[IEEE 2013 3rd International Conference on Electric Power and Energy Conversion Systems (EPECS) -...

2013 3rd International Conference on Electric Power and Energy Conversion Systems, Yildiz Technical University, Istanbul, Turkey, October 2-4, 2013

Online Voltage Stability Monitoring Based on PMU Measurements and System Topology

Dinh Thuc Duong and Kjetil Uhlen Department of Electric Power Engineering

Norwegian University of Science and Technology Trondheim, Norway

Abstract-This paper presents a new approach to estimate the

Thevenin impedance and the maximum power transfer at load

buses, which are criteria for voltage stability assessment. The

proposed algorithm utilizes system topology, load impedances

and phasors from phasor measurement unit (PMU) as means of

estimation, in which load impedances are integrated into the

admittance matrix of the power system to compute Thevenin

equivalent parameters and phasors at monitored load buses are

used to identify the maximum loadability point. The proposed

method shows good performance on monitoring voltage

stability. It is also applicable for online applications since PMUs

are required at buses of concern and other information is

available from existing state estimator. In case study section, the

proposed methodology is demonstrated and validated by

simulation on IEEE test systems.

Index Terms-Admittance matrix, load impedance, maximum

power transfer, Thevenin impedance, voltage stability.

I. INTRODUCTION

Voltage stability remains one of the major concerns in power system operation due to its severity, especially for those systems heavily loaded or highly penetrated by fluctuating renewable generation. Voltage stability limits cannot be found by normal load flow calculation because of singularity of the Jacobian matrix when the system is close to the limits. However, this problem can be solved by the continuation power flow method [I], whose main idea is that load flow calculation is run with increasing load until it does not converge. Then, the prediction of state variables is made based on the tangent vector at current state. Next, the state variables are corrected under the condition of fixed power or voltage, depending on the current location on the V-P curve. These two steps are iterated until the end of the V -P curve. Overall, the process imposes high demand of computation and availability of a good system-wide state estimator for use in on-line operation. Therefore, this approach is preferable in offline analysis. Likewise, there are some functions to determine voltage stability limits in some commercial simulation software for power system with similar features. To determine the V-P or V-Q curves of a specific load bus, its active or reactive loads are kept increasing until the load flow

This research is supported by the Nordic Energy Research project STRONgrid, hereby kindly acknowledged.

calculation no longer converge while power of the other load is unchanged. These tools are useful for operation planning and contingency analysis.

Recently, more and more phasor measurement units (PMUs) are deployed in power systems. With the capability of measuring synchronously phasors of voltage and current across the network, along with present communication infrastructure, PMUs pave the way for online monitoring applications, including dynamic voltage stability assessment. The essence of this is that the maximum power transfer or voltage instability point is the point where load impedance at one load bus is equal to the Thevenin equivalent impedance of the rest of the system seen at that bus. While the load impedance can be obtained by local voltage and current phasors, the estimation of the Thevenin equivalent poses challenges and remains the object of ongoing researches.

There are presently two main approaches described in the literature to estimate the Thevenin impedance. The first one is based on a local measurement, while the other pursues a method that considers both system topology and wide area measurements. In [2], the estimation is an adaptive process with a user-defined tuning factor. In addition, the Thevenin parameters, i.e. voltage ETh and impedance Xrh, are assumed constant during the short period of their identification. Their values are completely determined by local load voltage and current phasors and variation directions of both load impedance and XTh. The performance of this approach is significantly influenced by the tuning factor and quality of measurements such as noise.

On the other hand, [3] introduces the concept of "coupled single-port circuit" by pulling all generators and loads out of the grid, and formulating impacts of other load currents (Ecollpled) on the considered load. Here, the virtual impedance that is equal to the coupling term Ecollpled divided by the examined load current is modeled as an impedance in series with the equivalent impedance obtained from inversion of the admittance matrix; and the sum of the two impedances is considered as the Thevenin equivalent impedance seen at the study bus. However, it is noted that this methodology is tested with assumption that all loads increase simultaneously by the

978-1-4799-0688-8/13/$31.00 ©2013 IEEE


same scaling factor. Therefore, the virtual impedance remains quite constant during the test.

