preference of user demand. Based on these ideas, he then developed a novel distributed framework for demand response and verified the convergence and dynamic behaviour of his adaptive algorithm, namely distributed price feedback (DPF), by case studies. With this framework in place, a novel distributed EV charging method was proposed such that each EV user could adapt their charging rate according to their personal preferences, maximising their own benefits.

### 3.5 Enhanced distributed price feedback

As was the case with AIMD, the original DPF framework does not consider the impact of EV charging on grid parameters. To address this we introduced an enhanced DPF (EnDPF) implementation in [53] that includes additional functionality similar to that introduced in EnAIMD.

### 3.6 Ideal centralised instantaneous charging

*k*, all charging EVs are required to send their charging requests \(c_{ \rm{max} }^{i}\) to their local transformers. Each local transformer calculates the charge rate for each EV

*i*in area

*j*, taking into account the current local capacity \(TR_{ \rm{max} }^{j} - P_{\rm{sub}}^{j} (k),~ \forall i \in \phi_{c}^{j}\), and forwards the power requirements to the main substation. If the total amount of requested power exceeds the available power, the main substation TR(0) allocates the available power to each substation in proportion to the requested values. Each substation then updates their EV charge rates accordingly and broadcasts the information to the charge points. This strategy is feasible for solving the following optimisation problem:

Note that voltage constraints are not considered within this formulation for the sake of simplicity.

### 3.7 Optimal decentralised valley-filling charging

A novel decentralised temporal optimisation algorithm, namely optimal decentralised valley-filling (ODVF), was proposed in [46] to optimally schedule EV charging to perform valley filling through an iterative process. It has been shown that the charging profile for each EV can reach optimality within a few iterations and that the approach provides satisfactory performance and is robust to errors in users’ specifications and outdated signals.

*b*(

*k*) denotes the

*k*

^{th}element of base load profile

*. Details of the algorithm implementation could be referred in [46].*

**b**### 3.8 Decentralised selfish charging strategy

### 3.9 Centralised cost minimisation charging strategy

In this section, a centralised charging cost minimisation (CCCM) strategy based on linear programming is proposed to minimise the total cost of charging EVs. Using this method, the charge rate matrix of all vehicles * C* is determined at a centralised control center. Rather than updating the charge rate locally according to some feedback signals (e.g. price signal) at every time slot, centralised approaches are more amendable to fulfilling temporal based objectives (e.g. minimising total charging costs, valley-filling). We assume that at the beginning of the scheduling window, all essential information is provided to the optimisation program for computation in the control center. This information includes the predicted base load

*and the charging schedule of each EV, i.e., \(\tau_{\text{in}}^{i}\) and \(\tau_{\text{out}}^{i}\).*

**b**Comment: If the power system constraints are omitted, the solution to the optimisation problem defined in (24) is mathematically equivalent to the solution of (23). Thus, CCCM is an enhancement of DSC since it is not practical to incorporate power system constraints in the DSC method. However, this is at the expense of substantial communication and computation overhead.

### 3.10 Centralised load-variance-minimisation charging

In this case, the optimisation problem is formulated as a quadratic programming problem, the aim of which is to flatten the overall load profile, i.e., valley-filling. Compared to the ODVF method, as defined in Section 3.7, this charging strategy, denoted CLVM, gathers all the necessary information from both the grid and EV customers before solving the quadratic optimisation problem, in order to determine the optimal charge rate matrix * C* before charging commences. Mathematically, the optimisation problem is given by (13) and (14) as presented in Section 2.3.

## 4 Case studies and results

### 4.1 Simulation set-up

In order to compare the performance of the different charging strategies, a one day simulation was run with ∆*T* set to 5 min, i.e. *M* = 288. The simulation was conducted on a typical residential low voltage distribution network with *S* = 3 and *N* = 160. The houses are distributed evenly across phases with maximum 50% EV penetration randomly connected in three household areas.

This distribution network was modelled and implemented using a custom OpenDSS/ Matlab simulation platform. OpenDSS [54], an open source electric power system distribution system simulator, was used to simulate the power system and calculate the instantaneous power flows and voltage profiles for the test network. Matlab was used to simulate typical residential EV connection, SOC and disconnection patterns (randomly generated for each EV) and to create a wrapper programme to simulate the operation of the network over a period of time for varying household and EV loads based on various charging strategies. The topology of the network is given in Fig. 1.

The non-EV household load profiles for each scenario were generated based on residential customer smart meter electricity trial data provided by the Commission for Energy Regulation (CER) [55] in Ireland. The assumptions on EV travelling patterns, and hence SOC and plug-in/out probability distributions, were taken from [56]. For comparison purposes, the same plug-in/out and SOC values were used with each method considered. It was assumed that all EVs charged overnight and that once an EV was plugged in, it only physically plugged out at the scheduled plug-out time.

- 1)
Uncoordinated charging strategy

- 2)
Fairness based strategies: AIMD, DPF, EnAIMD, EnDPF and ICIC and their TOU price adjusted extensions

- 3)
Cost minimisation strategies: DSC and CCCM

- 4)
Valley-filling strategies: ODVF and CLVM

### 4.2 Evaluation of the uncoordinated charging strategy

### 4.3 Evaluation of fairness based strategies

*δ*in (20) was set to 10. The resultant power and voltage profiles, and the profile of the loading of the second transformer are presented in Figs. 4, 5, and 6, respectively.

