Methods of unbalanced voltage control in ... - Wiley Online Library

Received: 23 January 2016

Revised: 3 October 2016

Accepted: 9 October 2016

DOI: 10.1002/etep.2301

RESEARCH ARTICLE

Methods of unbalanced voltage control in nonsolidly grounded systems based on current injection Yongduan Xue1

| Jing Xie2 | Bingyin Xu3

1

College of Information and Control Engineer, China University of Petroleum, Qingdao, China

2

State Grid Yantai Power Supply Company, Yantai, China

3

Smart Grid Research Center, Shandong University of Technology, Zibo, China Correspondence Yongduan Xue, College of Information and Control Engineer, China University of Petroleum, Qingdao, 266580, China. Email: [email protected]

Summary In nonsolidly grounded systems, 3‐phase voltage imbalances endanger the system safety. On the basis of the principle of unbalanced voltage control through injecting current, the variation of unbalanced voltage along with the injected current's amplitude and phase was studied. Three practical closed‐loop unbalanced voltage control methods are given, such as the 2‐point method, the tracking method, and the synthetic method. The synthetic method combines the advantages of the 2‐point method and the tracking method and is fast to control with a high precision. Compared with traditional methods, the proposed methods advance in means such as higher precision, more flexible control, less investments, and better adaptation in systems with dynamic changes. The methods are verified in designed simulation. K E Y WO R D S

active compensation, current injection, nonsolidly grounded system, unbalanced voltage, voltage control

1 | IN T RO D U C T IO N The smart distribution grids call for greater demands on power quality and safety performance,1,2 and in such cases, unbalanced voltage control (UVC) is an important aspect in future distribution network studies. Factors including poor line transposition of overhead lines, unequal parameters (including length) of 3 single‐core cables in the 3‐phase cable lines, 2‐phase 2‐wire networks, and unbalanced installation of coupling capacitors can contribute to the imbalance of 3‐phase impedance and 3‐phase admittance to the ground.3–6 The neutral voltage would appear in nonsolidly grounded systems, which leads to 3‐phase voltage unbalances. In resonant grounded systems, the voltage resonance between the Petersen coil's inductance and the lines' 3‐phase capacitance to ground would increase the magnitude of the unbalanced voltage.7,8 This unbalanced voltage has no effect on user's power supply but can endanger power lines, power transformers, potential and current transformers, arresters, etc. The Chinese standard requires that the unbalanced voltage must

be less than 15% of the rated phase‐to‐ground voltage. Therefore, it is necessary to eliminate the unbalanced voltage. Existing unbalanced voltage limiting methods are as follows: (1) line transpositions and bus phase transposing—the former is difficult to achieve and needs heavy workload, and the latter produces poor effect; (2) installing coupling capacitors—it cannot adapt to the systems with frequent structure changes, and it can even aggravate 3‐phase voltage unbalances; and (3) deviating Petersen coil from the resonance point, or increasing the damping rate appropriately— this method has limited effect because it can only reduce the amplification effect of Petersen coil. At present, more flexible and more effective methods to control the unbalanced voltage are desiderated in this field. The active compensation is a novel technique that has been used in the arc suppression of single phase‐to‐earth fault in nonsolidly grounded systems.9–14 It can compensate both reactive and active components of fault current, whereas Petersen coil15,16 can only compensate the reactive component. This technique can reduce residual current of fault point to the

This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non‐commercial and no modifications or adaptations are made. © 2016 The Authors. International Transactions on Electrical Energy Systems published by John Wiley & Sons Ltd.

Int Trans Electr Energ Syst. 2017;27:e2301. https://doi.org/10.1002/etep.2301

wileyonlinelibrary.com/journal/etep

1 of 11

XUE

2 of 11

minimum and increase the probability of arc extinguishing. Active compensation devices can provide a path whose impedance is infinite (do not change the system grounding) for injected zero‐sequence current to eliminate unbalanced voltages. Publication17,18 proposed the base theory to control unbalanced voltage by current injection. However, its analysis of UVC principle ignores the impact of unbalanced 3‐phase impedance, so the relationship between the current injection and the unbalanced voltage is not accurate. In another hand, its calculation of current injection depends on the accurate 3‐phase admittance to ground, so its UVC method cannot adapt to the system with frequent structure changes. Thus, the base theory remains to be further improved. In Section 2 of this article, the analysis of UVC principle that considered the impact of both unbalanced 3‐phase impedance and 3‐phase admittance to ground is given. In Section 3, the accurate relationships between the unbalanced voltage along with the injected current's amplitude and phase are given separately. In Section 4, 3 practical close‐loop UVC methods free form 3‐phase impedance and 3‐phase admittance to ground are given to adapt to frequent system structure changes, such as the 2‐point method, the tracking method, and the synthetic method. And they have their own respective advantages. In Section 5, the UVC methods are verified by simulation.

