# Energy Management Method for Fast-Charging Stations with the Energy Storage System to Alleviate the Voltage Problem of the Observation Node

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. FCS Model

**κ**

_{j}stand for a set of time intervals when the EV’s charging load does not obey the normal distribution at the FCS j. Then ${P}_{j\kappa}^{\mathrm{E}\mathrm{V}}$ can be formulated as,

**κ**

_{j}, ${P}_{j\kappa}^{\mathrm{c}\mathrm{h}}$ is also a normal distribution variable, N(${\mu}_{j\kappa}$−${P}_{j\kappa}^{\mathrm{P}\mathrm{V}}$,${\sigma}_{j\kappa}^{2}$). Since the linear transformation of a normal distribution still follows a normal distribution, when κ ∈

**κ**

_{j}, the FCS model can be written as,

#### 2.2. Impact on the Radial Distribution Network

#### 2.3. Optimization Model

**x**) stands for the objective function calculated by the elements of the optimization vector

**x**,

**x**= [${x}_{1}$, …, ${x}_{m}$], u

_{j}(

**x**) stands for the jth inequality constraint. It is obvious that many intelligent algorithms can be used to solve Equation (6), such as particle swarm optimization (PSO) and the genetic algorithm (GA). Compared with other intelligent algorithms, the traditional GA is more robust and can effectively search the complex spaces, but it is easier to fall into a local optimum [19,20,21]. Therefore, an improved real-coded genetic algorithm (RCGA-rdn) is used to solve Equation (6), which can reduce the probability of falling into local optimum and improve the computational efficiency [22]. The constrained optimization model needs to be converted into an unconstrained optimization model before using the RCGA-rdn. In this paper we use the penalty function method to achieve this conversion, then Equation (6) can be rewritten as,

_{g}(.) stands for the penalty function of population g, and pp is the initial population number. For a detailed description of the optimization model, please refer to Section 4.

## 3. Analytical Assessment Model of the Nodal Voltage Change

#### 3.1. The Radial Distribution Network Model

_{s}stand for the phase voltage at the source node s, U

_{d}stand for the phase voltage at the load node d, and U

_{o}stand for the phase voltage at the observation node o. Based on the circuit superposition theorem, the typical radial distribution network can be decomposed into three different type sub-circuits as shown in Figure 4. The voltage at the observation node o (U

_{o}) can be expressed as,

_{od}is the shared impedance between the node d and observation node o from the source node s, and

**D**is a set of the load nodes.

_{d}to S

_{d}+ ∆S

_{d}and the nodal voltage change from U

_{d}to U

_{d}+ ∆U

_{d}, then the voltage change at the observation (∆U

_{o}) can be written as,

_{od}= R

_{od}+jX

_{od}, ∆S

_{d}= ∆${P}_{d}^{\mathrm{G}}$+j∆${Q}_{d}^{\mathrm{G}}$ and ∆U

_{d}= ∆${U}_{d}^{\mathrm{r}}$ + j∆${U}_{d}^{\mathrm{i}}$. By listing the real and imaginary parts of ∆U

_{o}separately, we can get,

_{o}. Since the phase angle of the phase voltage is usually small, it is reasonable to ignore the ∆${U}_{d}^{\mathrm{r}}$ and ∆${U}_{d}^{\mathrm{i}}$. Let U

_{d}= |U

_{d}| ∠ θ

_{d}, then Equation (11) can be rewritten as,

_{d}and Z

_{od}are known variables, A, B, C, and D are constants. Therefore, ∆${U}_{o}^{\mathrm{r}}$ and ∆${U}_{o}^{\mathrm{i}}$ can be approximated as a linear combination of ∆${P}_{d}^{\mathrm{G}}$ and ∆${Q}_{d}^{\mathrm{G}}$.

