Transmission Swtiching to Relieve Voltage Violations in Low Load Period

W. M. Guo, Q. Wei Henan Electric Power Research

Institute Zhengzhou 450052, Henan Province,

China. 719277394@qq.com

G. J. Liu , Y. Wang School of Electrical Engineering,

Shandong University Jinan 250061, Shandong Province,

China

X. K. Zhang Shandong Electric Power Corporation Jinan 250001, Shandong Province,

China

Abstract—With the development of modern power system, extra-high voltage (EHV) AC transmission lines are widely used. Because of large capacitive charging, long EHV AC lines may cause voltage violations in low load period. The leading-phase operation of synchronous generators and switching on shunt reactors are two common measures to relieve voltage violations in low load period. In this paper, we propose that opening some EHV lines in low load period, namely transmission switching is a more economic and efficient measure. The problem of finding an optimal topology to relieve voltage violations in low load period is formulated as a mixed integer non linear programming problem. After some simplifications, the model is converted to an easily solved mixed integer linear program. The modified IEEE 30-bus test system is used to evaluate the applicability and effectiveness of the proposed method.

Index Terms-- Transmission switching, voltage violations, EHV power system

I. INTRODUCTION In order to meet the need of long-distance and large-

capability electricity transmission, EHV transmission lines are widely used. While transmitting the load power, EHV transmission lines both produce and consume reactive power. In low load period, the amount of reactive power consumed by EHV transmission lines significantly reduces and the problem of reactive power surplus can be very serious, which may result in network voltage exceeding specified upper limit. For example, it has been reported that the risk of voltage violations is pretty high in some regional power grids of China during the Chinese New Year holiday when the load can decrease as much as 40%.

To absorb surplus reactive power, power system operators usually take the following measures, namely the leading-phase operation of synchronous generators and switching on shunt reactors. However, the leading-phase operation of synchronous generators can influence real power generation, resulting in that power system can’t operate in the most economical way [1]. Besides, the amount of reactive power absorbed by synchronous generators through the leading-phase operation is very limited due to generator static stability

constraints. As for switching on shunt reactors, it needs a large amount of investment. What’s more, a considerable percentage of shunt reactors is switched on only during the low load period and most of time they have to remain idle.

Compared with above two approaches, opening some EHV lines, namely transmission switching, can relieve voltage violations in a more economical way. When an EHV line is open through control of transmission line circuit breakers, the amount of charging power injection is reduced. In addition, as the power carried by this line is shifted to other remaining lines, other remaining lines will consume more reactive power. To implement transmission switching, only some communication and switching equipment are required, most of which have already been installed in nowadays power system.

In fact, optimizing power system operation through transmission switching is not a wholly new problem. In the early 1980s many studies have been conducted dealing with line and bus-bar switching to improve system security, reduce the network loss and/or generation cost, or to relieve voltage violations or line overloads caused by line or transformer outages[2]-[5]. In [6], the problem of finding an optimal generation dispatch and transmission topology to meet a specified load is formulated as a mixed integer linear program, with the binary variables representing the operating state of each transmission line. A significantly saving in generation cost can be achieved if transmission line can be switched offline temporarily compared with the dispatch over a fixed network. More importantly, transmission switching does not intrinsically cause security and reliability problems. In [7], it is proved that transmission switching can reduce generation cost while satisfying strict N-1 standards. In [8], a co-optimization formulation of the unit commitment and transmission switching while ensuring N-1 standards is presented.

Our investigation extends this literature by proving the degree to which transmission switching could relieve EHV network voltage violations in low load period. The difficulties of this problem mainly lie in the discrete performance of switching actions and AC constraints, which are hard to be modeled. Using a binary variable indicating the line is in

Project Supported by National Natural Foundation of China (NSFC) (51077087, 51007047,51177091)

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978-1-4799-2984-9/13/$31.00 ©2013 IEEE

2013 4th IEEE PES Innovative Smart Grid Technologies Europe (ISGT Europe), October 6-9, Copenhagen

service or out of service, in this paper the problem of relieving voltage violations in low load period through transmission switching is formulated as a mixed integer nonlinear program (MINLP). Due to its complexity, this MINLP model is very difficult to solve. In order to apply transmission switching to large scale EHV network, this model is converted to a mixed integer linear program (MIP) based on some simplifications.

