Size Estimation of Bulk Capacitor Removal Using Limited Power Quality Monitors in the Distribution Network

Abstract

With a large number of distributed generators (DG) and sensitive power loads connected to the distribution network, power quality issues have increasingly become the focus of users’ attention. Accurate and quick estimation of the amount of bulk capacitor removal that causes voltage sag is helpful to maintain power quality management equipment in time. This paper presents a novel size estimation of bulk capacitor removal using a limited power quality monitor (PQM) in the distribution network, including PQM deployment optimization, feeder localization, and capacitor removal amount calculation. The PQM placement is optimized by taking the estimated capacitance removal sizes of all buses as a constraint. The change of reactive power consumption before and after removing the capacitor at each power line is adopted to determine the feeder where the disturbance is located. Based on the impedance characteristics of the power grid components, the steady estimation method (SEM) is deduced using the fundamental voltage and current. Applying the sampling points of instantaneous voltage and current waveform, the transient estimation method (TEM) is constructed by data fitting. Case studies and index analysis for the IEEE 13 bus test work are presented to verify the reasonableness and accuracy of the proposed method for disturbed bus, capacitor size, load symmetry, disturbance duration, and DGs. SEM shows more stability and accuracy, while TEM performs faster and is more robust. The new methods provide a reliable and acceptable disturbance size estimation with several limited PQMs.

1. Introduction

With the gradual penetration of distributed generators (DG) in the distribution network, more and more capacitors have been installed at the distribution bus for the purpose of improving power quality [1,2,3]. Power disturbance resulting from inadvertent tripping of the bulk capacitor may cause adverse impacts on customer equipment such as tripping of adjustable speed drivers, damaging of the programmable logic controller, and misalignment of the precision instrument [4,5]. The first step for decreasing the adverse impacts of capacitor removal is to correlate them to the specific capacitor bank that has not been energized. Then, using some approaches, such as the quick toggle switch, pre-withdrawal reactor, and resistor, one can reduce the active and reactive power consumption [6,7]. For the above steps, quickly detecting the disturbance start time [8], determining the disturbance location [9], and estimating the capacitance removal are the important prerequisites [10,11,12]. In addition, deploying as few power quality monitors (PQM) as possible makes a lot of sense for the wide distribution network [6,13].

The frequent unexplained removal or damage of bulk capacitors will cause severe voltage sags, thereby exposing the equipment of the grid and the user to risk. It is important to quickly locate power lines with large-capacity trips and estimate the size of capacitor removal, which is helpful for operators to maintain equipment in a timely manner and prevent accidents from worsening. This is of great significance for the safe and stable operation of the power grid and the safety and normal production of users’ equipment.

There is some research on this subject. However, the estimation of bulk capacitors causing power disturbance under the limited PQMs is rarely considered [14,15]. Combining the PQM of the active power output of a limited number of generators and the impedance matrix of the system, a two-stage process method that can simultaneously estimate the time, size, and location of a disturbance was carried out in [6,16]. Based on that, a novel method of localization and size estimation for power loss was set forth by phasor measurements received in the control center within a reasonable wait time following the event inception in [17]. The above methods are only suitable for the transmission network that is fully symmetrical for all phases but not for the distribution networks. A capacitor loss measurement system for power electronics converters was pointed out in [18]. Using spread spectrum time domain reflectometry and dictionary matching in [19], a method for capacitance estimation has been presented. The methods in [18,19] are only suitable for estimating capacitance changes in numbers of μF and pF at very low voltage levels, respectively. An analytical expression of a switched capacitor as a function of transient frequency was presented in [20]. Unfortunately, this method does not lend itself to real-world applications, as it needs to find eigenvalues from system dynamic equations. The backward Kalman filter was applied in [21] to estimate the condition parameter of a switched capacitor bank by estimating the voltage rise of the capacitor bank. This solution is also impractical to implement because it is based on the assumption that an exact power system dynamic model exists. Based on the empirical observation and the assumption that all equivalent circuit elements [20], such as line inductance, resistance and capacitor bank size, and load levels, are already known, a technique for tracing the disturbed capacitor bank was developed in [22] by estimating the feeder line impedance. However, the full coverage of PQM required in this method is difficult to achieve in the distribution network [23].