#### 3.2. The AMM

_{o}) smaller than the limiting value (δ

_{o}) can be used to measure the impact of the FCSs on the voltage change at the observation node o. The larger P(|∆U

_{o}| < δ

_{o}) is, the smaller the impact of the FCSs on the voltage change will be. In this section, we derive the AMM which is used to evaluate the impact of the FCSs on the voltage change at any node in the LV radial distribution network. In this paper, we select the average load at the node without the FCS as the initial load at this node. Let the complex draw power change at the node without the FCS be zero, then we can obtain the nodal voltage change of the observation node, which is caused by charging load changes of FCSs. The reactive power change (∆Q) can be ignored at the node with the FCS since the power factor of the node with the FCS is usually close to 1 so that the reactive power change is usually very small. Hence, for an n-node LV distribution network system, Equation (12) can be rewritten as,

**κ**

_{j}and is a constant (${C}_{j\kappa}^{\mathrm{S}}$) when κ ∉

**κ**

_{j}. By plugging Equation (15) into Equation (1), ${P}_{j\kappa}^{\mathrm{G}}$ is,

**κ**

_{j}, ${P}_{j\kappa}^{\mathrm{G}}$ is still a normal variable at this time. Let ${\mu}_{j\kappa}^{\mathrm{G}}$ stand for the expectation of ${P}_{j\kappa}^{\mathrm{G}}$ when κ ∈

**κ**

_{j}. Select ${\mu}_{j\kappa}^{\mathrm{G}}$ as the initial nodal load for the FCS j, then ∆${P}_{j\kappa}^{\mathrm{G}}$ is a normal variable, ∆${P}_{j\kappa}^{\mathrm{G}}$~N(0, ${\gamma}_{j\kappa}^{2}$). By plugging Equation (16) into Equation (1), we can obtain the expression of ${P}_{j\kappa}^{\mathrm{G}}$ when κ ∉

**κ**

_{j},

**κ**

_{j}, ∆${P}_{j\kappa}^{\mathrm{G}}$ is zero at this time. Then Equation (14) can be rewritten as,

**E**is a set of nodes with the FCS. According to Equation (19), ∆${U}_{o\kappa}^{\mathrm{r}}$ and ∆${U}_{o\kappa}^{\mathrm{i}}$ are linear combinations of ∆${P}_{j\kappa}^{\mathrm{G}}$. Therefore, ∆${U}_{o\kappa}^{\mathrm{r}}$ and ∆${U}_{o\kappa}^{\mathrm{i}}$ can be regarded as normal distribution variables. We assume that the load at each node is independent of each other, then ∆${U}_{o\kappa}^{\mathrm{r}}$ and ∆${U}_{o\kappa}^{\mathrm{i}}$ can be expressed as,

**∆U**

_{o}= (∆${U}_{o}^{\mathrm{r}}$,∆${U}_{o}^{\mathrm{i}}$)

^{T}, ∆

**U**

_{o}~N(0,

**C**

_{o}), where

**C**

_{o}is a covariance matrix of

**∆U**

_{o}for observation node o.

**C**

_{o}can be written as,

**C**

_{o}can be diagonalization by eigenvalue decomposition as:

**W**

_{o}is an eigenmatrix of

**C**

_{o}, ${\lambda}_{1o}$ and ${\lambda}_{2o}$ are related eigenvalues and

**Λ**

_{o}is the diagonally similar matrix of

**C**

_{o}. Let

**V**

_{o}= ${W}_{o}^{\mathrm{T}}$∆

**U**

_{o}, where

**V**

_{o}= (V

_{1o},V

_{2o})

^{T}. Since Cov(V

_{1o},V

_{2o}) is equal to 0, V

_{1o}and V

_{2o}are independent mutual variables. ${V}_{o}^{\mathrm{T}}$

**V**

_{o}can be expanded as,

^{2}+ (∆${U}_{o}^{\mathrm{r}}$)

^{2}, ∆${U}_{o}^{2}$=${V}_{o}^{\mathrm{T}}$

**V**

_{o}. Therefore, ∆${U}_{o}^{2}$ is the sum of two independent weighed chi-square random variables. Then ∆${U}_{o}^{2}$ can approximately obey Gamma distribution (Γ(α

_{o}, β

_{o})) and the parameters of Γ (α

_{o}, β

_{o}) can be calculated as [22],

_{o}| < δ

_{o}) can be calculated as,

#### 3.3. The Control Characteristic of β_{o}

_{o}is a known constant, Fo(.) is decided by parameters α

_{o}and β

_{o}. We can increase Fo(.) by controlling the two parameters. In this section, we discuss the control characteristic of β

_{o}, which can give theoretical support for the VCOM. According to Equation (24), α