Indeed, transmission switching itself represents a step change in power systems, which can be considered to be similar to a contingency. Some may argue that transmission switching can cause potential threats to power system security and reliability, e.g., some unstable transients may get triggered. In this paper, opening some EHV lines is suggested only in low load period, when the possibility of the reduction of system security by such switching actions is relatively low. We are not suggesting relieving voltage violations through transmission switching at the expense of secure and reliable power system operations. EHV transmission lines that are open in low load period may be switched back into the system in other heavy load periods.

This paper is structured as follows: Section II provides a detailed analysis on relieving voltage violations through transmission switching in low load period using a simple 3-bus system. Section III gives a description on the basic MINLP formulation and the MIP formulation for this problem. Section IV presents the results and analysis for the modified IEEE 30-bus system. Section VII summarizes the main findings of this work.

II. RELIEVE VOLTAGE VIOLTATIONS THROUGH TRANSMISSION SWITCHING

The reactive power source in EHV power system mainly consists of synchronous generators, transmission lines, capacitors and reactors. Among them, synchronous generators can generate reactive power in normal operation states and absorb reactive power through leading phase operation. Transmission lines both produce and consume reactive power. Capacitors can only generate reactive power and reactors can only absorb reactive power. Here we mainly focus on transmission lines.

Generally, the reactive power produced by transmission lines is proportional to the square of the voltage and it can be assumed relatively constant considering that the voltage must be kept within about ± 5% of nominal voltage. The reactive power consumed by transmission lines is proportional to the square of the current. Since the current in low load period is much smaller than that in heavy load periods, a large amount of reactive power will be generated and the network voltage may exceed the specified upper limit.

Besides the leading-phase operation of generators and switching on shunt reactors, opening some EHV lines in low load period can also relieve voltage violations. To give a detailed analysis, a 3-bus system shown in Fig.1 is used as an example. The nominal voltage of this 3-bus system is 500kV and the network voltage upper limit is 525kV, namely 1.05pu. The line parameters are obtained from a real 500kV transmission line. Table I, II give the concrete data of the lines and buses respectively.

G

1 2

3

Fig.1. Three-bus system

TABLE I . LINE DATA

From bus To bus R(pu) X(pu) BCAP a(pu) 1 2 0.0022 0.0222 1.0050 1 3 0.0022 0.0222 1.0050 2 3 0.00111 0.0111 0.5025

a BCAP =half total line charging susceptance

TABLE II. BUS DATA

Bus number

Voltage schedule(pu)

Pload (pu MW)

Qload (pu MVAR)

1 1.03 0 0 2 0.5 0.5 3 0.3 0.3

Here we focus on the reactive power flow across transmission line and the Newton power flow results are shown in TABLE III.

TABLE III. TRANSMISSION LINES FLOWS

Bus i Bus j Reactive power flow

from bus i to bus j(pu MVar)

Reactive power flow from bus j to bus i (pu MVar)

1 2 -2.292 0.141 1 3 -2.331 0.180 2 3 -0.641 -0.48

As can be seen from TABLE III the net reactive power value generated by transmission lines reaches as much as 5.423pu. The sum of reactive power demand is 0.8pu and the generator at bus 1 has to absorb as much as 4.623pu reactive power, which may exceed its reactive power absorption capability. Besides, in the given condition, the voltage of bus 2 is 1.0556pu and the voltage of bus 3 is 1.0565pu, both of which have already been above the specified 1.05pu. To relieve voltage violations, some uneconomic generators have to be started or additional shunt reactors have to be switched on.

