Assessment for Voltage Violations considering Reactive Power Compensation Provided by Smart Inverters in Distribution Network

Due to the high proportion of renewable energies, traditional voltage regulation methods such as on-load tap changers (OLTCs) and switching capacitors (SCs) are currently facing the challenge of providing fast, step-less, and low-cost reactive power to reduce the increasing risks of voltage violations in distribution networks (DNs). To meet such increasing demand for voltage regulation, smart inverters, including photovoltaics (PVs) and electric vehicle (EV) chargers, stand out as a feasible approach for reactive power compensation. This paper aims to assess the voltage violation risks in DNs considering the reactive power response of smart inverters. Firstly, reactive power compensation models of PVs and EV chargers are investigated and voltage deviation indexes of the regulation results are proposed. Moreover, kernel density estimation (KDE) and slice sampling are adopted to provide the PV output and EV charging demand samples. Then, the risk assessment is carried out with a voltage regulation model utilizing OLTCs, SCs, and available smart inverters. Numerical studies demonstrate that the reactive power support from smart inverters can significantly mitigate the voltage violation risks and reduce the switching and cost of OLTCs and capacitors in DNs.


INTRODUCTION
The development of power electronics in distribution networks (DNs) brings prosperity for distributed photovoltaics (PVs), electric vehicle (EV) chargers, and other devices with AC-DC inverters. However, DNs are more prone to voltage violation problems nowadays because of the uncertainty of PV outputs and the load impact of EV charging (Kekatos et al., 2015). Traditional reactive power compensation facilities such as on-load tap changers (OLTCs) and switching capacitors (SCs) are only able to provide step-wise and high-delay reactive power at the feeder head, which limits the regulation effect (Kekatos et al., 2015), while the high cost of Distribution Static Synchronous Compensators (D-STATCOMs) restricts its application in DNs (Chen et al., 2018). Therefore, it is of vital importance to implement step-less compensation facilities at the feeder terminal.
Meanwhile, the two-way reactive power ability of smart inverters (i.e., PVs and EV chargers in this paper) enables these terminal end power electronics in DNs to participate in voltage regulation. There have already been some investigations to achieve reactive power delivery from smart inverters. Sharma and Das (2020) and Feng et al. (2018) extend the reactive power exchange for PV inverters and help to balance the active and reactive power transmission of each phase. Buja et al. (2017) analyze and validate the reactive power compensation abilities of EV chargers theoretically and experimentally. Moreover, Varma and Siavashi (2018), Abeywardana et al. (2018), and Kesler et al. (2014) point out that the reactive power compensation process of PVs and EV chargers does not intervene in the active power delivery or cause damage to EV batteries, which expands the implementation of EV chargers to a large extent. To integrate these flexible power electronics, Varma and Siavashi (2018) present a novel smart inverter PV-STATCOM which controls PV inverters as a dynamic reactive power compensator. Furthermore, Singh et al. (2019) achieve voltage regulation through smart inverters of PVs and EV charging stations in the global as well as local domain. Smart inverters have been well investigated and developed to compensate reactive power (Ustun et al., 2020;Gush et al., 2021). Therefore, it is feasible to resort to smart inverters for accurate and fast voltage regulation in DNs.
The application of smart inverters to regulate voltage quality has been investigated. Zeraati et al. (2019) develop a distributed voltage regulation scheme to utilize the reactive power capability of PV inverters. Quirós-Tortós et al. (2016) and Singh et al. (2019) achieve reactive power control in DNs with the support of EV chargers and PVs. However, the voltage regulation techniques using smart inverters are still under research and need practical implementations. Therefore, it is essential to assess the voltage regulation effects of smart inverters to provide references for the market price of their reactive power compensation.
To assess the reliability of power systems, the Monte Carlo simulation is usually utilized to sample from the target distribution when considering the uncertainty of renewable energies or power demands (Zhou et al., 2016). It is applicable for common Monte Carlo methods, such as importance sampling (Tómasson and Sö) and acceptance-rejection sampling (Hu et al., 2017), to sample from a standard target distribution. However, the irregular PV outputs and EV loads require non-parametric estimation techniques to acquire their probability density functions (PDFs) and more universal sampling methods suitable for any non-standard PDF (Huang et al., 2020). Slice sampling is an advanced method of Markov Chain Monte Carlo (MCMC) simulation (Neal, 2003). It is feasible for PV outputs and EV loads owing to its ability to sample from irregular PDFs efficiently.
Based on the above analysis, this paper proposes a voltage violation assessment model considering the participation of smart inverters. Samples of PV outputs and EV loads are generated. The voltage violation assessment is carried out to evaluate the performance of smart inverters. The main contributions of this paper can be summarized as two-fold: 1. The quantitative compensation ability assessment of smart inverters is proposed with the available reactive power capacities under active power constraints. The optimization model for voltage regulation is established considering both traditional reactive power compensation facilities and smart inverters. 2. The voltage violation assessment under the uncertainty of PV outputs and EV behaviors is achieved based on the proposed voltage deviation indexes, non-parametric kernel density estimation (KDE), and slice sampling. The non-standard PDFs of PVs and EVs are fitted accurately by KDE. Besides, the automated step width selection for slice  sampling is adopted to efficiently generate samples from the obtained PDFs. The results demonstrate that smart inverters have better voltage regulation effects and are able to reduce the operation cost of OLTCs and SCs.
The remaining of this paper is organized as follows. Section 2 describes the basic models and voltage deviation indexes for voltage regulation. KDE and slice sampling are introduced in Section 3. Section 4 illustrates the risk assessment process for voltage violations. Numerical studies are presented in Section 5. Finally, Section 6 concludes this paper.

