Abstract— In this paper, we proposed a novel index for monitoring and prediction of oscillatory instability (Hopf Bifurcation) in power systems. Considering modern control techniques, the index uses damping information of the whole power system. Therefore, we call it as DMI (Damping Matrix Index). It easily uses power system available signals such as electro-mechanical torques, speeds and angles of synchronous machines to predict oscillatory instability. Since the values of each monitoring index hides behind its estimation method and the proposed index is based on state space model of power system, we use Subspace System Identification (SSI) algorithms to estimate the proposed index. Based on SSI techniques and the proposed index, we suggest an algorithm for power system monitoring. Tests and simulations have been conducted using the proposed index on simulated measurements of a two-area 4-machine power system. Results express good performance of DMI in comparison with other well-known oscillatory instability indices. Keywords: Small Signal Stability; Subspace System Identification (SSI); Power System;

I. INTRODUCTION. Stresses engaging in power system operation are usually

arisen from a wide range of errors such as increasing or decreasing of loads, lines short circuit, machines loading or unloading and undesired harmonics. It seems that monitoring of power system behavior is always a prerequisite for improving of system performance [1] - [3].

The nature of system is usually guiding us to the method of monitoring. It normally provides the researcher with the adequate indices of system monitoring. Estimation and identification issues arise as soon as the researcher needs to apply the power system monitoring index. Monitoring should be fast and accurate as possible as it can be. Therefore, both the index and estimation method should be chosen carefully.

Power system is normally a nonlinear system. But its linearized model may be very useful in the case of monitoring, since we can suggest effective linear model identification methods which are fast and accurate enough in the sense of small signal analysis and computation efforts.

In this research, we want to limit the monitoring field to

Aliakbar Mohammadi is a PhD student in Department of Electrical

Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran. (Corresponding author phone: +989173009915; e-mail: aa.mohammadi@srbiau.ac.ir).

Hamid Khaloozadeh is with Khaje Nasir University of Technology Tehran, Iran. (e-mail: h_khaloozadeh@kntu.ac.ir)

Roya Amjadifard is with Tarbiat Moallem University, Tehran, Iran. (e-mail: amjadifard@tmu.ac.ir)

small signal stability of power system. Small signal stability is the power system ability for preserving synchronization when small disturbances arise. Such disturbances may appear because of small variations in power generation and load [4]. Small signal analysis also lets us to track small variations of power system parameters, specifically those related to loads.

In different papers, we can find footsteps of indices related to small signal analysis which can provide us with instability or oscillatory instability margin detection. They usually use a linear model of power system in order to offer information about an especial operating point [5] , [6].

Indices are usually ranked by their computation easiness and rapidity of estimation and assessment. Therefore, not only we should care about attractive mathematical methods of realizing a power system monitoring method, but also we should deeply think of the identification and estimation method which may be applied.

Subspace System Identification (SSI) methods have good and wonder properties which caused them to be applied in different fields of industry since the time of appearance [7] , [8]. We may include an introduction of SSI methods in the following sections to clarify their benefits. SSI methods are tools of our work when we want to estimate proposed oscillatory instability index.

In this paper, we want to propose a novel index for assessment of oscillatory instability margin. The proposed index is based on linearization of power system and application of modern control system theory to the linear model. We will estimate the proposed index using a modified version of subspace system identification methods. In order to investigate the advantages of proposed index, we use it for monitoring of several power systems in different conditions.

Different sections of the paper are arranged as follows; an introduction of oscillatory instability will be presented in the next section. The proposed index will be included in section three and then there will be an investigation and modification of subspace system identification methods. The innovated oscillatory instability index will be estimated using SSI methods in section four. A comparison of different indices will be provided in the fifth section using different conditions of a test power system.

II. OSCILLATORY INSTABILITY INDICES Oscillatory instability (Hopf bifurcation) of a power

system will occur when a pair of its complex poles travels from left to right in s-plane and crosses the imaginary axes.

Power System Monitoring Using Subspace Identification Aliakbar Mohammadi, Hamid Khaloozadeh, Roya Amjadifard

2011 2nd International Conference on Control, Instrumentation and Automation (ICCIA)

978-1-4673-1690-3/12/$31.00©2011 IEEE 19

Among well-known indices for Hopf bifurcation, the EVI [5] index is reintroduced here for further investigations; Eigen Value Index (EVI): Suppose the most critical eigenvalue of power system is

c c cj (1) Then we can define one of the Hopf bifurcation indices as

cEVI (2) There are some difficulties in application of above index;

this index is a general index for power system stability, not only an indicator of oscillatory behavior. Moreover, it is necessary to monitor critical eigenvalue of power system which is itself a challenging problem [9, 10].