Considering transmission lines are lightly loaded in low load period, opening some transmission lines won’t cause many threats to the security of the system. For example, the line from bus 2 to bus 3 can be open if the remaining two lines could work reliably. After opening the line from bus 2 to bus 3, the reactive power flow is shown in TABLE IV,.

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TABLE IV. TRANSMISSION LINES FLOWS

Bus i Bus j Reactive power flow

from bus i to bus j(pu MVar)

Reactive power flow from bus j to bus i (pu MVar)

1 2 -1.644 -0.5 1 3 -1.852 -0.3

From the results, it is known that the net reactive power generated by transmission lines drops to 4.296pu. Now the voltage of bus 2 is 1.0415pu and the voltage of bus 3 is 1.0463pu. Both of them are within the specified range. Under this condition, generator at bus 1 absorbs 3.496pu reactive power, which is only 76% of that when all the three lines are closed.

In fact, the network voltage can be very different depending on which line is switched off. The comparison result for the 3-bus system is summarized in TABLE V.

TABLE V. VOLTAGE PROFILE AT EACH BUS

Bus number

Voltage when line 2-3 is open(pu)

Voltage when line 1-2 is open(pu)

Voltage when line 1-3 is open(pu)

1 1.03 1.03 1.03 2 1.0415 1.0587 1.0586 3 1.0463 1.0586 1.0611 As can be seen from TABLE V, relieving voltage

violations in low load period through transmission switching can work only if the right line is open. For example, opening the line from bus 1 to bus 2 or the line from bus 1 to bus 3 will make voltage violations more serious. So it is very necessary to give a systemic search method to find out the sets of lines that should be switched off in low load period.

III. FORMULATION AND MODELING Most of the existing ACOPF methods optimize generator

outputs, transformer taps and reactive power compensation over a fixed network to minimize system operation cost or transmission losses. Transmission switching to relieve voltage violations can also be incorporated into ACOPF with a binary variable zk representing the line is open or closed. Considering AC constraints and binary variables, a basic formulation can be defined as follows:

minMinimize: no ali

i PQV V

∈

−� (1)

. . :s t 2 2 2 2, ,( )k k k ij k ij k kz S P Q z S− ≤ + ≤ (2)

2 2 2 2, ,( )k k k ji k ji k kz S P Q z S− ≤ + ≤ (3)

2, ( cos

sin )k ij k i k i j k ij

i j k ij

P z V g VV g

VV b

θθ

= −

− (4)

2, ( cos

sin )k ji k j k i j k ji

i j k ji

P z V g VV gVV b

θθ

= −

− (5)

2 2, ( sin

cos )k ij k i k i k i j k ij

i j k ij

Q z V b V B VV gVV b

θθ

= − − −

+ (6)

2 2, ( sin

cos )k ji k j k j k i j k ji

i j k ji

Q z V b V B VV gVV b

θθ

= − − −

+ (7)

, ,, ,

g k n k n ndg n k n k n

P P P P∗ ∗∈ ∗ ∗

+ − =� � � (8)

, ,, ,

g k n k n ndg n k n k n

Q Q Q Q∗ ∗∈ ∗ ∗

+ − =� � � (9)

min maxiV V V≤ ≤ (10)

min maxiθ θ θ≤ ≤ (11)

min maxg g gQ Q Q≤ ≤ (12)

Where, Vi denotes the voltage of bus i; Vnominal denotes system nominal voltage and a typical value is 1.0pu; zk denotes the status of transmission line k(open, zk =0;closed, zk =1); Sk denotes the MVA rating of line k; Pk,ij denotes the real power across line k from bus i to bus j; Qk,ij denotes the reactive power across line k from bus i to bus j; gk denotes the conductance of line k; bk denotes the susceptance of line k; Pg denotes the real power generated by generator g; Pnd denotes the real power demand at bus n; Qg denotes the reactive power generated by generator g; Qnd denotes the reactive power demand at bus n; Vmin and Vmax denote minimum and maximum system voltage; �min and �max denote minimum and maximum voltage angle; Qg

min and Qgmax denote the reactive

power generation limits of generator g.