2 | P R I NC I P L E O F UVC BAS E D O N C UR RE N T I NJ E CT IO N The 3‐phase admittance to ground is the only path for the grounding current during normal operations in isolated neutral systems. An unbalanced voltage is generated to keep the sum of unbalanced 3 phase‐to‐ground currents to zero when the 3‐phase admittance to ground is unbalanced. The

FIGURE 1

Distribution network containing active compensation device

ET AL.

Petersen coil becomes a new path connecting the system neutral point and ground during normal operations of resonant grounded systems. A part of the unbalanced 3 phase‐to‐ ground current flows through the new path. However, the unbalanced voltage is increased because of the voltage resonance between the Petersen coil's inductance and the lines' 3‐phase capacitance to ground. The unbalanced voltage will be limited to zero if a zero impedance path between the neutral point of the system and the ground is provided for the unbalanced 3 phase‐to‐ground currents. However, the grounding mode is changed to the solidly grounded mode under this method. Thus, the ideal UVC method should provide a path whose the impedance is infinite between the neutral point of the system and the ground for the unbalanced 3 phase‐to‐ ground currents and eliminate the unbalanced voltage without changing the grounding mode at the same time. As a current source, active compensation devices could meet the previously mentioned requirements. The distribution network containing active compensation devices is shown in Figure 1. Here, vectors E_ A, E_ B, and E_ C are supply phase voltages of the grid. Vectors U_ A , U_ B , and U_ C are phase‐to‐ground voltages, respectively. Vector U_ 0 is the unbalanced voltage. K is the switch of Petersen coil. gL and L are the loss conductance and the tuning inductance of the Petersen coil, respectively. Vector I_ i is the injected current from the active compensation device. The system has n feeders. Vectors I_ ALi , I_ BLi , and I_ CLi are phase load currents of feeder i. Vectors I_ Ai , I_ Bi , and I_ Ci are phase cur_ Bi, and U′ _ Ci are phase‐ _ Ai, U′ rents at the head of feeder i. Vectors U′ to‐ground voltages at the end of feeder i. RAi, RBi, and RCi and LAi, LBi, and LCi are phase resistance and phase inductance of feeder i, respectively. gAi, gBi, and gCi and CAi, CBi, and CCi are phase conductance to ground and phase capacitance to ground of feeder i, respectively. I_ Ai , I_ Bi , and I_ Ci can be expressed as

XUE

ET AL.

3 of 11

8  1
> _ Ai ðgAi þ jωC Ai Þ; > > I_ Ai ¼ I_ ALi þ U_ A þ U′ > > 2 > <  1
_ Bi ðgBi þ jωC Bi Þ; I_ Bi ¼ I_ BLi þ U_ B þ U′ > 2 > > >  > 1
> _ Ci ðgCi þ jωC Ci Þ: : I_ Ci ¼ I_ CLi þ U_ C þ U′ 2

G¼ (1)

1 n ∑ ðg þ jωCAi Þ2 ðRAi þ jωLAi Þ 4 i¼1 Ai 1 n þ ∑ ðgBi þ jωC Bi Þ2 ðRBi þ jωLBi Þ 4 i¼1 1 n þ ∑ ðgCi þ jωC Ci Þ2 ðRCi þ jωLCi Þ 4 i¼1

(8)

2.1 | UVC in isolated neutral system with K open Vector I_ ∑ is the sum of all feeders' phase‐to‐ground current, which can be expressed as follows:  1 n
_ Ai ðgAi þ jωC Ai Þ I_ ∑ ¼ ∑ U_ A þ U′ 2 i¼1  1 n
_ Bi ðgBi þ jωC Bi Þ þ ∑ U_ B þ U′ 2 i¼1  1 n
_ Ci ðgCi þ jωC Ci Þ þ ∑ U_ C þ U′ 2 i¼1

d0 ¼

(3)

Here, U_ A ¼ E_ A þ U_ 0 ; U_ B ¼ E_ B þ U_ 0 ; U_ C ¼ E_ C þ U_ 0

ðgA þ jωC A Þ þ a2 ðgB þ jωCB Þ þ aðgC þ jωC C Þ (9) jωC ∑

(2)

Without injecting current, according to the Kirchhoff's law, I_ ∑ is equal to zero: I_ ∑ ¼ 0

u_ 0 ¼

(4)

g∑ ωC ∑

(10)

Equations 7 and 8, which are parts of Equation 6, are represented by I_ ia and G, because Equation 6 is too long to be expressed completely. Vector u_ 0 is the unbalance degree. d0 is the damping rate of the isolated neutral system and usually       equals to 1.5% to 2.0%. U ph ¼ E_ A  ¼ E_ B  ¼ E_ C  is the magnitude of the rated phase supply voltage and is a constant value when the voltage level of distribution network is fixed. pffiffiffi pffiffiffi a ¼ −1=2 þ j 3=2 and a2 ¼ −1=2−j 3=2. C∑ = CA + CB + CC and g∑ = gA + gB + gC are the sum of 3‐phase capacitance to ground and the sum of 3‐phase ground conductance n

n

n

respectively. C A ¼ ∑ C Ai , C B ¼ ∑ C Bi , and C C ¼ ∑ C Ci i¼1

  8 1_ > _ _ _ > > U′Ai ¼ U A − I Ai −2U A ðgAi þ jωC Ai Þ ðRAi þ jωLAi Þ; > > > >   < 1_ _ _ _ U′Bi ¼ U B − I Bi − U B ðgBi þ jωC Bi Þ ðRBi þ jωLBi Þ; > 2 > >   > > > > _ Ci ¼ U_ C − I_ Ci −1U_ C ðgCi þ jωC Ci Þ ðRCi þ jωLCi Þ: : U′ 2