_{o}and β

_{o}are functions of λ

_{1o}and λ

_{2o}. We next note that λ

_{1o}and λ

_{2o}can be expanded as,

_{oj}and we can obtain the expansion of Z

_{oj}as follows,

_{oj}is the impedance angle of the shared impedance ${Z}_{oj}$. Therefore, Equation (26) can be simplified as,

_{o}is approximately equal to 0.5, which is not affected by the change of the nodal load change. Let α

_{o}= 0.5 and t = β

_{o}x, then ${F}_{o}$(.) can be rewritten as,

_{o}> 0, ${F}_{o}$(.) increases as the increase of β

_{o}until ${F}_{o}$(.) reaches 1. In other words, the smaller the value of β

_{o}is, the smaller the nodal voltage change at the observation will be.

## 4. The VCOM

**O**is a set of observation nodes. The inequality constraint of ${P}_{j\kappa}^{\mathrm{S}}$ can be written as,

_{jκ}= ${\gamma}_{j\kappa}$/δ

_{jκ}when κ ∈

**κ**

_{j}, we can rewrite Equation (15) by plugging k

_{jκ}= ${\gamma}_{j\kappa}$/δ

_{jκ}in Equation (15),

_{oκ}can be designed as a constraint which is used to restrain the voltage change at the observation node in the optimization framework as follows:

_{oκ}(.) increases with the increase of ${\beta}_{o\kappa}$ until it reaches 1 when δ

_{o}is a known constant, we can obtain ${\beta}_{o\kappa}^{\mathrm{min}}$ by the Algorithm 1 as follows:

Algorithm 1: Calculate${\mathit{\beta}}_{\mathit{o}\mathit{\kappa}}^{\mathbf{min}}$ |

1: INPUT: This algorithm knows the acceptable probability of the nodal voltage change at the observation node (${F}_{o\kappa}^{\mathrm{min}}$) at time interval κ, the initial nodal load at each node at time interval κ, the adjustment step size of ${\beta}_{o\kappa}^{\mathrm{min}}$ (∆), basic information of the distribution network, which can be used to calculate power flow by using traditional Newton power flow method. |

2: OUTPUT: ${\beta}_{o\kappa}^{\mathrm{min}}$ = [${\beta}_{1\kappa}^{\mathrm{min}}$, ${\beta}_{2\kappa}^{\mathrm{min}}$,…, ${\beta}_{n\kappa}^{\mathrm{min}}$] |

3: PROCEDURE: |

4: Obtain the nodal voltage at each node by using traditional Newton power flow method. |

4: for o = 1 to n do |

5: Obtain ${\beta}_{o\kappa}$ according to Equation (31) |

6: Obtain F_{oκ}(.) according to Equation (32) |

7: for I = 1 to N do |

8: if ${F}_{o\kappa}$(.) ≥ ${F}_{o\kappa}^{\mathrm{min}}$ |

9: ${\beta}_{o\kappa}^{\mathrm{min}}$ ←${\beta}_{o\kappa}$—∆ |

10: Update F_{oκ}(.) by plugging ${\beta}_{o\kappa}^{\mathrm{min}}$ into Equation (32) |

11: else |

12 ${\beta}_{o\kappa}^{\mathrm{min}}$←${\beta}_{o\kappa}$ |

13: end if |

14: end for |

15: end for |

_{j}is a random number in the range [0, 1].

## 5. Case Study and Discussion

^{®}Core™ i7-6700 Processor and 8 GB of memory, in which the CPU works at the nominal frequency of 3.4 GHz. The base voltage of the system is 12.66 kV, and the base capacity is 100 MW. The reference voltage is set to 0.95.

#### 5.1. The Performance of the AMM

#### 5.2. The Effect of the Proposed Day-Ahead ESS Strategy

- (i)
- Case 1: the traditional PV-ESS complementarity strategy is used for the ESS.
- (ii)
- Case 2: the strategy proposed in this paper is used for the ESS.