In [24], a technique based on bounded exhaustive search (BES) and connectivity and symmetry constraints was introduced to determine the possible optimal placement of the PQM, which reduced the search space in solving a minimum number of PQM allocations whilst maintaining full system observability [25]. A multi-objective optimization involving the number of PQM, monitor overlapping index, and sag severity index was solved in [26]. In [27], a multi-objective evolutionary algorithm with table (MEAT) was adopted to allocate the PQM considering a minimize cost of monitoring. In [28], an optimal monitoring of voltage sag for all types of faults was achieved. Moreover, there is no PQM optimization aimed at estimating capacitance loss.

Therefore, given the challenges of estimating bulk capacitance removal in the distribution network with an extremely wide area, this paper proposes a new method for determining the disturbance location and estimating the bulk capacitor loss with a limited number of PQMs. On the premise that the capacitance estimation method proposed in this paper can estimate the capacitance removal of the multi-level adjacent buses through the PQM installed at one bus, a PQM optimization configuration strategy aiming at capacitance estimation is proposed. The location of the disturbance source is realized through the change of reactive power consumed by each power line before and after the capacitor is removed. Following the location, two capacitor loss estimation methods are proposed for quickly detecting and estimating the size of capacitor removal. The bulk capacitance removal amount in the distribution network can be estimated by the two together.

The method improves over existing methods: (1) the bulk capacitor removal amount can be quickly estimated with acceptable errors, (2) avoiding the use of complex signal processing algorithms to monitor the start and end times of power quality disturbances, and (3) only a limited number of buses must be equipped with PQMs.

The main body of this paper is structured as follows: the methodology and steps of capacitor loss estimation are presented in Section 2. The results and case studies are discussed in Section 3. Finally, the paper is concluded in Section 4.

2. Methodology

2.1. Capacitor Removal Estimation

The methodology of bulk capacitor removal estimation is divided into three steps (see Figure 1): (1) optimizing the PQM deployments; (2) localizing the power line where capacitor disturbance occurs; and (3) calculating the amount of capacitor removal.

2.1.1. PQM Deployment Optimization

The capacitance disturbance that occurred at the adjacent buses can be obtained by using the PQM data when adopting the capacitor removal estimation method proposed in this paper (see Section 2.1.3 for details). The PQM deployment aimed at capacitor removal amount estimation is modified based on the covering and packing problem described in [29].

The optimized PQM allocation can be expressed in Equations (1) and (2) as follows.

$$min\sum _{i=1}^{N_B}{c}_i{x}_i={\mathbf{C}}^{\mathbf{T}}\mathbf{X}$$

(1)

$$subjectto\mathbf{DX}\ge \mathbf{U}$$

(2)

where $N_B$ is the number of buses in the power network, and $\mathbf{X}$$(N_B\times 1)$ is the optimization variable, which means that a PQM has been installed at bus i if $x_i$ = 1; otherwise, there is no PQM installed at bus i if $x_i$ is equal to 0. $\mathbf{C}$$(N_B\times 1)$ is the corresponding cost of installing a PQM at each bus; other factors have little effect compared to the value of PQM. All $c_i$ are considered to be the same in this paper. $\mathbf{D}$$(N_B\times N_B)$ is the density matrix, which depends on the power network, and its element is zero or unity. $\mathbf{U}$$(N_B\times 1)$ is a vector whose elements are all ones.

Basically, the elements of $x_i$ and $d_{ij}$ can refer to [30]. Compared with the transmission network, the power lines in the distribution network are extremely short, and the impedance of the power line is much smaller than that of the load. In such cases, the elements of $\mathbf{D}$ can be further improved to Equation (3) when the estimation accuracy of the capacitance is not high, which greatly decreased the number of PQM.

$$d_{ij}=\left\{\begin{array}{c}1,\left\{\begin{array}{c}i=j\hfill \\ iisadjacenttoj\hfill \\ iisadjacenttokandjisadjacenttok\hfill \end{array}\right.\hfill \\ 0,otherwise\hfill \end{array}\right.$$

(3)

$Gurobi$, a new generation, large-scale mathematical programming optimizer developed by the American Gurobi Company, is adopted to solve the above binary integer programming problem in this paper.

2.1.2. Disturbed Power Line Location

Due to the existence of single-phase loads in the distribution network, the minimum unit of the capacitance disturbance location and removal amount estimation in this paper is set to a certain phase of the power line monitored by PQMs. The format of PQM data used for analysis is stored as shown in

Table 1. The format of PQM data used for analysis.