Reactive Power Compensation Mechanism of Smart Inverters
The AC-DC inverters of EV chargers and PVs are composed of controllable power electronics with a three-phase six-pulse topology. The equivalent smart inverter models are depicted in Figure 1. This structure provides these smart inverters with the flexibility to operate their output voltages. By changing the amplitude and phase angle of the output voltage, the inverter can adjust its active and reactive power transmission. To illustrate the reactive power compensation mechanism of smart inverters, Figure 2 shows two typical operation modes, which provide pure capacitive or inductive reactive power to the grid.
As shown in Figure 2, the reactive power compensation modes are determined by the voltage of smart inverters. The yellow circle range in Figure 2 is drawn based on the limitation of the rated current, which protects the inverter from over-loading. As the inverter voltage varies within this yellow circle, the current changes accordingly. Then, the inverter gives out both active power P and reactive power Q according to the current and the grid voltage. The operation range of P and Q of smart inverters is thus further determined as illustrated in Figure 1 (Kisacikoglu et al., 2013;Su et al., 2014). However, EV chargers and PVs have different operation ranges. EV chargers can operate in four quadrants, which can absorb or give out both P and Q, while PVs do not consume any active power. Therefore, the operation range can be expressed as: where S max is the maximum value of the inverter capacity; P EV and P PV are the active power of EV chargers and PVs respectively; P EV max and P EV min denote the maximum and minimum allowed active charging power for EV chargers respectively; P PV output is the active power generated by PVs.
Eq. 1 indicates that when chargers and PVs are idle, these smart inverters are capable of providing considerable reactive power compensation. This advantage is illustrated by the orange rectangle in Figure 1, where the inverter has a vast operation range under low active power level.

Models of On-Load Tap Changers and Switching Capacitors
OLTCs and SCs are fundamental voltage regulation resources in DNs, which are usually equipped at the transformer substation. The reactive power compensation mechanism for OLTCs is to change the turn ratio by adjusting the tap position. The model of the OLTCs can be expressed as (Wu et al., 2017) where k ij is the turns ratio of the transformer between node i and node j; k min ij is the minimum turns ratio; tap ij is the tap position of the OLTC; tap ij is the maximum value of the tap position; Δk ij is the ratio change per tap.
SCs provide capacitive reactive power to maintain the voltage level. According to the capacity of each SC connected to the grid, the model of SCs can be described as where Q SC is the total reactive power provided by SCs; N SC is the total number of the SC types; Q SC i and n SC i are the capacity and the number of SCs that belong to type i; n SC i is the maximum value of n SC i .