_

1−jd0 þ

G jωC ∑

n

i¼1

i¼1

i¼1

of phase ground capacitances and the sum of phase ground conductance, respectively. With injecting current, the relationship between I_ i and _I ∑ is

The unbalanced voltage can be derived from Equations 2–5: U_ 0 ¼ −

i¼1

n

and gA ¼ ∑ gAi , gB ¼ ∑ gBi , and gC ¼ ∑ gCi are the sum

(5)

I ia u_ 0 U ph þ jωC ∑

i¼1

n

(6)

I_ i ¼ I_ ∑

(11)

According to Equations 2, 4, 5, and 11, the relationship between I_ i and U_ 0 is
I_ i ¼ U_ 0 gΣ þ jωC ∑ þ GÞ þ u_ 0 U ph ⋅jωC ∑ þ I_ ia

n  n n 1 I_ ia ¼ U ph ∑ ðgAi þ jωC Ai Þ2 ðRAi þ jωLAi Þ þ a2 ∑ ðgBi þ jωC Bi Þ2 ðRBi þ jωLBi Þ; þa∑ ðgCi þ jωCCi Þ2 ðRCi þ jωLCi Þ 4 i¼1 i¼1 i¼1   n n 1 n _ _ _ − ∑ I Ai ðgAi þ jωC Ai ÞðRAi þ jωLAi Þ þ ∑ I Bi ðgBi þ jωC Bi ÞðRBi þ jωLBi Þ þ ∑ I Ci ðgCi þ jωC Ci ÞðRCi þ jωLCi Þ 2 i¼1 i¼1 i¼1

(12)

(7)

XUE

4 of 11

2.2 | UVC in resonant grounded system with K closed The unbalanced voltage in resonant grounded system is represented as vector U_ 0P to be distinguished from U_ 0 in isolated neutral system. U_ 0P is _

U_ 0P ¼ −

I ia u_ 0 U ph þ jωC ∑ G v−jd þ jωC ∑

(13)

where v is the out‐of‐resonance degree of Petersen coil. d is the damping rate of the resonant grounded system. According to Equation 13, the amplitude of U_ 0P increases with the decrease of |v| and d. Petersen coil has amplification effect on unbalanced voltage. With the current injection, the relationship between I_ i and _I ∑ is I_ i ¼ U_ 0P



1 1 þ jωL rL



þ I_ ∑

(14)

According to Equations 2, 4, 5, and 14, the relationship between I_ i and U_ 0P is  1 I_ i ¼ U_ 0P gL þ þ gΣ þ jωC∑ þ G þ u_ 0 U ph ⋅ jωC ∑ þ I_ ia jωL

ET AL.

magnitude of I_ ia is equal to the parts of Δ I˙, which are determined by phase current I_ Ai, I_ Bi, and I_ Ci and phase supply voltage E_ A, E_ B, and E_ C. On the basis of Equations 1, 6, and 13, the load current affects the unbalanced voltage through voltage drop on line impedance. Because of the 3‐phase ground conductance is very small, the effect of unbalanced 3‐phase conductance to ground can be disregarded. Thus, it can be treated approximately as gA = gB = gC, and u_ 0 can be simplified to u_ 0 ¼

C A þ a2 C B þ aC C C∑

(19)

If ignoring the impedance of lines, I_ ia ¼ 0 and G = 0. Equations 12, 15, and 16 can be simplified as
I_ i ¼ U_ 0 gΣ þ jωC ∑ Þ þ u_ 0 U ph ⋅ jωC ∑

(20)

 1 I_ i ¼ U_ 0P gL þ þ gΣ þ jωC ∑ þ u_ 0 U ph ⋅jωC ∑ (21) jωL I_ i ¼ u_ 0 U ph ⋅jωC ∑

(22)

(15) On the basis of Equations 12 and 15, we can control the unbalanced voltage to any value by changing the amplitude and phase of the injected current. The I_ i to control both U_ 0 in isolated neutral systems and U_ 0P in resonant grounded systems to zero is I_ i ¼ u_ 0 U ph ⋅jωC ∑ þ I_ ia

(16)

