# Understanding Complexity in Charging Infrastructure through the Lens of Social Supply–Demand Systems

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## Abstract

**:**

## 1. Introduction

## 2. General Abstraction of Infrastructures as Systems of Supply and Demand

#### 2.1. Environment

#### 2.2. Resource Access Points

#### 2.3. Users

#### 2.4. Transactions

#### 2.5. Research Methodologies for Charging Infrastructure as Social Supply–Demand System

## 3. Charging Infrastructure as a Complex System

#### 3.1. Feedback Loops

#### 3.2. Learning and Adaptation

#### 3.3. Emergence

#### 3.4. Network Formation and Resilience

#### 3.5. Self-Organisation

#### 3.6. Non-Linearity

#### 3.7. Path Depencency

#### 3.8. Phase Transitions

## 4. Analyzing the Complexity of the EV System

#### 4.1. How Complex Is the EV System?

#### 4.1.1. Ratio 1: Necessity of Sharing Resources

_{1}, in line with node clustering metrics in bipartite graphs [96]. First, we set a timeframe ($\tau $) to $\tau +{\delta}_{t}$, with a transaction set of ${T}_{t,t+{\delta}_{t}}\subset T$. We define $A$ as the set of all pairs of transactions${t}_{x}^{a},{t}_{y}^{b}$from two different users ${u}_{a},{u}_{b}$, where ${\tau}_{{t}_{x}^{a}},{\tau}_{{t}_{y}^{b}}\in {T}_{t,t+{\delta}_{t}}\wedge {u}_{a}\ne {u}_{b}$ at their given location ${l}_{x},{l}_{y}$, based on the location of the resource access point. We consider the set of considered resources${R}_{{l}_{x}}^{{u}_{a}},{R}_{{l}_{y}}^{{u}_{b}}$ of users and observable resources per location ${R}_{{l}_{x}},{R}_{{l}_{y}}.$ We calculate Ratio

_{1}as follows:

_{1}approaches 0 as (i)$\left|{R}_{{l}_{x}}^{{u}_{a}}\cap {R}_{{l}_{y}}^{{u}_{b}}\right|$ is zero, in most cases, since each user has its own CP (Figure 5a). On the other hand, for DC fast charging, all users may potentially select any of the available outlets at arrival, leading to a value close to 1 (Figure 5c). For public charging infrastructure, we see that the set of overlapping $\left|{R}_{{l}_{x}}^{{u}_{a}}\cap {R}_{{l}_{y}}^{{u}_{b}}\right|$is smaller than $|{R}_{{l}_{x}}\cup {R}_{{l}_{y}}|$, which leads to a value significantly lower than 1 and higher than 0. For semi-public charging, we expect values close to DC fast charging, since both ${R}_{{l}_{x}}^{{u}_{a}}~{R}_{{l}_{y}}^{{u}_{b}}$ and ${R}_{{l}_{x}}~{R}_{{l}_{y}}$. We expect that complexity decreases at the limits of this ratio (0,1).

#### 4.1.2. Ratio 2: Probabilities of Queuing

_{2}) specifically focuses on the complexity of behavior in the system, related to the time aspect (e.g., service rate of supply and demand). This ratio builds upon the concept of supply and demand rates, which has been studied in queueing theory [97]. In queueing theory, the ratio between $\frac{\lambda}{\mu}$ determines the probability of a queue’s existence. In this equation, $\lambda $ is the arrival process, assumed to be a Poisson distribution $\lambda $ (defined in our SSDS as ${\lambda}_{j}$), and $\mu $ is the exponentially distributed service time ($\delta \tau $ in our SSDS) [97]. The literature on DC fast charging infrastructure has modeled outlets as a multi-server network queue [98,99]. Yet, for AC public charging, the following aspects make queuing models complex in highly occupied systems [97]: (i) $\mu $ is not a property of the outlet alone, as it is related to parking behavior, rather than transactions size; (ii) EV users have shown to be strongly habitual, causing multi-modal Gaussian arrival patterns, rather per ${u}_{j}$ [3]; and (iii) when EV users arrive at a location, they have individual preferences for ${R}_{{u}_{j}}$, even if they overlap in locality.

_{2}as the duration time of a transaction divided by the mean time between arrivals at ${r}_{\iota}$. To take locality into account, we average over each set of local observable resources ${R}_{{l}_{k}}$, rather than over the whole system at once (e.g., $\frac{\overline{\lambda}}{\overline{\mu}}$).