A comparison of simulation results for various EV charging strategies (50% EV penetration)

Strategies | Ave costs (cents/kWh) | Ave rate (kW) | Min volt (p.u.) | Max TR loading (%) | Overload duration (h) | Ave time (h) | Std time (h) | Min time (h) | Max time (h) | Overall assessment |
---|---|---|---|---|---|---|---|---|---|---|

No EV | n/a | n/a | 0.954 | 112.93 | 2.25 | n/a | n/a | n/a | n/a | n/a |

Uncontrolled | 22.32 | 3.70 | 0.915 | 177.68 | 14.83 | 2.80 | 0.38 | 1.83 | 3.83 | Bad |

AIMD | 13.36 | 1.38 | 0.940 | 122.49 | 9.33 | 7.53 | 0.88 | 5.42 | 9.17 | Bad |

AIMD(P) | 11.21 | 1.25 | 0.941 | 113.92 | 6.58 | 8.34 | 1.05 | 5.92 | 10.08 | Bad |

EnAIMD | 12.74 | 1.36 | 0.953 | 112.93 | 2.75 | 7.89 | 1.63 | 4.58 | 10.83 | Good |

EnAIMD(P) | 11.18 | 1.23 | 0.950 | 112.93 | 2.25 | 8.56 | 1.29 | 5.83 | 11.00 | Good |

DPF | 13.93 | 1.36 | 0.946 | 122.17 | 9.33 | 7.86 | 1.84 | 4.42 | 11.42 | Bad |

DPF(P) | 11.30 | 1.22 | 0.947 | 113.91 | 5.83 | 8.71 | 1.81 | 5.00 | 12.33 | Bad |

EnDPF | 13.63 | 1.35 | 0.953 | 112.93 | 5.17 | 8.00 | 2.42 | 3.92 | 14.50 | Good |

EnDPF(P) | 11.22 | 1.16 | 0.954 | 112.93 | 2.33 | 9.19 | 2.02 | 5.83 | 14.58 | Good |

ICIC | 13.84 | 1.41 | 0.951 | 124.47 | 9.92 | 7.41 | 1.11 | 4.67 | 9.25 | Ugly |

ICIC(P) | 11.20 | 1.27 | 0.945 | 116.01 | 5.75 | 8.24 | 1.10 | 5.67 | 10.08 | Ugly |

ODVF | 10.46 | 1.36 | 0.958 | 112.93 | 2.25 | 7.31 | 0.74 | 5.92 | 8.58 | Good |

CLVM | 10.46 | 1.36 | 0.958 | 112.93 | 2.25 | 7.31 | 0.74 | 5.92 | 8.58 | Ugly |

DSC | 10.00 | 1.13 | 0.954 | 112.93 | 2.61 | 8.88 | 0.26 | 7.92 | 9.00 | Ugly |

CCCM | 10.00 | 1.11 | 0.954 | 112.93 | 2.25 | 8.89 | 0.27 | 7.92 | 9.00 | Ugly |

### 4.4 Evaluation of cost minimisation strategies

### 4.5 Evaluation of valley filling charging strategies

ODVF computation time for different iteration counts (50% EV penetration)

ODVF | Time (s) |
---|---|

5 | 6.08 |

10 | 11.41 |

15 | 16.96 |

20 | 22.79 |

CLVM computation times with different levels of EV penetration

CLVM (%) | Time (s) |
---|---|

10 | 20.87 |

20 | 104.72 |

30 | 236.02 |

40 | 789.23 |

50 | 2105.78 |

The performance of the ODVF method was also evaluated from the perspective of the mean square error (MSE) between the iterative load profile and the optimal aggregated load profile using the CLVM method. This showed that ODVF is able to converge to within 90% of the CLVM optimum in less than 6 iterations.

## 5 Discussion

In general, uncontrolled charging provides the best performance in terms of its objective, i.e., minimising customer charging times. Similarly, by applying the basic AIMD and DPF based charging strategies, charging fairness can be achieved for each customer under appropriate assumptions, while the basic DPF method can adjust this fairness according to the WTP specified by individual customers.

However, typically these approaches violate grid constraints, as can be seen in Table 1 and Figs. 2 and 3. The addition of constraints allows the respective charging goals to be met while respecting grid constraints, as shown in Table 1 and Figs. 4, 5 and 6. While centralised algorithms achieve the best performance in terms of their stated goals, including satisfying power system constraints, they require global information and also do not scale well, as mentioned previously. For example, here, the CLVM method has the longest simulation times. Decentralised algorithms, like the ODVF algorithm, approach the performance of centralised algorithms and are scalable. However, there is a large communication overhead associated with such algorithms.

Therefore, the modified AIMD and DPF algorithms show much promise. These algorithms provide a trade-off between a small communication overhead and almost optimal performance in terms of their grid objectives, and can be coordinated to approximately achieve some temporal objectives without the necessity of accurate load prediction, i.e. by applying a price-adjusted available power heuristic.

## 6 Conclusions

A mathematical framework for formulating EV charging problems has been presented that incorporates both power system and charging infrastructure constraints and caters for both instantaneous and temporal optimisation objectives. Within this framework, several charging strategies were evaluated. These algorithms were tested on a power system distribution level testbed using a hybrid Matlab/OpenDSS platform, and using realistic demand and charging profiles. Of the algorithms considered, it was found that those algorithms that ignored system constraints typically violated them for large EV penetrations (BAD). Of the algorithms that considered power system constraints, the modified AIMD and DPF algorithms provided the best trade off in terms of achieving almost optimal performance in terms of their grid objectives while satisfying constraints and maintaining a small communication overhead (GOOD) in comparison to the other communication based optimal algorithms considered (UGLY).

## Notes

### Acknowledgment

The authors would like to thank the Irish Social Science Data Archive (ISSDA) for providing access to the CER Smart Metering Project data. The authors also gratefully acknowledge funding for this research provided by Science Foundation Ireland (Grant 11/PI/1177 and Grant 09/SRC/E1780).

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