_{o}| < δ

_{o}) at each node under different cases in a day. It is obvious that the day-ahead ESS strategy can increase P(|∆U

_{o}| < δ

_{o}) in comparison with the traditional PV-ESS complementarity strategy. Voltages under case 2 are closer to the reference voltage than other cases as shown in Figure 11a, which is more intuitively displayed in Figure 11b. Simulation results above demonstrate the effect of the proposed day-ahead ESS strategy. It can be seen from Figure 12a that the ESS charges during periods of low EV charging load and provides active power compensation by discharge during periods of large EV charging load to reduce the voltage change between actual and reference voltages. In addition, we can find that due to the constraint of the SOC of the ESS, the ESS cannot always compensate for the active power. For example, from 6.00 p.m. to 9.00 p.m., since the SOC of the ESS is close to its lower boundary, the ESS can hardly compensate the active power, which results in the failure of the day-ahead ESS strategy. A similar result can also be obtained according to Figure 13. Therefore, to give full play to the effect of the day-ahead ESS strategy, it is inseparable from reasonable energy storage system planning.

## 6. Conclusions

- A voltage change optimization model (VCOM) considering the randomness of the EV load is constructed to alleviate the voltage change problem caused by EV fast charging and it can be easily solved by traditional intelligent algorithms, such as GA.
- An analytical assessment model (AAM) of the nodal voltage change with shorter computational time and higher reliability is proposed.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${P}_{j\kappa}^{\mathrm{G}}$ | Total grid load at the FCS j at κth time interval. |

${\mu}_{j\kappa}^{\mathrm{G}}$ | Expectation of ${P}_{j\kappa}^{\mathrm{G}}$. |

${\gamma}_{j\kappa}$. | Standard change of ${P}_{j\kappa}^{\mathrm{G}}$ |

${P}_{j\kappa}^{\mathrm{PV}}$ | Output power of the PV at the FCS j at κth time interval. |

${P}_{j\kappa}^{\mathrm{S}}$ | Charge–discharge active power of the ESS at the FCS j at κth time interval. |

${P}_{j\kappa}^{\mathrm{EV}}$ | The NFCL at the node j with the FCS at κth time interval. |

${P}_{j\kappa}^{\mathrm{EV}}$ | The EV’s charging load at the FCS j at κth time interval. |

${\mu}_{j\kappa}$ | Expectation of ${P}_{j\kappa}^{\mathrm{E}\mathrm{V}}$. |

${\sigma}_{j\kappa}$ | Standard deviation of ${P}_{j\kappa}^{\mathrm{E}\mathrm{V}}$. |

${S}_{j}^{\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}}$ | Rated capacity of the service transformer at the FCS j. |

${\delta}_{o\kappa}$ | Nodal voltage change limit at the observation node o at κth time interval. |

${\alpha}_{o\kappa}$ | Shape parameters for the nodal voltage change at observation node o at κth time interval. |

${\beta}_{o\kappa}$ | Control parameter for the nodal voltage change at observation node o at κth time interval. |

x_{κ} | Optimization variable at κth time interval. |

pp | Initial population number. |

U_{s} | Phase voltage at the source node s. |

U_{d} | Phase voltage at the load node d. |

U_{o} | Phase voltage at the observation node o. |

${S}_{d}^{*}$ | Conjugate complex draw power at node d. |

${U}_{d}^{*}$ | Conjugate phase voltage at node d. |

Z_{od} | Shared impedance between node d and observation node o from the source node s. |

${P}_{j}^{\mathrm{S}+}$ | Rated upper limit of the charge-discharge active power of the ESS at the FCS j. |

${P}_{j}^{\mathrm{S}-}$ | Rated lower limit of the charge-discharge active power of the ESS at the FCS j. |

${P}_{j\kappa}^{\mathrm{Sd}}$ | Dynamic lower limit of the charge-discharge active power of the ESS at the FCS j at the κth time interval. |

${P}_{j\kappa}^{\mathrm{S}}$. | Active power output of the ESS at the FCS j at κth time interval. |

${\xi}_{j}^{\mathrm{Ps}}$ | A preset constant for restraining ${P}_{j\kappa}^{\mathrm{S}}$. |

SOC | State of charge. |

${\mathrm{SOC}}_{j0}$ | The initial SOC of the ESS at the FCS j at the beginning of the day. |

${\xi}_{j}^{\mathrm{SOC}}$ | A preset constant for restraining the SOC for the FCS j. |

${SOC}_{j}^{+}$ | Rated upper limit of the SOC. |

${SOC}_{j}^{-}$ | Rated lower limit of the SOC. |

${SOC}_{j\kappa}$ | Value of the SOC of the ESS at the FCS j at κth time interval. |