The objective is to minimize the sum of bus voltage deviations. Here, it is assumed that Pg is constant and the control actions are generator reactive outputs and the status of each transmission line. For the sake of simplicity, N-1 security constraints are not included in this model, but there is no reason that they could not be included other than the usual problems with convergence and computation time.

Clearly, the above model is a mixed integer nonlinear program, which is very complex to solve. As far as we know, existing solvers are mainly to solve mixed integer linear programming problems. As for such a complex MINLP with non-convex constraints, there are no solvers to give a satisfactory optimal solution, especially for large scale EHV power systems.

One common approach is to use a linear approximation of this problem. In order to convert the objective into a linear form, two variables �i

+ and �i- are introduced to replace the

absolute value form in (1). These two variables satisfy the following constraints:

( )( )

min

min

max 0,

max 0,

no ali i

no ali i

V V

V V

ω

ω

+

−

� ≥ −��

≥ −�� (13)

Thus the objective changes into the flowing expressions,

Minimize: ( )i ii PQ

ω ω+ −

∈

+�

(14)

After a simple analysis, it can be found that the objective (14) is equivalent to (1).

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Because transmission lines are lightly loaded in low load period, it is assumed that the possibility of line flow violations is pretty low when a limited number of lines are open. Therefore, constraints (2)-(5), (8) and (11) are ignored. To ensure system security, power flow constraints are checked in the end of problem solving procedure. Thus, (6)-(10) and (12) are incorporated into the last model. Among them, (6) and (7) are nonlinear expressions.

For convenience, firstly all transmission lines are assumed to be closed. In (6) and (7), gk can be assumed to be zero as line resistance is much smaller than line reactance for EHV transmission lines. (6) and (7) can be written as:

2 2, cosk ij i k i k i j k ijQ V b V B VV b θ= − − +

(15)

2 2, cosk ji j k j k i j k jiQ V b V B VV b θ= − − +

(16)

As can be seen from (15) and (16), the relation between reactive power flow and bus voltage is nonlinear due to the square of variable Vi and the product of the variables Vi and Vj.

Using the method of Taylor series expansion, Vi 2 and Vi Vj can be written as:

2 2 1.0i iV V= − (17)

1.0i j i jVV V V= + − (18)

In (15) and (16), cos�ij and cos�ji can be assumed to be 1.0 as �ij and �ji are usually small. Substituting (17) and (18) into (15) and (16),

, ( ) 2k ij i j k i k kQ V V b V B B= − − − + (19)

, ( ) 2k ji j i k j k kQ V V b V B B= − − − + (20)

(19) and (20) give the linear approximation of the relation between reactive power flow and bus voltage. From numerical experiments, it is found that they are accurate enough to give a satisfactory solution. What’s more, we will run a Newton power flow program to obtain the exact power flow results after the optimization procedure.

Taking the binary variable zk into account, (19) and (20) can be written as,

, ( ( ) 2 )k ij k i j k i k kQ z V V b V B B= − − − + (21)

, ( ( ) 2 )k ji k j i k j k kQ z V V b V B B= − − − + (22)

They are nonlinear relations for containing the product of continuous and binary variables. In [9], a method to convert this kind of nonlinear relations into linear forms is proposed. Taking zkVi as an example, it can be found that zkVi can be replaced by si , which is a new continuous variable. si satisfies the following constraints:

min maxk i kV z s V z≤ ≤

(23)

max0 (1 )i i kV s z V≤ − ≤ − (24)

After a simple analysis, it can be found that zkVi is equivalent to si. A further detailed description can be found in [9].To limit the number of lines allowed to open, we have,

(1 )kk

z j− ≤�

(25)

Where, j is the number of lines allowed to be open.