The method presented in this article is to approximate Thevenin equivalence and the maximum power transfer, which functions as the criterion to assess voltage stability at load nodes. The methodology is based on a combination system topology and PMU measurements.

In this paper, the structure is as follows. Section II presents an overview of voltage stability assessment based on impedance matching condition and maximum power transfer, followed by the proposed algorithm to detennine Thevenin parameters in Section III. Section IV shows the results of test cases, and conclusion is finally drawn in Section V.

II. REVIEW OF VOLTAGE STABILITY ASSESSMENT BASED ON IMPEDANCE MATCHING

CRITERION

To examine voltage stability of a particular load bus in power system, let us consider a common example of the twonode grid as shown in Fig. 1, where ETh is a voltage source and ZTh is an impedance. The load is represented by the impedance ZL, which is the ratio of the bus voltage VL to the current h. It has been proven that the maximwn power transferred to the load is reached when the magnitude of the load impedance is equal to that of the impedance ZTh,

(1)

Figure I. Thevenin equivalent circuit.

If the ZTh is known, the ETh is formulated as

ETh=VL+ILZTh' (2)

where VL and h are voltage and current phasors of the load.

Then we can form the Impedance Stability Index (lSI) at the load bus as [5]

lSI JZTh l

I ZLI ' (3)

and estimate the maximwn power transfer at the load node by [3]

IE�h I [I ZTh 1- (imag ( ZTh ) sin 0 + real ( ZTh ) cos 0) ] Smax = 2 ' 2 [imag( ZTh ) coso -real( ZTh ) sin 0]

(4)

where 0 is the load power angle.

Consequently, the power margin of the present operation point is [3]

S -S margin = //lax Lxi 00% S '

L

where SL is the current apparent power of the load.

(5)

In summary, real-time voltage stability monitoring presented in the literature is the process based on continuous comparison of impedance and apparent power of load to the estimated Thevenin impedance and maximwn power transfer, which are proposed and presented in the next section.

III. DESRIPTION OF PROPOSED METHOD

A. Estimation ofThevenin equivalent paramters State estimators have been widely used by many TSOs for

long period of time. In these estimators, system topology and its changes after events like breaker operation, change of tap position of on-load tap changer are available and updated. With recent development of communication infrastructure, refresh rate of state estimators has been improved significantly. Obviously, state estimators are working well and would not be superseded in near future. On the other hand, regarding voltage stability issue, it is noticed that only some load buses are vulnerable to this problem, and need closer monitoring. Solutions that require PMU installation only at these concerned buses would be cost effective and viable for practical implementation. Therefore, the main idea of the proposed approach is to estimate equivalent Thevenin parameters at load buses, and consequently maximwn power transfer as means to evaluate voltage stability, based on the existing information of system topology extracted from state estimator and PMU measurements at load nodes.

Given a simple grid in Fig. 2, the generator G 1 is connected to bus B 1 through the step-up transfonner n. There are two loads, namely Ll and L2, at bus Bl and B2 respectively; meanwhile Line 1 is the connection between B 1 and B2. Assume that G 1 is capable of keeping its terminal voltage unchanged. At one instant of time, the electric equivalent circuit of this system is depicted in Fig. 3, where n, Line 1, L, and L2 are represented by impedances ZT, Z[,

Zu, and ZL2 respectively. For the sake of illustration, shunt admittances of Line 1 are neglected.

Tl Bl B2

Line 1

L2 Gl

Ll

Figure 2. A simple two-node grid.


Figure 3. Equivalent circuit of the two-node grid.

Obviously, the Thevenin equivalent impedance of the system seen at bus B2 is

(6)

On the other hand, from Fig. 2, we can classically form the admittance matrix of the grid as

(7)

Let us modify the diagonal elements of the matrix Y by adding the two admittances of Tl and Ll to establish a new matrix

is

Y =

l

dL + dT

+ Z� I

-dL ] eq

1 1 -- -ZL ZL

(8)

Defme the impedance matrix Zeq as the inversion of Yeq,

Z =y-I eq eq •

(9)

The diagonal element in the second row and column of Zeq

( ) ZLIZT Zeq 2, 2 = ZL + ,

ZLI +ZT (10)

which is exactly equal to the Thevenin impedance seen at bus B2 in (6). Indeed, it is generally valid that the diagonal element indexed (i,i) of the impedance matrix is the Thevenin impedance seen at the equivalent i'

h bus in the grid [4].