Power Line	Voltage	Current	Phase Difference
1	$u_{1A}\left(t\right)$	$i_{1A}\left(t\right)$	${\varphi }_{1A}$
2	$u_{1B}\left(t\right)$	$i_{1B}\left(t\right)$	${\varphi }_{1B}$
3	$u_{1C}\left(t\right)$	$i_{1C}\left(t\right)$	${\varphi }_{1C}$
4	$u_{2A}\left(t\right)$	$i_{2A}\left(t\right)$	${\varphi }_{2A}$
⋮	⋮	⋮	⋮

. For each voltage waveform, Hilbert–Huang transform (HHT) is adopted to detect the start time of disturbance.

After reading the data of the file in the above format, its reactive powers before and after removing the capacitor are calculated by taking the serial number of the power line as the unit. The power line with the largest reactive power change before and after the disturbance is selected to be the potential candidate where the capacitor removal may exist. All phases of power lines corresponding to potential candidates are considered as the location of the disturbed power lines.

2.1.3. Capacitor Removal Calculation

The effects of resistance and/or reactance in series with the capacitance are ignored in this paper, since its impedance is much smaller than that of the latter. As shown in Figure 2, the PQM installed at bus 1 can monitor all the power lines connected it. So, here, we take bus 2 as an example only as the capacitors removal at bus 1 can be easily calculated if it becomes true at bus 2.

A.

Steady-state estimation

For a long duration disturbance, a steady-state estimation method (SEM) is adopted. Before and during the capacitor disturbance, the relationship between steady-state voltage $\dot{U}$ and current $\dot{I}$ at bus 1 can be expressed as follows:

$$\dot{U}=\dot{I}\left[(R_{12}+j\omega L_{12})+\frac{1}{\frac{1}{(R+j\omega L)}+j\omega C}\right]$$

(4)

where $R_{12}$ and $L_{12}$ are the equivalent resistance and inductance of the power line, and $\omega$ is the angular frequency of the power system. R, L and C are the whole equivalent resistances, inductances, and capacitors connected at bus 2.

$R_{12}$ and $L_{12}$ are usually negligible since they have extremely small values compared to the latter ones. Therefore, Equation (4) can be approximated by Equation (5), and the steady-state form before and during capacitance removal can be expressed in Equation (6).

$$\frac{\dot{I}}{\dot{U}}=\frac{1}{R+j\omega L}+j\omega C$$

(5)

$$\left\{\begin{array}{c}\frac{{\dot{I}}_1}{{\dot{U}}_1}=\frac{1}{R+j\omega L}+j\omega C_1\hfill \\ \frac{{\dot{I}}_0}{{\dot{U}}_0}=\frac{1}{R+j\omega L}+j\omega C_0\hfill \end{array}\right.$$

(6)

where $\dot{X}=X\angle \varphi$ (X = U, I), and $\varphi$ is the angle of the voltage that leads the current. ${\dot{U}}_1$ and ${\dot{I}}_1$ are the voltage and current, respectively, monitored by PQM at bus 1 during the bulk capacitor removal, ${\dot{U}}_0$ and ${\dot{I}}_0$ are the corresponding parameters in steady state, and $C_0$ and $C_1$ are the before and during equivalent capacitors separately.

The capacitor removal amount ($\mathsf{\Delta }$C in Equation (7)) can be obtained from Equation (6), as shown below:

$$\mathsf{\Delta }C=C_0-C_1=\frac{\frac{{\dot{I}}_0}{{\dot{U}}_0}-\frac{{\dot{I}}_1}{{\dot{U}}_1}}{j\omega }$$

(7)

The real part of $\mathsf{\Delta }C$ of each phase is considered as its capacitor removal amount. The accuracy of the capacitor removal amount is high when a long disturbance (more than one power frequency cycle) occurred, because the stable voltage and current vectors in Equation (7) can be achieved during the duration. However, it does not work well in the case of short disturbance. Hence, the following solution is used to solve those cases.

B.