In Figure 1, vector Δ I˙, which is the loss of 3 phase‐to‐ ground currents caused by the voltage drop on line impedance, can be expressed as   n
 1 1 ΔI_ ¼ ∑ I_ Ai − ðgAi þ jωC Ai Þ U_ 0 þ E_ A ðRAi þ jωLAi Þ⋅ ðgAi þ jωC Ai Þ; 2 2 i¼1   n
 1 1 þ∑ I_ Bi − ðgBi þ jωCBi Þ U_ 0 þ E_ B ðRBi þ jωLBi Þ⋅ ðgBi þ jωC Bi Þ; 2 2 i¼1   n
 1 1 þ∑ I_ Ci − ðgCi þ jωCCi Þ U_ 0 þ E_ C ðRCi þ jωLCi Þ⋅ ðgCi þ jωC Ci Þ: 2 2 i¼1

(17) It has E_ A ¼ U ph, E_ B ¼ a2 U ph, E_ C ¼ aU ph. Thus, (17) can be also expressed as ΔI_ ¼ −I_ ia −U_ 0P G

(18)

In Equation 18, U_ 0P can be replaced by U_ 0 in isolated neutral system. On the basis of Equations 7, 17, and 18, the

The calculation of Equations 20–22 is simplified, but certain errors exist in the results. Using the injected current will not control the unbalanced voltage up to the intended target precisely. 3 | VA RIAT ION O F U NBA LANC ED VO LTAG E AL O NG WI TH I NJ E CTE D CU RR EN T The injected current only needs to compensate the unbalanced part of 3 phase‐to‐ground currents. It's much smaller than the feeders' phase current, which contains load currents and phase‐to‐ground currents. During a short time, the parameters of lines and loads in the distribution system can be considered unchanged. I_ Ai , I_ Bi , and I_ Ci in Equation 7 can be treated as constants. Thus, considering impedance of lines, I_ ia can be approximated as a constant phasor and G is a constant admittance. Comparing Equations 12 and 20, or Equations 15 and 21, the variation tendency of unbalanced voltages along with the injected current when considering that the impedance of lines is the same with the variation tendency when ignoring line impedances, only their amplitudes of unbalanced voltages are different under the same injected current. According to Equations 20 and 21, the variation tendency of unbalanced voltage along with injected current is same in isolated neutral system and resonant grounded systems. The only difference is that, corresponding to any I_ i, the amplitude

XUE

ET AL.

5 of 11

of U_ 0P is k times as bigger as the amplitude of U_ 0 . k is the amplification factor of unbalanced voltage. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ d 20 k¼ v2 þ d 2

(23)

Take the condition of ignoring the impedance of lines in resonant grounded system, for example, to simplify the analysis process. The analysis results can be applied to different situations. To be more intuitive and easier to understand, a 35‐kV distribution network shown in Figure 2 is taken as an example during analysis process, and the parameter‐ changing curves of the unbalanced voltage as well as the amplitude and phase of the injected current are shown pictorially. The network in Figure 2 consists of 4 overhead lines and one 3‐phase cable line constituted by 3 single‐core cables. The overhead lines are all not transposed, and parameters of the 3‐phase single‐core cables are the same except for the length. The parameters of overhead lines are zero‐sequence resistance R0 = 0.23Ω/km, zero‐sequence inductance L0 = 5.475mH/km, and 3‐phase capacitance to ground CA = 322.613nF, CB = 311.231nF, CC = 299.848nF. The cable line's parameters are zero‐sequence resistance R0 = 2.7Ω/km, zero‐sequence inductance L0 = 0.250mH/km, and 3‐phase capacitance to ground CA = 2518.722nF, CB = CC = 2798.581nF (the lengths of single‐core cables in phase A, B and C are 9 km, 10 km, 10 km respectively). Both the phase impedance and the ground admittance are unbalanced under this condition, but the phase combination has minimized the unbalance degree. The out‐of‐resonance degree of the Petersen coil v is −8%. The damping rate of the Petersen coil d is 6%. The unbalance degree u_ 0 is 0.0291 ∠ − 177.9°, and the injected current I_ i to control unbalanced voltage to zero is 1.669 ∠ − 87.9 ° A. From Equation 21, the amplitude of U_ 0P can be expressed as

FIGURE 2

A 35‐kV distribution network

U 0P

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2 I 2i þ u20 U 2ph ωC ∑ −2I i u0 U ph ωC ∑ sinðφi −φu0 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r

¼ 2
 1 2 gL þ g∑ þ ωC ∑ − ωL (24)

Here, Ii and φi are the amplitude and phase of I_ i , respectively. u0 and φu0 are the amplitude and phase of u_ 0. The variation of U0P along with the amplitude and phase of the injected current is shown in Figure 3. 3.1 | Variation of unbalanced voltage along with amplitude of injected current Take the derivative of U0P with respect to Ii in Equation 24:

I i −u0 U ph ωC ∑ sinðφi −φu0 Þ ∂U 0P ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
∂I i  1 2 gL þ g∑ þ ωC ∑ − ωL h

(25)

i−12
2 I 2i þ u20 U 2ph ωC ∑ −2I i u0 U ph ωC ∑ sinðφi −φu0 Þ

According to Equations 24 and 25, the variation of U0P along with Ii under different given phases of the injected current is shown as curves C1 to C7 in Figure 4. 1. If φu0 < φi < φu0 + π, the variation is described as follows: a. If 0 ≤ Ii < u0UphωC∑ sin (φi − φu0), ∂U∂I0Pi u0UphωC∑ sin (φi − φu0), ∂U∂I0Pi >0 . U0P increases with increasing Ii (such as the monotone increasing parts of C1 to C3 in Figure 4). c. If Ii = u0UphωC∑ sin (φi − φu0), ∂U∂I0Pi ¼ 0 . U0P gets the minimum Ii (such as points on C8 in Figure 4).