_{2}is, therefore, calculated as follows:

_{2}and, consequently, limited changes in the complexity for AC public charging.

#### 4.1.3. Ratio 3: Impact of Transactions on Others

_{3}, in line with literature on cascading failures, as follows [103]. In line with the other ratios, we set a timeframe ($\tau $) to $\tau +\delta \tau $ for a transaction subset of ${T}_{t,t+\delta \tau}\subset T$. For $\forall {t}_{p}{}_{{r}_{\iota}}^{{u}_{j}}\in {T}_{t,t+\delta \tau}$, we initiate the set $S{P}_{{t}_{j}}=\left\{\right\}$, which we fill with transactions ${t}_{l}{}_{{r}_{i}}^{{u}_{j}}$affected by the alternative decisions that ${u}_{j}$ may have made for transaction ${t}_{l}$ and initiate $AO=\left\{\right\},$as the set of alternative ${r}_{j}$, which we have taken into consideration during calculation. We iteratively perform the following calculations.

- (1)
- $\forall {t}_{l}{}_{{r}_{\iota}}^{{u}_{j}}$ of ${u}_{j}$we regard as the set of alternatives minus the already considered $r\in OA$, in using ${R}_{{u}_{j}}-OA$;
- (2)
- For all ${r}_{l}\in {R}_{{u}_{j}}-OA$, we add the set of affected transactions at ${r}_{l}$ during the ${t}_{l}$ interval $[{\tau}_{{t}_{l}{}_{{r}_{k}}^{{u}_{j}}},{\tau}_{{t}_{l}{}_{{r}_{k}}^{{u}_{j}}}+\delta {\tau}_{{t}_{l}{}_{{r}_{k}}^{{u}_{j}}}]$ to $S{P}_{{t}_{j}}$;
- (3)
- For $\forall t\prime \in S{P}_{{t}_{j}}$, we return to (1), while adding transactions to the initial $S{P}_{{t}_{j}},$until ${R}_{{u}_{i}}-OA=\varnothing $.

_{3}to be low for DC fast charging. For private home charging, the spread takes, as a maximum, the value of the number of EVs that share the home charging CPs. The Ratio

_{3}value is expected to be between 0 and 1. For semi-public charging, we also expect this value to be between 0 and 1. For the mature public charging infrastructure, we expect this value to be above 1 [104].

#### 4.2. Conclusions on Complex System Ratios

_{1}is normalized over all pairs. We expect ${R}_{{t}_{k}}$ and ${R}_{{u}_{i}}$ to increase as the density of the CPs increases. We also expect $|{R}_{{u}_{a}}\cap {R}_{{u}_{b}}|$ to decrease as more users have nearby subsets of options. We expect $|{R}_{{t}_{x}}\cup {R}_{{t}_{y}}|$ to increase as the set of alternatives grows. As such, we expect a slow decrease of Ratio

_{1}over time and the complexity to slightly decrease. On the other hand, an increase of ${R}_{{u}_{a}}$ and $\left|U\right|$ may lead to an increase of Ratio

_{3}and the spread of effects of alternative decisions. In the end, this may add to the complexity of the behavior in the system. We, therefore, expect that AC charging infrastructure (separate from its public DC counterpart) will remain within the complex regime.

## 5. Analyzing Interactions and Performance of Complex Systems

#### 5.1. Systemic Metrics for Charging Infrastructure

_{3}), is high.

#### 5.2. Modeling Charging Infrastructure as Complex System

#### 5.3. Proposed Model for Analyzing Charging Infrastructure as Complex System

## 6. Discussion and Implications for Policy Makers and Research

#### 6.1. Implications for Policy Makers

#### 6.2. Implications for Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Glossary

Symbol | Example | Description |

$A$ | Set of all pairs of transactions${t}_{x}^{a},{t}_{y}^{b}$ | |

$c\_i$ | $ci$ | Maximum transaction speed at $ri$ |

$\delta {\tau}_{p}$ | $\delta {\tau}_{p}$ | Duration of transactions $p$ |

$E$ | $E\subset {\mathbb{N}}^{2}$, $lk\in E$. | Environment contains a set of locations ${l}_{k}$ |