Comparisons of Different Compensation Strategies
Since OLTCs and SCs are commonly located in the transformer substation, traditional compensation methods regulate the voltage levels by injecting reactive power at the head of the feeders, resulting in the difficulty of balancing the voltage of Frontiers in Energy Research | www.frontiersin.org July 2021 | Volume 9 | Article 713510 the whole feeder line. Figure 3 illustrates the effect of different compensation strategies. When the voltage violation occurs, the voltage profile with no compensation adopted drops seriously at the feeder terminal. Although the OLTC and SCs are able to raise the voltage profile as shown in the blue line, they cause over-voltage at the head of feeders because of too much reactive power injection. Smart inverters can regulate the voltages at the demand side in a local way, as they are distributed close to terminal users in DNs. Therefore, smart inverters are capable of compensating reactive power locally as the orange line in Figure 3 depicts. However, in severe voltage violations, smart inverters may not be able to raise all voltages above the safe line due to the limited capacities of demand-side power electronics.
The green line in Figure 3 illustrates that by using both traditional compensation resources and smart inverters, a satisfactory compensation effect can be achieved without causing over-compensation or being constrained by capacities, which is feasible for voltage regulation in DNs.

Voltage Deviation Indexes
To further evaluate the voltage deviation, the voltage violation probability P vio and the expected comprehensive deviation of voltage violations E dev are proposed in this paper. The definition of voltage violations in this paper is the scenario where any nodal voltage exceeds the safe range, which is set as 0.95 p. u. to 1.05 p. u. In MCMC simulation analysis, P vio can be expressed as the proportion of voltage violation scenarios in all simulated scenarios: L vio i 1 , the voltage violation happens 0 , no voltage violation happens (6) where N sim is the number of all simulated scenarios and L vio i marks the simulation result in the i th scenario. E dev describes the deviation degree of the voltage violation. The expression of E dev is where D i is the comprehensive voltage deviation of the i th scenario; N node is the number of nodes in the DN; V j (i) is the nodal voltage of node j in the i th scenario; V ref is the reference nodal voltage, which is set to 1 p. u. in this paper.

SIMULATION METHODS FOR VOLTAGE VIOLATION ASSESSMENT 3.1 Kernel Density Estimation of Electric Vehicle Loads and Photovoltaics Outputs
The KDE method is utilized to model the PDFs of EV loads and PV outputs. Its basic idea is to treat each sample as a kernel function and sum these functions to form an overall PDF. Several load samples and the normal distribution kernel function are selected in Figure 4 to illustrate the mechanism of KDE. Different kernel functions create different PDFs of the sample data. With the kernel function defined as K(·), the PDF estimated by KDE is (Bowman and Azzalini, 1997) f where N sam is the number of sample data; h is the bandwidth; X 1 , X 2 , . . . , X n are the sample data from the target distribution. The standard normal distribution kernel function is adopted in this paper. Hence, Eq. 9 can be specified as To avoid over-smooth or under-smooth in KDE, the bandwidth h is determined according to the formula provided by Silverman (2018).
where σ is the standard deviation of the sample data.