XUE

6 of 11

FIGURE 3

ET AL.

Relationship between unbalanced voltage and injected current

Relationship between unbalanced voltage and amplitude of injected current

FIGURE 4

U 0P min


 u0 U ph ωC ∑  cos φi −φu0  ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
 1 2 gL þ g∑ þ ωC ∑ − ωL

(26)

U0Pmin and its corresponding Ii change as cosine and sine function with the symmetry axis φi ¼ φu0 þ 12 π, respectively. They are shown in Figure 5. When φi ¼ φu0 þ 12 π, Ii gets the maximum

2. If φu0 + π ≤ φi ≤ φu0 + 2π, sin(φi − φu0) < 0 and ∂U∂I0Pi > 0. U0P increases with increasing Ii (such as the monotone increasing parts of C4 to C7 in Figure 4). Under this condition, the unbalanced voltage would be larger than the initial unbalanced voltage without the injected current. 3.2 | Variation of unbalanced voltage along with phase of injected current Take the derivative of U0P with respect to φi in Equation 24:

I i max ¼ u0 U ph ωC∑

(27)

At this point, U0Pmin = 0. The unbalanced voltage is eliminated.

−I i u0 U ph ωC ∑ cosðφi −φu0 Þ ∂U 0P ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (28)

2
∂φi  1 2 gL þ g∑ þ ωC ∑ − ωL h i−12
2 I 2i þ u20 U 2ph ωC ∑ −2I i u0 U ph ωC ∑ sinðφi −φu0 Þ

Minimum of unbalanced voltage and corresponding amplitude of injected current with different given phase of injected current

FIGURE 5

XUE

ET AL.

7 of 11

According to Equations 24 and 28, variations of U0P along with φi under different given amplitudes of injected current are shown as curves C1 to C6 in Figure 6. 0P 1. If φu0 − 12 π < U_ 0P2 ¼ I_ i2 −u_ 0 U ph ⋅ jωC ∑ > I_ i −u_ 0 U ph ⋅jωC ∑ U_ 0PT > > > ¼ : _ U 0P2 I_ i2 −u_ 0 U ph ⋅jωC ∑

(31)


 U_ 0PT I_ i1 −I_ i2 þ U_ 0P1 I_ i2 −U_ 0P2 I_ i1 I_ i ¼ U_ 0P1 −U_ 0P2

(32)

(30)

If Ii = u0UphωC∑, U0Pmin = 0 (such as minimum point of C4 in Figure 6). If Ii > u0UphωC∑, U0Pmin increases with increasing Ii (such as minimum points of C5–C6 in Figure 6). If 0 < Ii < u0UphωC∑, U0Pmin increases with decreasing Ii (such as minimum points of C2–C3 in Figure 6).

FIGURE 6

of such methods is difficult. The following methods can be used to improve the practicability and precision.

When U_ 0PT ¼ 0, U_ 0P1 I_ i2 −U_ 0P2 I_ i1 I_ i ¼ u_ 0 U ph ⋅jωC ∑ ¼ U_ 0P1 −U_ 0P2

(33)

If the unbalanced voltage changes during the current injection, it means that the unbalance condition of the system is affected by structure and parameter changes in the system. The present injected current and changed unbalanced voltage can be chosen as I_ i1 (I_ i1 can also be zero at the initial state) and U_ 0P1. The injected current only needs to be changed once to calculate I_ i . Considering the impedance of lines, I_ ia can be approximated as a constant phasor, and Equation 31 can be modified as

XUE

8 of 11

8
 > U_ 0P1 I_ i1 − u_ 0 U ph ⋅ jωC ∑ þ I_ ia > >
 > < U_ 0P2 ¼ I_ i2 − u_ 0 U ph ⋅ jωC ∑ þ I_ ia
 > I_ i − u_ 0 U ph ⋅jωC ∑ þ I_ ia U_ 0PT > > >
 ¼ : _ U 0P2 I_ i2 − u_ 0 U ph ⋅jωC ∑ þ I_ ia

(34)