Once the Thevenin impedance at the load bus is determined, the Thevenin equivalent circuit of this system is equivalent the circuit in Fig. 1 where load L2 is the considered as load. Then, the equivalent Thevenin voltage can be computed by

(11)

where VL and hare phasors of voltage and current at observed bus obtained from PMU measurements.

It is noticed that to calculate the Thevenin impedance seen at bus Bl , one can repeat the above procedure, including inversion of the admittance matrix. However, matrix inversion takes tremendous efforts of computation, especially in large power system. Let us make a small modification of the matrix Yeq in (8) by adding the admittance of load L2 into bus B2, forming

-- ] ZL

1 l'

ZL +

ZL2

(12)

After calculating the impedance matrix by (9), we obtain the equivalent circuit for bus B2 as shown in Fig. 4 where E'

Th and z'

Th are new Thevenin voltage and impedance respectively.

Bl B2

Figure 4. Thevenin equivalent circuit, including all load impedances.

However, the parameter we intend to fmd is ZTh, not z'Th.

By contrasting the two equivalent circuits in Fig. 1 and Fig. 4, one can recognize that the z'

Th is equal to the equivalent impedance of the two parallel impedances ZTh and Zu. Therefore, the ZTh is formulated as

(13)

With this small adjustment, we can estimate all Thevenin impedances seen by load buses with only one inversion the modified admittance matrix.

B. Network reduction

In large power systems, inversion of the modified admittance matrix of the whole grid as mentioned above requires tremendous computation effort. However, one can notice that there are many parts of the system that are linked to other areas by single connection as depicted in Fig. 5. In this case, in order to estimate Thevenin parameters at buses in the study area, we can follow these steps:

First, consider each external area as a subsystem, and then, use the above proposed method to compute Thevenin impedance seen at the boundary with respect to the study area. Here they are B 1, B2 and B3. Second, integrate newly estimated Thevenin impedances shown in Fig. 6 into the admittance matrix of the study grid and continue calculation process.


This feature also makes it possible to investigate voltage stability of a small part of the grid, for example branch 3, which has weak connection to the rest of the system.

Figure 5. Simplification of system with radial branches.

study area

Area 3::"",/ \

. ZTh3

'

) ,-�-----�

Figure 6. Equivalence of radial connections.

C. Effects of generator excitation limit

When the system is approaching voltage collapse, demand of reactive is very high and hence it may lead to the situation that nearby generators hit maximwn reactive power limit imposed by overexcitation limiter, resulting in acceleration of voltage collapse phenomenon. In this case, the internal voltage of generator cannot continue increasing; hence the terminal voltage is no longer constant and drops significantly due to the voltage drop on the synchronous reactance. Therefore, this synchronous reactance must be added in series with the stepup transformer impedance into the admittance matrix. Now, the electric circuit of the two-node grid in Fig. 3 is modified as depicted in Fig. 7.

Bl B2

ZL2

Figure 7. Equivalent circuit as generator reaches excitation limit.

D. Summary of proposed methodology

Overall, the proposed voltage stability assessment is summarized as following:

1) Update the admittance matrix of the grid: When system topology changes are detected, the admittance matrix

must be updated to represent the true configuration of the grid. Breaker statuses could be communicated from SCADA or EMS/state estimators at regular intervals or preferrably be event based.

2) Calculate all load impedances and admittances: For buses where PMUs are installed, load impedance is the ratio between phasors of bus voltage and load current. Instead, voltage and power injections obtained from state estimation are utilized. PMUs are required at load buses involved in voltage stability monitoring scheme.

3) ModifY the admittance matrix for the observable part

of the grid: Add load admittances to corresponding elements on the diagonal of the admittance matrix by (12). Note that terminal bus of generators is considered as the reference bus in Thevenin equivalent. Thus, impedance of step-up transformer, if any, must be added to the bus at the high voltage side. For those generators that hit excitation limits, their synchronous reactances must be added in series of stepup transformer impedance.