Transient Estimation

For the extreme short duration disturbance, a transient estimation method (TEM) is achieved. The time domain expression of Equation (4) is shown in Equation (8). Same as the previous, its simplified form is shown in Equation (9) by ignoring the effect of $R_{12}$ and $L_{12}$.

$$\begin{array}{c}\hfill u\left(t\right)+RC\frac{du\left(t\right)}{dt}+LC\frac{d^{2}u\left(t\right)}{d{t}^2}=(R+R_{12})i\left(t\right)+(L+L_{12}+R{R}_{12}C)\frac{di\left(t\right)}{dt}\\ \hfill +(R{L}_{12}+R_{12}L)C\frac{d^{2}i\left(t\right)}{d{t}^2}+L{L}_{12}C\frac{d^{3}i\left(t\right)}{d{t}^3}\end{array}$$

(8)

$$u\left(t\right)+RC\frac{du\left(t\right)}{dt}+LC\frac{d^{2}u\left(t\right)}{d{t}^2}=Ri\left(t\right)+L\frac{di\left(t\right)}{dt}$$

(9)

where $u\left(t\right)$ and $i\left(t\right)$ are the time domain expressions of $\dot{U}$ and $\dot{I}$, and $\frac{d^{m}x\left(t\right)}{d{t}^m}$ is the mth (m = 1, 2) derivative of $x\left(t\right)$ (x = u, i).

As expressed in Equation (A1), R and L can be deduced by using $x\left(n\right)$, the discrete values of $x\left(t\right)$. Furthermore, R and L can be eliminated by comparing two adjacent $x\left(n\right)$ with equal length, resulting in a system of equations containing only one unknown C, as shown in Equation (A2). As a result, C is calculated by Equation (A3) using the least squares method.

The respective steady-state capacitors are calculated using the data before and during the disturbance according to Equation (A3). The amount of capacitor removal can be obtained by subtracting the above two capacitors.

2.2. Method Application

2.2.1. Limited PQM Placement

Figure 3 shows the IEEE 13 bus distribution network system preset by Real System Computer Aided Design (RSCAD) software, in which bus 671a is different from the conventional topology. The $\mathbf{D}$ matrix in Equation (2) corresponding to this system is as follows:

$$\mathbf{D}=\left[\begin{array}{ccccccccccccccc}\mathrm{Bus}& \mathbf{650}& \mathbf{632}& \mathbf{633}& \mathbf{645}& \mathbf{671}\mathbf{a}& \mathbf{646}& \mathbf{671}& \mathbf{684}& \mathbf{680}& \mathbf{692}& \mathbf{611}& \mathbf{652}& \mathbf{675}& \mathbf{634}\\ \mathbf{650}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{632}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\\ \mathbf{633}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\\ \mathbf{645}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{671}\mathbf{a}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{646}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{671}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}\\ \mathbf{684}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}\\ \mathbf{680}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{692}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}\\ \mathbf{611}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}\\ \mathbf{652}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}\\ \mathbf{675}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}\\ \mathbf{634}& \mathbf{0}& \mathbf{1}& \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\end{array}\right]$$

(10)

The optimal placement results for PQM is listed in

Table 2. The optimal placement results for PQM.

Bus	650	632	633	645	671a	646	671	684	680	692	611	652	675	634
X	0	1	0	0	0	0	1	0	0	0	0	0	0	0
DX	1	2	1	1	2	1	2	1	1	1	1	1	1	1

. The element of $\mathbf{X}$ to be 1 means the corresponding bus is configured with a PQM. An element no less than one in the covering result vector $\mathbf{DX}$ means its capacitor removal amount can be estimated by at least one PQM. The bulk capacitor removal for any bus in the network (shows in Figure 3) can be obtained after the limited PQMs are installed at buses 632 and 671.

2.2.2. Disturbance Localization

This example, for the IEEE 13 bus system in Figure 3, demonstrates the localization of disturbance that occurred at bus 692. The parameters setting of the test system is listed in

Table 3. The parameters setting of the test system.

Parameter	Value	Parameter	Value	Parameter	Value
rated voltage (L-L)	4.16 kV	system frequency	60 Hz	duration	2$\mathrm{s}$
capacitor removal	3000 $\mathsf{\mu }$F	sampling frequency	16,667 Hz	* start phase	0${}^{\circ }$

* start phase: voltage phase angle of bus 632 A-phase when capacitance removal is performed.

.

The reactive power changes of the power lines are listed in

Table 4. The reactive power changes of the power lines.