The solution of I_ i is still the same to Equation 31. That is, Equation 32 has spontaneously considered the error current I_ ia caused by line impedances. Thus, the 2‐point method is also applicable to models considering line impedances. 4.2 | Tracking method of UVC Figure 4 shows that there is 1 and only 1 amplitude of the injected current corresponding to the minimum unbalanced voltage under any fixed phase of the injected current. Figure 6 shows that the phases of the injected current corresponding to the minimum unbalanced voltage are the same under different fixed amplitudes of the injected current. We start with an arbitrary initial injected current I_ 1. First, fix the amplitude of injected current and change its phase gradually, the phase amplitude that leads to the local minimum unbalanced voltage is found and denoted as φf. Second, fix the phase found in the first step and change its amplitude gradually, the phase amplitude that leads to the global minimum unbalanced voltage is found and denote it as If. I_ f (amplitude If, and phase φf) is the injected current to control the unbalanced voltage to zero. The previously mentioned tracking method can also be used to control the unbalanced voltage to any nonzero target value. 4.3 | Synthetic method of UVC The 2‐point method and the tracking method have their own advantages and disadvantages. To fully use the advantages of them and make up for their disadvantages, the synthetic method combines the 2 methods: set the injected current calculated by the 2‐point method as the initial injected current I_ 1 of the tracking method. The error of the 2‐point method can be corrected by the tracking method. The unbalanced voltage can be controlled to any target value quickly. The injected current calculated by Equation 22 can also be set to the initial injected current I_ 1 if the phase‐to‐ground admittance can be measured. For the calculation error of Equation 22 is larger than that of the 2‐point method, its control time is longer. 4.4 | Performance analysis In the 2‐point method, measurements of the ground admittance and the phase impedance are not required. It has simple calculation, fast control speed and can be adapted to dynamic

ET AL.

system structure changes. However, the control precision may be affected by errors from active compensation devices, potential transformers, and other equipment. In the tracking method, the phase and amplitude of the injected current should been changed step by step based on the variation of unbalanced voltage along with the injected current and control the unbalanced voltage to zero. This method also does not need ground admittance and phase impedance measurements. It could overcome errors from active compensation devices, potential transformers, and other equipment. It can be applied to models that considering line impedances as well and has high precision and stability. However, in this method, the injected current needs to be changed step by step. The control time may be long if the choice of the initial injected current is unreasonable. The control precision may also be affected by adjustment step length of the injected current. The synthetic method can reduce the control time by decreasing the change range of the injected current, improve the control precision by reducing the adjustment step length of the injected current, and overcome the errors caused by equipment to guarantee control precision. The result of synthetic method will be better than the others. When the topology and parameters of distribution systems changed, amplitudes of unbalanced voltages are different under the same injected current, but the variation tendency of unbalanced voltage along with the injected current is same. The unbalanced voltage can be controlled to the target value according to methods in this article. In resonant grounded systems, the measurement and automatic tuning of some Petersen coils relay on unbalanced voltages. Thus, the target value of the unbalanced voltage should be set as small as possible on the premise that the unbalanced voltage meets the requirement of Petersen coil. The zero‐sequence voltages caused by different causes at the neutral point should call for different control method. If the phase‐to‐ground parameters of lines are unbalanced in normal steady states, the neutral voltage should be controlled small to keep the balance of 3 phase‐to‐ground voltages. If the phase‐to‐ground fault occurs, the neutral voltage should be controlled high enough to make the voltage of fault point close to zero and reduce the residual current to minimum, or to enable the normal operation of protection. Thus, the causes of neutral voltage should be found before using methods in this article. The phase‐to‐ground conductance of the system can be significantly increased by the transition resistance when the phase‐to‐ground fault occurs in the system. This can be used to distinguish the causes of neutral voltage. The methods proposed can be implemented by developing software based on existing active compensation devices such as residual current compensation mentioned by Winter,12 the new type principal‐auxiliary arc suppression coil mentioned by Qu et al,14 and other similar devices.

XUE

ET AL.

9 of 11

5 | SIMULATION AND V ER IF ICAT IO N The simulation model of a 35‐kV distribution network is shown in Figure 2. Three groups parameters under different unbalance conditions are given in Table 1. The comparison performance of the 3 methods proposed in this article and the method used Equation 22 based on the results of EMTP/ATP are given in Table 2. Here, the lengths of single‐core cables in 3‐phase are all 10 km, and only the phase admittance to ground is unbalanced in group 1. The lengths are 9.25, 10.25, and 10 km, respectively, in group 2, and 9, 10, and 10 km, respectively, in group 3, with both the phase impedance and the admittance to ground unbalanced. Here, method I is injecting the current calculated by Equation 22 into the system immediately. Method II is the 2‐point method. Method III is the tracking method with the initial injected current calculated by method I. Method IV is the synthetic method. The step size of the injected current in methods III and IV are the same: phase Δφ = 0.1° and magnitude ΔI = 0.1 mA. The control time depends on the initial injected current and the step size. The calculation of the control time in the simulation is the product of the change times of the injected current and the average time (approximately 0.4 s in simulation) required by the unbalanced voltage to reach the steady state after injecting the current in each time. The control time in method III is the longest for the largest error between the initial injected current and the final injected current and the small step size of the injected current. Take group 3 in Table 1, for example, Figure 7 shows waveforms of unbalanced voltages and 3‐phase voltages after injecting the final current immediately in method IV, and