Slice Sampling for Voltage Violation Assessment
It is essential to accurately sample from the PDF obtained by KDE to generate adequate data for MCMC simulation. Slice sampling is an efficient method for handling the continuous PDF with an irregular shape and is mainly composed of two procedures as illustrated in Figure 5 (Neal, 2003).
The first procedure is defined as the step-out process in Figure 5A. Assume x k is the previous sampling result and y k is an auxiliary variable randomly drawn from U[0,f (x k )]. The step-out process is to extend the slice range from the initial point Frontiers in Energy Research | www.frontiersin.org July 2021 | Volume 9 | Article 713510 of x k step by step in a step width of ω until the probability densities of both ends are all below y k . Therefore, the eventual slice range completely contains the PDF range which is above y k . Then, the shrinkage process is carried out to obtain the next sample x k+1 that satisfiesf (x k+1 ) ≥ y k . In Figure 5B, L max 0 and R max 0 are the initial values of the left end and the right end of the slice range respectively. The first sample is drawn from U[L max 0 , R max 0 ] with its corresponding PDF value calculated. If the value is below y k , the new slice range boundary will be updated and the next sample will be drawn according to the new range until the sample satisfiesf (x k+1 ) ≥ y k . The step-out and shrinkage procedures continue to provide samples until the simulation converges.

Automated Step Width Selection for Slice Sampling
The inappropriate value of the step width ω can decrease the sampling efficiency significantly. Therefore, the automated selection mechanism is introduced in this paper to find the optimal ω (Tibbits et al., 2014).
The selection algorithm is shown in Figure 6. This algorithm helps to minimize the effort of step-out and shrinkage operations with several pre-sampling iterations. In each iteration, the numbers of step-out and shrinkage operations are recorded to optimize ω until the tuning iteration converges. After that, the optimal ω is utilized in slice sampling to accelerate the MCMC simulation.

RISK ASSESSMENT FOR VOLTAGE VIOLATION
In this section, the tap position of the OLTC and the reactive power from smart inverters and SCs are regarded as manipulated variables to regulate voltage violations in DNs. As different EV loads and PV outputs are sampled from the slice sampling methods, these manipulated variables are optimized to minimize the voltage deviations in each simulation. P vio and E dev are also calculated during the simulation. When P vio and E dev converge, the MCMC simulation stops and outputs the final risk assessment results.

Voltage Regulation Models
The voltage regulation models proposed in this paper only utilize the remaining reactive power capacities of smart inverters. The total reactive power compensation provided from node i is where Q i j is the reactive power by the j th smart inverter of node i and N SI i is the number of smart inverters in node i. Besides, the tap position and n SC i of SCs are also controlled in the voltage regulation. Their models are expressed in Eq. 3 and Eq. 4. Based on the above equations, the power flow model is (Saadat, 2011) where P load i and Q load i are the active and reactive power consumed by loads in node i respectively; V i is the nodal  voltage of node i; θ ij is the voltage phase angle difference of node i and node j; G ij and B ij are the conductance and susceptance of the nodal admittance matrix elements in row i and column j respectively.

Voltage Regulation Constraints
Eq. 3 and Eq. 4 show the operational constraints of the OLTC and SCs. The operation range of smart inverters is where (S j,max i ) 2 is the maximum capacity of the j th smart inverter of node i and P i j is the active power from the j th smart inverter of node i. Apart from constraints of reactive power compensation resources, the voltage regulation model is also limited by voltage amplitude constraints and transmission capacity constraints: where V i and V i are the minimum and maximum allowed nodal voltages respectively; S ij , P ij , and Q ij are the line apparent power, active power, and reactive power between node i and node j respectively; S ij and S ij are the minimum and maximum allowed line capacity between node i and node j respectively. The voltage regulation aims to minimize the nodal voltage deviation of all nodes with reactive power resources, which can be expressed as

Risk Assessment Process
With the models and methods mentioned above, the risk assessment process is proposed and shown in Figure 7. Firstly, the KDE is carried out to derive the PDFs of EV loads and PV outputs. Then, these PDFs are utilized by the slice sampling with automated width selection to produce data samples for voltage regulation. During the voltage regulation, smart inverters are considered the first choice to reduce the operation cost of OLTCs and SCs. Moreover, voltage deviation indexes are calculated to describe the performance of regulation and estimate the convergence of MCMC simulation. If the convergence requirements are not met, the slice sampling will continue to provide samples for risk assessment. The convergence requirements are satisfied when the accuracy indexes σ(F) of both P vio and E dev are below the threshold σ (Green et al., 2010): where var(F) is the variance calculation function of F.