${\varphi}_{j}$ | Demand rate user j max uptake speed | |

${l}_{\iota}$ | ${l}_{\iota}\in E$ | Location of resource $\iota $ |

${\lambda}_{j}$ | Mean demand rate for user j in frequency time | |

${\mathsf{\Lambda}}_{t}$ | Local average $\lambda $ used in Ratio_{2} | |

Mj | Memory of connections for user j | |

$\mathbb{N}$ | Environment size | |

${o}_{\iota}$ | ${o}_{\iota}$ =$\left\{{o}_{1}..{o}_{\eta}\right\}$ | Set of outlets at ${r}_{\iota}$ |

${q}_{\iota}$ | Quality of resources at access point $\iota $ | |

${\mathsf{\rho}}_{\mathrm{j}}$ | Demand rate user j in terms of quality | |

$R$ | $R=\left\{{r}_{1},\dots ,rn\right\}$ | Set of resource access points |

${R}^{{u}_{j}}$ | Set of user preferred resource access points | |

${R}_{{l}_{k}}^{{u}_{j}}$ | Set of user preferred resource access points at the given location | |

${R}_{lk}$ | Set of local observable resource access points at the location | |

${r}_{\iota}$ | ${r}_{\iota}=\left\{{l}_{\iota},{o}_{\iota},{c}_{\iota},{q}_{\iota}\right\}$ | Resource access point tuple |

$S$ | $S=\left\{E,R,U,T\right\}$ | System |

$S{c}_{j}$ | Maximum resource uptake speed of user $j$ | |

$S{t}_{j}$ | Storage capacity user $j$ | |

$\tau $ | Time | |

$T$ | $T=\left\{{t}_{1},\dots ,tO\right\}$ | Transactions |

${T}^{{u}_{j}}$ | $T\_j=\left\{{t}_{1},\dots ,t\_J\right\}$ | Set of transactions for user $j$ |

${T}_{{r}_{i}}^{{u}_{j}}$ | Set of transactions for user $j$ at resource access point $\iota $ | |

${t}_{p}$ | ${t}_{p}=\left\{{u}_{j},{o}_{\iota},{r}_{\iota},\tau {s}_{p},\delta {\tau}_{p},{V}_{p}\right\}$ | Transaction tuple |

$\tau {s}_{p}$ | Start time of transaction p | |

$U$ | $U=\left\{{u}_{1},\dots ,{u}_{m}\right\}$ | Set of users |

${u}_{j}$ | ${u}_{j}=\left\{{\mathrm{St}}_{\mathrm{j}},{\mathrm{Sc}}_{\mathrm{j}},{\lambda}_{j},{\varphi}_{j},{\mathrm{T}}_{\mathrm{j}},{\mathsf{\rho}}_{\mathrm{j}}\right\}$ | User tuple |

${V}_{p}$ | Transaction size |

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**Figure 1.**Illustration of feedback loop, related to policy makers’ charging point deployment strategies.

**Figure 5.**Overview of different configurations for ratio1. (

**a**) Public charging; (

**b**) private charging; (

**c**) DC fast charging.

**Figure 6.**Illustration of arrival and departure patterns for different type of charging infrastructure.

**Figure 7.**The path of the spread of effect for (

**a**) public charging, (

**b**) private charging, and (

**c**) DC fast charging (green dot indicates start of cascade).

System | Supply Side/Resources | Demand Side/Users | Environment | Transactions | |
---|---|---|---|---|---|

Education | School, classes | Students/pupils | Geospatial map | Knowledge transfer | [28] |

Healthcare | Hospitals, care centers | Patients | Geospatial map | Health Care | [29,30] |

Electricity | Electricity generators: central and decentral | Household | Network of High mid and low voltage grid | Energy | [31,32,33] |

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**MDPI and ACS Style**

Helmus, J.; Lees, M.; van den Hoed, R.
Understanding Complexity in Charging Infrastructure through the Lens of Social Supply–Demand Systems. *World Electr. Veh. J.* **2022**, *13*, 44.
https://doi.org/10.3390/wevj13030044

**AMA Style**

Helmus J, Lees M, van den Hoed R.
Understanding Complexity in Charging Infrastructure through the Lens of Social Supply–Demand Systems. *World Electric Vehicle Journal*. 2022; 13(3):44.
https://doi.org/10.3390/wevj13030044

**Chicago/Turabian Style**

Helmus, Jurjen, Mike Lees, and Robert van den Hoed.
2022. "Understanding Complexity in Charging Infrastructure through the Lens of Social Supply–Demand Systems" *World Electric Vehicle Journal* 13, no. 3: 44.
https://doi.org/10.3390/wevj13030044