TABLE 1

Table 3 shows the performance of traditional control methods mentioned in the introduction section under the unbalance condition of group 3. Here, transposing bus line phases changes the phase combination of lines to decrease the degree of imbalance. Because the line phase combination under the unbalanced condition of group 3 in Table 1 has already made the unbalance degree to minimum, it would only increase the unbalanced voltage. Three‐phase coupling capacitors are installed at the head of lines. With tuning Petersen coil to v = − 50% and d = 20%, the amplification factor of the unbalanced voltage k is reduced from 10 to 1.86. It does not consider the phase impedance and loads in Equation 22, so method I in Table 2 cannot control the unbalanced voltage to zero and will produce large errors. Methods II, III, and IV can control the unbalanced voltage to approximately zero because line parameter measurements are not required in these methods and can overcome errors caused by phase impedances and loads. In practical applications, although the control time of method II is short, errors caused by equipment have effect on its precision. Method III can overcome the disadvantage of method II, but the time consumption is much higher when the recovery time of unbalanced voltage and the control time of active compensation device are considered. Method IV can greatly shorten control time and overcome all kinds of errors. Its performance is superior to method I, II, or III. Comparing Table 3 with group 3 in Table 2, the performance of practical closed‐loop control methods of unbalanced voltages, particularly the synthetic method, is much better than traditional methods. Transposing bus line phases can only adjust the 3‐phase unbalanced parameters roughly

Three groups of parameters under different unbalance conditions A‐phase capacitance to ground CA, nF

B‐phase capacitance to ground CB, nF

C‐phase capacitance to ground CC, nF

Unbalance degree, %

1

3098.429

3121.669

3121.669

0.0025∠−180.0°

2

2888.535

3238.358

3098.429

0.0331∠−156.6°

3

2841.335

3109.812

3098.429

0.0291∠−177.9°

Group

TABLE 2

Group

Simulation results Method

Final injected current, A

Unbalanced voltage, V U_ 0P before injected current

U_ ′ 0P after injected current

_′  U 0P  U_ 0P ,

Control time, s

%

1

I II III IV

0.1475 0.1451 0.1452 0.1452

∠−90.0° ∠−94.2° ∠−94.3° ∠−94.3°

476.59∠134.8°

37.58 ∠34.5° 0.72 ∠42.3° 0.29 ∠−69.1° 0.29 ∠−69.1°

7.89 0.15 0.06 0.06

0.4 0.8 28.0 2.0

2

I II III IV

1.9360 1.6119 1.6094 1.6094

∠−66.6° ∠−70.9° ∠−70.8° ∠−70.8°

5354.21∠158.2°

1169.55 ∠2.2° 10.92 ∠−63.2° 2.12 ∠75.4° 2.12 ∠75.4°

21.84 0.20 0.04 0.04

0.4 0.8 1324.8 12.4

3

I II III IV

1.6694 1.3780 1.3792 1.3792

∠−87.9° ∠−91.4° ∠−91.4° ∠−91.4°

4690.24 ∠137.8°

1038.74 ∠−22.4° 5.09 ∠99.6° 3.20 ∠46.7° 3.20 ∠46.7°

22.15 0.11 0.07 0.07

0.4 0.8 1181.2 6.0

XUE

10 of 11

FIGURE 7

TABLE 3

ET AL.

Unbalanced voltage and 3‐phase voltage around injecting current

Simulation results of traditional methods

Method

Transpose bus line phases

Unbalanced voltage (V) U_ 0P before taking measure

U_ ′ 0P after taking measure

4690.24∠137.8°

More than 4690.24

More than 100.00

1243.09∠−54.4°

26.50

Install 3‐phase coupling capacitors Tune Petersen coil

_′  U 0P  U_ 0P , %

900.15∠156.0°

and cannot adapt to the frequent changes of system structures. It has no effect when the phase combination of lines has already made the unbalance degree to minimum. Coupling capacitors are generally installed in station, and the phase capacitance to ground of system needs to be measured. This method also cannot overcome errors caused by phase capacitance and loads and also cannot adapt to the dynamic change of system structures. Its effect is similar to method I's effect. Tuning Petersen coil can only reduce its amplification factor k, which is more than 1 in general instead of eliminating the unbalanced voltage. Although its effect is better than the effect of installing 3‐phase coupling capacitors in simulation, it would be limited in practical applications. Because of arc suppression in single‐phase earth fault, Petersen coil could not deviate so far from resonance point. 6 | C O NC LUS I O N Active compensation devices, which have already been used to compensate fault current in single‐phase earth fault, can control unbalanced voltages with injecting a controllable current. The injected current is far less than the algebraic sum of the 3 phase‐to‐ground capacitive currents. It has the advantage of lower investments and easier realization.