4) Inverse the admittance matrix and compute Thevenin

impedance at concerned load buses: Conduct inversion of the admittance matrix to achieve the impedance matrix. Note that elements on the diagonal of this matrix are not Thevenin impedances at each load buses. Use (13) to obtain the equivalent impedance for every bus of the observed network.

5) Establish lSI for each load bus: Form lSI for each load bus by (3). The ISIs are dynamic and depend on operating point of power system; the load bus with maximum lSI is the weakest one in the grid. In addition, the maximum power transfer - the voltage instability, condition is the point at which lSI is equal to 1.

6) Determine maximum power transfer and margin: It is more convenient, especially for system operators, to express the distance to the limit in power. By applying (4) and (5), one can determine proximity to limits in terms of power. Those buses with small margin are the most critical ones.

IV. TEST CASES

To validate the proposed approach, we use the RMS Simulation function of the commercial software DIg SILENT PowerFactory. All generators have both governor and automatic voltage regulator. Voltage collapse is emulated by gradually ramping up one load while the others are modeled as mixture of constant impedance (60%), constant current (20%) and constant power (20%).

The first test is carried out on the IEEE 9-bus test system as shown in Fig. 8. All generators do not have limits in excitation system. The load at bus #7 is arbitrarily chosen for projection to voltage collapse although the procedure is defmitely applicable to other loads. First, from the base operation point, the load at bus #5 increases in both active and reactive power by 200% during the period from 2s to 7s. This variation is seen at bus #7 by a slight drop of the Thevenin impedance in Fig. 9.


Figure 8. The IEEE 9-bus test system.

In next step, the load at bus #7 is projected to voltage collapse. In Fig. 10, the load impedance is plotted versus the estimated Thevenin impedance while Fig. 11 depicts the load apparent power and the estimated maximum power transfer at the bus #7. It can be seen that when the two impedances are equal, the actual maximum of the load roughly matches the estimated power transfer computed from proposed method. In other words, the method has successfully estimated the power limit at the bus.

E 37.075 .<::: Q. 37.07 <J) u c

37.065 '" "0 <J) 0. .£; 37.06 c 'c <J) 37.055 > <J) .<::: f-

2 3 4 5 6 7

Tirre (5)

Figure 9. Thevenin impedance seen at bus #7 during load #5 variation.

80,------,-------,,---------------------,

E .<::: 60 Q.

<J) u c '" "0 40 <J) 0.

E

20 10 11

............ Thevenin irrpedance

--- Load irrpedance

12 13

Tirre (5)

14

Figure 10. Estimated Thevenin impedance and the load #7.

� > � Q; ;: 480 0 0. C � 460 '" 0. 0. «

11 12 13

Tirre (5)

14

15

15

Figure I I. Estimated maximum power transfer and apparent power at bus #7.

In the second case study, we use the IEEE 39-bus test system shown in Fig. 12. The load at bus #15 is arbitrarily chosen for emulation of voltage instability. Again, excitation limit is not considered. The Thevenin impedance at the bus # 15 is estimated by two approaches. The first one compute equivalence parameters based on the admittance matrix of the entire system. Meanwhile the second treats the dashed area as external system and identifies its Thevenin impedance seen at bus #16. This impedance is then integrated to the admittance matrix of rest of system to compute the Thevenin impedance seen at bus #15. As can be seen in Fig. 13, the impedances estimated by two methods show the same performance. Regarding Fig. 14, the maximum apparent power of load #15 matches the estimated maximum power transfer at the instant of time the two impedances are equal as seen in Fig. 13.

';'---""r----r-'"

I I I I I I I I I /1

I I

, , , , , , , , I I I I I I I I I I I I

Figure 12. The IEEE 39-bus test system.

40,-��--�---------,--,_--------------,

I 30

Q. <J) g 20 '" "0 � 10

- - - - - Estimated Zth1

•••••••••• Estimated Zth2

-- Load

O�------�---------L--------�------� o 500 1000

Tirre (5)

1500 2000

Figure 13. Estimated Thevenin impedance versus the load at bus # 15.