Power Line	Power/kVar	Power Line	Power/kVar	Power Line	Power/kVar
632to633A	0.003	632to633B	0.002	632to633C	0.003
632to671aA	10.125	632to671aB	14.91	632to671aC	16.893
632to650A	−10.128	632to650B	−14.908	632to650C	−16.752
∖	∖	632to645B	−0.05	632to645C	−0.144
671to692A	12.489	671to692B	19.891	671to692C	23.531
671toLdA	0.016	671toLdB	0.007	671toLdC	−0.023
671toLyA	0.0	671toLyB	0.0	671toLyC	−0.001
671to671aA	−12.401	671to671aB	−19.908	671to671aC	−23.432
671to680A	0.0	671to680B	0.0	671to680C	0.0
671to684A	−0.108	∖	∖	671to684C	−0.092

(the two regions separated by the dotted line represent the two PQMs) and the max value, which is 23.502 kVar at power line 671to692C. Hence, the three phases at 671to692 are all selected as the candidate locations.

2.2.3. Capacitor Removal Amount

In this subsection, the two solutions for capacitor removal amount estimating are adopted.

Table 5. The capacitor removal amount under different estimation methods.

Estimated Method	671to692A	671to692B	671to692C
SEM	3002.062 + 47.488 j	2997.811 + 66.432 j	3012.031 + 42.31 j
TEM	2915.89	2990.901	2964.647

j: unit of the imaginary part of the complex number. The unit of the estimated capacitor in the table is $\mathsf{\mu }$F

lists the statistical results for capacitor removals when the conditions are consistent with the previous.

It can be concluded that the capacitor removal at power line 671to692 is very close to the actual value under the above two estimation methods, and the value derived from the steady-state method is closest.

3. Result and Discussion

3.1. Worked Example

Simulations of the IEEE 13-bus test system, as shown in Figure 4, are used here to verify the performance of the proposed method for the location and estimation of capacitor removal under the limited PQMs. Different from Figure 3, the photovoltaic (PV) and energy storage system (ES) are connected at bus 634 in Figure 4 to simulate the extensive new energy access in the current distribution network. The connections of each bus are also marked. The other parameters of the components in the system are the default values.

Using the location method mentioned above in this paper, 10 groups of capacitance disturbance values of 1000–10,000 $\mathsf{\mu }$ F are set in steps of 1000 $\mathsf{\mu }$F, which is turned off at each of the 14 buses shown in Figure 4 when it has been turned on for 10 s. The forward zero-crossing points of the voltage (A-phase voltage of bus 632) and current (A-phase current of three-phase capacitor or the own current of single-phase) are selected as the switch time for turning on and off the capacitor. The positioning accuracy of the disturbed power line is 100% for the above 140 disturbances. Therefore, the rest of this section mainly discusses the performance of the capacitance removal estimation method.

3.2. Impacts on Estimation Method

3.2.1. Disturbance Bus

Taking three different types of bus (bus 671, bus 692 and bus 675) monitored by PQM2 as the research object, each value between 500 and 10,000 $\mathsf{\mu }$F was set as the capacitance disturbance in steps of 200 $\mathsf{\mu }$F. Figure 5 shows the estimation error of TEM and SEM under each capacitor.

According to Figure 5, the following observations can be concluded: (1) the maximum relative errors of TEM and SEM are 0.158 and 0.131, respectively, and the maximum error of SEM is significantly lower than that of TEM; (2) the estimated value of TEM is slightly smaller than the true one in most of the cases, and the estimations of SEM at the buses closest to PQM2 are consistent with the deviation direction of TEM; it shows a tendency to be larger than the true value when the buses are farther from PQM2 for SEM; (3) the estimation errors at bus 692 are small for both TEM and SEM.

3.2.2. Capacitor Size

The impacts of capacitor size on TEM and SEM can be derived from the above case. When the size of the capacitor removal increases from 500 $\mathsf{\mu }$F to 1000 $\mathsf{\mu }$F, the overall errors of the TEM are larger, and the law that it is affected by the capacitance size is not obvious. The relative error of SEM is mostly within 0.1, and it shows a trend of increasing with capacitance removal size.

3.2.3. Load Symmetry

Figure 6 shows the impacts of the load symmetry on the aforementioned estimation methods with a capacitor removal of 5000 $\mathsf{\mu }$F at bus 692. It can be concluded that the load symmetry has little effect on the estimation values of TEM and SEM. Compared with the load balance state, the response time estimated by TEM has different degrees of lag when the load is unbalanced, but that for SEM is only slightly affected.

3.2.4. Disturbance Duration

In this example, the capacitor removal amount of 2000 $\mathsf{\mu }$F occurred at bus 633. All breakers for renewable energies are turned off, and the point on wave for cutting off the capacitor is 0 ${}^{\circ }$. We take the duration 0.01 s (shorter than a circle) and 1 s (longer than a circle) as examples; the disturbance power line locations are both extremely correct, and Figure 7 shows the voltage waveform and the responding capacitor removal amounts estimated by TEM and SEM (the widths of the shaded boxes in (a) and (b) represent the disturbances duration).