Γ(.) | Gamma function. |

Φ(.) | Normal probability distribution function. |

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**Figure 11.**(

**a**) Voltage curves of node 18 under different cases in a day. (

**b**) Differences between actual and reference voltages under different cases in a day.

**Figure 12.**(

**a**) The active power of the ESS at the FCS A under case 2. (

**b**) The SOC of the ESS at the FCS A under case 2.

**Figure 13.**(

**a**) The active power of the ESS at the FCS B under case 2. (

**b**) The SOC of the ESS at the FCS B under case 2.

Parameter | Description | Unit | Value |
---|---|---|---|

S^{Trans} | Rated capacity of the service transformer. | kVA | 1000 |

S^{PV} | Rated capacity of the PV. | kWp | 50 |

P^{S+} | Rated upper boundary of the charge-discharge active power of the ESS. | kW | 250 |

P^{S−} | Rated lower boundary of the charge-discharge active power of the ESS. | kW | −250 |

SOC_{0} | The initial SOC for the ESS at the beginning of the day. | % | 50 |

SOC^{+} | Rated upper boundary of the SOC of the ESS. | % | 30 |

SOC^{−} | Rated lower boundary of the SOC of the ESS. | % | 80 |

E^{S} | Rated capacity of the ESS. | kWh | 1000 |

**Table 2.**The probability distribution of EV load change at the FCS A in different time intervals in a day.

κ | The EV Load at κth Time Interval/kW | |||
---|---|---|---|---|

1~4 | N(789,192^{2}) | N(532,218^{2}) | N(167,121^{2}) | E(40) |

5~8 | E(40) | E(40) | N(223,40^{2}) | N(108,74^{2}) |

9~12 | N(107,95^{2}) | N(190,132^{2}) | N(294,159^{2}) | N(406,153^{2}) |

13~16 | N(547,144^{2}) | N(572,169^{2}) | N(575,185^{2}) | N(352,198^{2}) |

17~20 | N(207,163^{2}) | N(306,145^{2}) | N(312,163^{2}) | N(309,180^{2}) |

21~24 | N(328,180^{2}) | N(350,143^{2}) | N(494,154^{2}) | N(690,180^{2}) |

**Table 3.**The probability distribution of EV load change at the FCS B in different time intervals in a day.

κ | The EV Load at κth Time Interval/kW | |||
---|---|---|---|---|

1~4 | N(789,192^{2}) | N(532,218^{2}) | N(167,121^{2}) | E(40) |

5~8 | E(40) | E(40) | E(40) | N(108,74^{2}) |

9~12 | N(107,95^{2}) | N(190,132^{2}) | N(294,159^{2}) | N(406,153^{2}) |

13~16 | N(547,144^{2}) | N(572,169^{2}) | N(575,185^{2}) | N(352,198^{2}) |

17~20 | N(207,163^{2}) | N(306,145^{2}) | N(312,163^{2}) | N(309,180^{2}) |

21~24 | N(328,180^{2}) | N(350,143^{2}) | E(40) | N(690,180^{2}) |

Observation Node | The Correlation Coefficient | |||||||
---|---|---|---|---|---|---|---|---|

2~9 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

10~17 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

18~25 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

26~33 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

Methods | Average Time, s | CPU, % |
---|---|---|

MCPFL | 1.527 | 2 |

AMM | 20.313 | 11 |

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## Share and Cite

**MDPI and ACS Style**

Ye, R.; Huang, X.; Yang, Z.
Energy Management Method for Fast-Charging Stations with the Energy Storage System to Alleviate the Voltage Problem of the Observation Node. *World Electr. Veh. J.* **2021**, *12*, 234.
https://doi.org/10.3390/wevj12040234

**AMA Style**

Ye R, Huang X, Yang Z.
Energy Management Method for Fast-Charging Stations with the Energy Storage System to Alleviate the Voltage Problem of the Observation Node. *World Electric Vehicle Journal*. 2021; 12(4):234.
https://doi.org/10.3390/wevj12040234

**Chicago/Turabian Style**

Ye, Rui, Xueliang Huang, and Zexin Yang.
2021. "Energy Management Method for Fast-Charging Stations with the Energy Storage System to Alleviate the Voltage Problem of the Observation Node" *World Electric Vehicle Journal* 12, no. 4: 234.
https://doi.org/10.3390/wevj12040234