� > � Q; ;: 0 0. C <J) Co 0. 0. «

0 0 500 1000

Tirre (5)

-- Load

- - - - - Estimated Smax 1

•••••••••• Estimated Samx2

1500 2000

Figure 14. Estimated maximum power transfer and apparent power at bus #15.


The third study examines the impact of excitation limit of generators on maximwn power transfer. During the ramp of load #15, most of generators, except G7, hit excitation limits as seen in Fig. 15. The effect of these limits on Thevenin impedance and maximum power transfer are presented in Fig. 16 and Fig. 17. Obviously, excitation constraints result in increase of the Thevenin impedance seen at the load bus, and consequently drop of the maximum power that the source can supply the load, which is well estimated by the proposed method as depicted in Fig. 17.

6 -- E1

5 S S 4 OJ 0>

� -- E2 -- E3 -- E4

Jll 3 0 -- E5

> c;; -- E6 c 2 2 � -- EB E E9

-- E10

o o 200 400 600 800 1000 1200 1400 1600

Tirre (s)

Figure 15. Internal voltage of generators under overexcitation limiter

E .<:::

Q. OJ (.) C co '0 OJ 0. .§

60

40

20

1_- Thevenin irrpedance

----- Load

O�--�----�----�--�-----L----�--� 200 400 600 800 1000 1 200 1400 1600

Tirre (s)

Figure 16. Estimated Thevenin impedance versus the load at bus # 15 in case of lacking reactive support from generators.

3000,---.----,----.----.---,----.----,----, ....... ......

2500

� ......... "' .... ..

... � ...... .

� 2000 ...... \ ..... \ ..

I E OJ � 0. <{

•..•..•... , Estimated Smax

--- Load

o�--�--�----�--�--�----�--�--� o 200 400 600 800 1 000 1200 1400 1600

Tirre (s)

Figure 17. Estimated maximum power transfer and load power at bus # 15 in case of lacking reactive support from generators.

DISCUSSION

Estimation of Thevenin equivalent parameters based on system topology and local phasor measurements at load bus shows stable performance and good approximation of the maximwn power transfer. The method is feasible for practical implementation since information of system topology is available from state estimation, and PMU is only needed at those buses where voltage stability is of concern.

Since changes of topology affect the estimated equivalent parameters, a fast track of system structure is essential in this approach. Regarding load impedances at buses that are out of concern, their values are nonnally much larger than that of power lines and transfonners. Variations of these impedances have small impact on the estimation of Thevenin equivalence and maximwn power transfer. Thus, it does not require fast refresh rate. Update frequency of existing state estimators can be sufficient.

It can be noticed that voltage stability is rather a local problem, and impact of remote generators is small. It is not necessary to take the entire system into calculation, especially in large interconnected system. Here network reduction promises an appropriate tool to facilitate the proposed method. This topic is under ongoing research.

V. CONCLUSION

The paper has presented the new approach to estimate the Thevenin equivalent parameters and the maximum power transfer based on the combination of system topology and PMU measurements. Based on available information from existing state estimation and minimum requirements of PMU installation at monitored buses, the algorithm can be implemented in real time without high demand of resources. The results from simulations have, in addition, validated the performance of this approach.

REFERENCES

[I] v. Aiiarapu and C. Christy, "The continuation power flow: a tool for steady state voltage stability analysis," IEEE Trans. Power Syst., vol. 7, pp. 416-423, Feb. 1992.

[2] S. Corsi and G. N. Taranto, "A real-time voltage instability identification algorithm based on local phasor measurements," IEEE Trans. Power Syst., vol. 23, pp. 1271-1279, Aug. 2008.

[3] Y. Wang, 1. R. Pordanjani, W. Li, W. Xu, T. Chen, E. Vaahedi and 1. Gurney, "Voltage stability monitoring based on the concept of coupled single-port circuit," IEEE Trans. Power Syst., vol. 26, pp. 2154-2163, Nov. 2011.

[4] 1. J. Grainger and W. D. Stevenson, Power System Analysis, Singapore: McGraw-Hill, 1994, pp. 283-294.

[5] M. Begovic, B. Milosevic and D. Novosel, "A novel method for voltage instability protection," in Proc. 2002 IEEE Computer Society System Sciences Can!, pp. 802-811.

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