It can be see from Figure 7 that: (1) the estimation values of SEM and TEM are both close to 2000 (true capacitor removal amount) when the duration is 1 s; (2) TEM estimates the credible value faster than SEM under the long duration; (3) SEM is invalid when a short duration disturbance occurred, while TEM shows much better robustness.

3.2.5. Distributed Generator

In order to research the influence of the DG connected to the distribution network on the capacitance estimation algorithm, photovoltaic and energy storage systems with rated capacities of 1 MW and 2 MW are connected to the bus 634, respectively. Each value between 1000 and 10,000 $\mathsf{\mu }$F was set as the capacitance disturbance in steps of 500 $\mathsf{\mu }$F at bus 633 when the state-of-charge of the energy storage system stays in the 60–80% range. The reactive power outputs of the DGs are fixed to 0, while the reference solar intensity and temperature of the photovoltaic system are set to 1000 Watts/m${}^2$ and 25 ${}^{\circ }$C, respectively. The maximum power point tracking (MPPT) is activated for the active power output mode. Figure 8 shows the errors of TEM and SEM when the charge–discharge power of the energy storage system changes from 0 to the rated power when the capacitor removal amount is 500 $\mathsf{\mu }$F.

It can be seen from Figure 8 that: (1) the relative errors decrease with the increasing capacitance removal size; the accuracy of SEM and TEM when DG is connected is lower than that without DG comparing Figure 5 (about 0.01 at bus 692) and Figure 8; SEM and TEM are little affected by the DGs operating mode and power.

3.3. Discussion

In this subsection, some important considerations are discussed for better real-world application of the proposed method. The method proposed in this paper has the following four limitations: (1) There is a small degree of lag among the three phases in unbalanced load for TEM. (2) The accuracy of the method proposed in this paper will be greatly reduced when the size of capacitance removal is too small. (3) The result of the proposed localization method is a feeder connected to the bus with the PQM installed, not a certain bus. (4) Since the resistance and inductance of the feeder are ignored, the accuracy of this method will be greatly affected when used on long lines.

It takes time to obtain the stable amplitude of voltage and current in Equation (7), which results in a time of one power frequency cycle to obtain the stable capacitance disturbance. For all cases, the moment of forwarding zero-crossing of the current was selected as the switch time for the capacitor removal. Each sampling point can affect the result of TEM, which leads to the different degrees of lag in unbalanced load. However, for SEM, one power frequency cycle of time is needed to obtain the voltage and current amplitude, which show more powerful inertia. Despite its fast response time as well as high and stable accuracy in most situations, the maximum error of TEM reaches about 0.16, which has a lot to do with the ignoring of inductance and resistance of power lines ($R_{12}$ and $L_{12}$ in Figure 2). The location of the capacitor removal result from the PQM data is a feeder rather than a bus; a more precise localization algorithm under limited PQM will be discussed in future study.

In practical implementation, the stability and accuracy of SEM are recommend to combine with the fastness and robustness of SEM. In addition, it can be seen from Figure 6 and Figure 7 that the fast response of TEM to abnormal events can be used as an effective detection method for the start time of power disturbance.

4. Conclusions

This paper proposes a novel algorithm for bulk capacitor removal locating and estimating in distribution systems. The following conclusions have been reached:

A capacitance removal estimation method including PQM deployments optimization, feeder localization and capacitor removal size calculation is pointed out. Based on the impedance characteristics of the power grid components, the calculation method for the size of the capacitor removal named SEM is deduced using the fundamental voltage and current in the stable process of the power grid before and after removing the capacitor. Using the sampling points of instantaneous voltage and current waveform, the capacitance removal estimation method named TEM is constructed by data fitting.

A comparison study regarding the estimation of disturbance bus, capacitor size, load symmetry, disturbance duration and DG between SEM and TEM is performed for a limited number of PQMs. SEM shows more stability and accuracy, while TEM is faster and more robust. In conclusion, the new methods provide a reliable and acceptable disturbance size estimation with fewer PQMs.

In future research, we will further optimize the localization method to obtain a smaller range of perturbed locations. At the same time, we will focus on the effect of the parameters of feeders on the size of capacitance removal.