CASE STUDIES
In this section, the proposed risk assessment is carried out in a modified IEEE 33-bus distribution system (Baran and Wu, 1989). Seven charging stations and 6 PVs are randomly installed along the feeder as Figure 8 shows. Besides, the OLTC and SCs are equipped at the head of the feeder. The capacity of each smart inverter is 550 kVA and the total capacity of SCs is 5 Mvar in this case study.

Kernel Density Estimation Results
The EV loads and PV output measurements are derived from Huang et al. (2020) and UK Power Networks (2017) respectively. The data are normalized and estimated by KDE to obtain the PDFs of EV loads and PV outputs. Figure 9 illustrates the estimation results of KDE. To examine the goodness-of-fit, the chi-square test is performed with the significant level set at 0.05 and the degree of freedom at 49 (Su et al., 2020). The test results of the EV and PV are 23.53 and 42.40 respectively, which are both below the critical value 66.34 and validate the accuracy of KDE.

Performance of Slice Sampling
To compare the sampling efficiency of slice sampling and the classic Monte Carlo simulation (Zhao et al., 2009), both methods are tested on an i5-4590T 2.00 GHz CPU computer through Matlab R2020a and MATPOWER 7.1 (Zimmerman et al., 2011;Zimmerman and Murillo-Sánchez, 2020). The convergence threshold is 0.01. The optimal step widths ω of EV PDF and PV PDF for slice sampling are 0.761 and 0.343 respectively. The simulation results are listed in Table 1.
The slice sampling only consumes 28.7% of the time of the Monte Carlo method to accomplish almost the same number of iterations and the same results of indexes. It takes more time for the classic Monte Carlo method to complete the iterations because this method rejects many unqualified samples during simulation. By contrast, no samples are discarded in slice sampling, which improves the sampling efficiency. Table 2 illustrates three typical cases where only the OLTC and SCs, only smart inverters, and all these facilities are respectively utilized in voltage regulation. Compared with OLTC and SCs, and E dev by using smart inverters decrease 81.52 and 34.28% respectively, which shows that it is more effective to compensate reactive power along the feeder via smart inverters. With the OLTC, SCs, and smart inverters all adopted, P vio drops significantly to almost 0 and E dev also gets mitigated. Besides, p OLTC SC in Table 2 demonstrates that smart inverters are capable of reducing the utilization of the OLTC and SCs, which help to save the operation cost of these traditional compensation facilities.

Sensitivity Analysis
In the sensitivity analysis, the capacities of smart inverters and SCs are set to increase 10% of the original value respectively. Figure 10 depicts E dev under different increments of smart inverters and SCs. P vio is not illustrated since its value approximates zero and rarely varies during the simulation. If the capacity of smart inverters increases 10%, E dev drops by 14.24%, while E dev only drops by 5.52% if the SCs increases 10%. It can be concluded from Figure 10 that under the same increment of capacities, smart inverters perform better in voltage regulation than SCs.

CONCLUSION
This paper establishes a voltage violation assessment model considering the support provided by smart inverters. With the active power samples from KDE and slice sampling, the reactive power compensation process is modeled and implemented to minimize the voltage deviation. Based on the voltage deviation indexes proposed in this paper, the case studies demonstrate that  slice sampling takes less time to converge than the Monte Carlo method. Besides, while OLTCs, SCs, and smart inverters are all essential to avoid voltage violations, smart inverters are more effective than the traditional facilities, which can save the operation cost of these facilities as well. Moreover, the sensitivity analysis shows that it is more beneficial to develop smart inverters than SCs, which validates the prospect of smart inverters for voltage regulation.
In future works, the cooperative control strategies of smart inverters for voltage regulation will be studied. The integration of both active and reactive power deliveries will be considered to further implement the four-quadrant operation characteristic of smart inverters.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.