To ensure system security, j will be increased gradually so that every time there is only one new line to open, then a transient stability simulation and a Newton power flow program will be run. If there are any security violations, the line will be kept closed and the MIP model will be run again.

IV. IEEE 30-BUS TEST CASE-RESULTS AND ANALYSIS To evaluate the applicability and effectiveness of the

proposed method, the IEEE 30-bus test system is used. The susceptance of transmission line in the standard IEEE 30-bus test system is not as large as that of real EHV transmission lines. In order to simulate an EHV power system, the susceptance of transmission line is intentionally increased. The modified line data is given in TABLE VI.

TABLE VI . MODIFIED LINE DATA

From bus To bus BCAP (pu) From bus To bus BCAP (pu) 1 2 0.005750 15 18 0.002185 1 3 0.001852 18 19 0.001292 2 4 0.001737 19 20 0.006800 3 4 0.003790 10 20 0.002090 2 5 0.001983 10 17 0.008450 2 6 0.001763 10 21 0.007490 4 6 0.004140 10 22 0.001499 5 7 0.001160 21 22 0.002360 6 7 0.008200 15 23 0.002020 6 8 0.004200 22 24 0.001790 9 6 0.002080 23 24 0.002700 6 10 0.005560 24 25 0.003292 9 11 0.002080 25 26 0.003800 9 10 0.001100 25 27 0.002087

12 4 0.002560 28 27 0.003960 12 13 0.001400 27 29 0.004153 12 14 0.002559 27 30 0.006027 12 15 0.001304 29 30 0.004533 12 16 0.001987 8 28 0.002000 14 15 0.001997 6 28 0.005990 16 17 0.001932 Firstly, a comparison is made between the bus voltages

obtained using (17) and (18) and those obtained using the Newton power flow method, which can be considered as actual value. Taking the original network that all lines are closed as an example, the results are shown in Fig.2.

4

Bus number

Vol

tage

Newton method using Eq(19)and Eq(20)

Fig.2. Voltage using two methods

As can be seen from Fig.2, the bus voltages obtained using (19) and (20) are quite close to those obtained using Newton method. The average error is 0.00371pu. As shown in Fig.2, we will also run a Newton power flow program after the optimization. Hence, the method based on (19) and (20) are accurate enough to give the voltage variation trend as transmission lines are open.

To show the viability and effectiveness of transmission switching in reliving voltage violations, here it's assumed that generator’s reactive power outputs are constant. Thus the only control action during the optimization procedure is changing the status of transmission line.

For further testing the correctness of this solution, these 41 lines are opened one at a time, and a Newton power flow program is run to calculate the sum of bus voltage profile deviation after a line is open. After removing the non-convergence solutions, Fig.3 gives the top ten minimum deviation solutions.

Fig.3. the top ten minimum deviation solutions

As shown in Fig.3, the minimum deviation corresponds to opening the line 9-10. Then the number of lines allowed to be open in (25) is set to be 1, the results of the proposed model shows that line 27-28 should be open. The two solutions are different. In fact, opening the line 9-10 is infeasible for the voltage of bus 9 is 1.064pu after the line 9-10 is open, which exceed the upper limit. Consequently, the optimal feasible solution is to open line 27-28, which is identical to the proposed model.

As the increase of the number of the lines allowed to open, the sum of bus voltage deviation will decrease, but at a decreasing rate. (See Fig.4).

the

sum

of v

olta

ge d

evia

tion

the number of lines allowed to open

Fig.4. Sequences of line openings

V. CONCLUSIONS In this paper it is proposed that transmission switching can

be used to relive voltage violations in low load period for EHV power system. As only some communication and switching equipment are required, transmission switching can relieve voltage violations in a more economical way. A basic MINLP formulation is given. To avoid the computational intractability, this model is converted to a mixed integer linear program (MIP) based on some simplifications. This paper is a first step in analyzing the potential benefits of transmission switching to relieve voltage violations in low load period. More researches, particularly in the areas of stability need to be done in the future.

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