19.19

The following 3 practical closed‐loop control methods can be adopted to eliminate unbalanced voltage: 1. Two‐point method. Injected currents eliminating unbalanced voltage can be calculated by 2 different injected currents and their corresponding unbalanced voltages. Although the errors caused by equipment may affect the control precision, it can overcome errors caused by impedance and loads and has good control effect. The method also does not require line parameter measurements and has simple calculation and fast control speed. 2. Tracking method. Change the phase and amplitude of the injected current gradually in turn to control unbalanced voltages to zero. Although its control time may be longer, it has high precision and stability. The method can be applied without measuring phase capacitance to ground. 3. Synthetic method. Set the injected current calculated by the 2‐point method as the initial injected current in the tracking method. It combines all advantages of the 2‐point method and the tracking method, and avoids their disadvantages. Its control performance is best. Compared with traditional control methods, the 3 control methods based on current injection have advantages of low

XUE

ET AL.

investments, flexible control, high accuracy, and adapt to the dynamic change of system structures automatically. The main works of next step are analysis of the transient characteristics as well as development and application of devices. AC KNOWLEDGMENTS

The authors thank the National Natural Science Foundation of China (NSFC) (grant no. 51477184) and the National High Technology Research and Development Program of China (863 program) (grant no. 2012AA050213) for their financial support for this study. The authors extend special thanks to the anonymous reviewers for their constructive comments and suggestions in improving the quality of this article. REFERENCES 1. Han D, Yan Z. Evaluating the impact of smart grid technologies on generation expansion planning under uncertainties. Int Trans Electr Energy Syst. 2015. doi: 10.1002/etep.2115 2. Staroszczyk ZT. Smart grids instrumentation‐obtaining subgrid impedance information. Int Trans Electr Energy Syst. 2015. doi: 10.1002/etep.2077 3. Toman P, Drapela J, Orsagova J. Solution of voltage asymmetry and reduction of outage time in MV compensated networks. Proceedings of the 13th IEEE ICHQP, Wollongong, Australia; 2008:1–7. 4. Kalyuzhny A. Analysis of temporary overvoltages during open‐phase faults in distribution networks with resonant grounding. IEEE Trans Power Del. 2015;30(1):420–427. 5. Gustavsen B, Walseth JA. A case of abnormal overvoltages in a Petersen grounded 132‐kV system caused by broken conductor. IEEE Trans Power Del. 2003;18(1):195–200. 6. Niemi R, Lund PD. Alternative ways for voltage control in smart grids with distributed electricity generation. Int J Energy Res. 2011. doi: 10.1002/ er.1865 7. Chen Z, Wang H, Chen F. Research on damping ratio and off‐resonant degree of compensation network[C]. Proc APPEEC. 2011;1–4.

11 of 11

8. Leitloff V, Pierrat L, Feuillet R. Study of the neutral‐to‐ground voltage in a compensated power system. Eur Trans Electr Power Eng. 1994;4 (2):145–153. 9. Yoshihide H. Chapter 8: neutral grounding methods. In: Handbook of Power System Engineering; 2007:143–150. 10. Al‐Abdulwahab AS, Winter KM, Winter N. Reliability assessment of distribution system with innovative Smart Grid technology implementation. In: Proceedings of the 2011 IEEE PES Conference on Innovative Smart Grid Technologies—Middle East, Jeddah, 2011; 1–6. 11. Huang S, Chen Z, Zhuang Y, Bi T. Adaptive residual current compensation for robust fault‐type selection in mho elements. IEEE Trans Power Del. 2005;20(2):573–578. 12. Winter KM. The RCC ground fault neutralizer—a novel scheme for fast earth‐ fault protection. In: Proceedings of the CIRED 2005, Turin, Italy; 2005:1–4. 13. Janssen M, Kraemer S, Schmidt R, Winter K. Residual current compensation (RCC) for resonant grounded transmission systems using high performance voltage source inverter. Proceedings of the 2003 IEEE PES Transmission and Distribution Conference and Exposition; 2003, 2:7–12. 14. Qu Y, Tan W, Yang Y. H‐Infinity control theory apply to new type arc‐ suppression coil system. Proceedings of the IEEE International Conference on Power Electronics and Drive Systems (PEDS'07), Bangkok; 2007:1753–1757. 15. Chaari S, Bastard P, Meunier M. Prony's method: an efficient tool for the analysis of earth fault currents in Petersen‐coil‐protected networks. IEEE Trans Power Del. 1995;10(3):1234–1241. 16. Lin X, Huang J, Ke S. Faulty feeder detection and fault self‐extinguishing by adaptive Petersen coil control. IEEE Trans Power Del. 2011;26 (2):1290–1291. 17. Xue Y, Xu B. Voltage control method in power system with non‐effectively earthed neutral. CN. Patent CN201210196420, Oct. 17, 2012. 18. Zeng X, Hu J, Wang Y, et al. Suppression method of three‐phase unbalanced overvoltage based on distribution networks flexible grounding control. Proc CESS. 2014;34(4):678–684. (in Chinese)

How to cite this article: Xue Y, Xie J, Xu B. Methods of unbalanced voltage control in nonsolidly grounded systems based on current injection. Int Trans Electr Energ Syst. 2017;27:e2301. https://doi.org/ 10.1002/etep.2301