III. PROPOSED OSCILLATORY INSTABILITY INDEX Time domain small variations in electro-mechanical

torque, eT , of a synchronous machine can be expressed as a function of small variations of angular velocity, r , and angle, :

0( ) ( ) ( )e d r s

damping synchronizing

T t K t K t (3)

That part of electro-mechanical torque, which relies on angular velocity is called damping torque and the part which relies on angle is called synchronous torque [11]. Since

0 r ( s ) s ( s ) (4) Therefore, we can rewrite (3) in the s-domain as

( )( )

( )e

d s

T sH s K s K

s (5)

where . . / / secdK p u rad is called damping factor,

. .sK p u is synchronizing factor and 0 / secrad is rated frequency of power system. Above factors can truly represent damping and frequency n of electro-mechanical modes for a single machine power system [9]:

0 dn s

0 s

K1K ,

2 H 2 2 H K (6)

Oscillations of a multi-machine system may be affected by different modes, since oscillations of each machine are linear combination of all system modes. Thus, factors in (5) can also be introduced as a measure of oscillatory instability margin. In order to provide damping and synchronizing, one can use a linear model of power system such as following:

( ) ( )( ) ( )

x A x

y C x

t tt t

(7)

Where nx is states vector and it contains generator states, load and controllers of power system. my is vector of outputs. A and C are constant matrices which define the linearized system.

If we use modal transformation x Uz , we can rewrite outputs vector as:

10y s C( ) U sI z (8)

It is obvious that if we excite the initial modal content, z0, in the direction of mode i , then we can write variations of output signal yk in the Laplace domain as following:

k ik

i

0 iCy

sU z

(9)

where Ck is the kth row of C. Ui is the ith mode eigen vector. Therefore, if only the ith mode is excited, then one can provide the following transfer function between small variations of angle and electro-mechanical torque of jth generator of power system;

i

i

ej Tej iij s

j j is

T ( s ) C UH ( s )

( s ) C U

(10)

where CTej and jC are those rows of C which provide electro-mechanical torque and angle of jth machine. Hij is actually a complex number (Hij=Re(Hij)+jIm(Hij)).

If we replace i i is j in (5) and (10), and then we compare evaluated values, the following equations will be provided:

Im( )ijijd

i

HK (11)

Re( ) Im( )ij is ij ij

i

K H H (12)

where ijdK is a measure of ith mode damping in jth machine

signals. Thus, ijsK is a measure of ith mode synchronizing in

the signals of jth machine. In accordance to the above points, one can provide a

torque-angle block diagram such as Fig 1 for each synchronous generator. It expresses that when ith mode is excited, if the gains ij

dK and ijsK become zero or negative,

power system will be instable. Actually, if ijdK becomes

zero, power system will be oscillatory instable (Hopf Bifurcation occurs).

We can gather all of such gains in the following arrays:

ij

d d

ijs s

K K

K K (13)

where dK is a damping matrix and sK is a synchronizing matrix.

As we express previously, if a mode-machine damping decreases, the whole system damping will also decrease.

Fig 1: Torque-angle block diagram of a

synchronous generator.

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Therefore, if . is a measure of matrix norm, one can

introduce dK as a measure of secure margin to oscillatory instability of system. Thus, we can express that the smaller dK , the higher possibility for oscillatory

instability. If dK becomes zero, oscillatory instability or Hopf bifurcation may occurs, because the system damping goes to zero (Fig 1).

Power system is usually work near its limits and it is usually stable, but its operating point may travel from a stable point to an oscillatory instable and then exponentially instable because of small variations of loads or other components. dK can be calculated in different time intervals using power system measurements. Therefore, one can monitor the power system behavior using dK .

We know that oscillatory instability is equal to Hopf bifurcation in power system, thus above proposed index can also be introduced as a Hopf bifurcation index.

IV. ESTIMATION OF PROPOSED OSCILLATORY INSTABILITY INDEX

In order to estimate the proposed oscillatory instability index, we need to provide the system matrix; A. System matrix can be extracted from state space model of power system as depicted in (7). Therefore, we need to identify a state space model of power system.

If the power system is engaged in small and stochastic oscillations of load and/or other internal phenomena, one can provide a linear model of power system using measurements of some system signals and application of subspace system identification (SSI) algorithm [12, 13].

Since electromechanical torques of power system are affected by angles and speeds of different machines, it is obvious that we should use measurements of these two signals variations and electro-mechanical power or torque variations for identification of a state space model of power system. Therefore, the model may content those modes which are more affected by the mentioned signals [9, 10].

Sampling proper signals of power system, we can estimate state matrix using the following stages of a subspace identification algorithm;

A. Model The considered system model is;

( 1) ( ) ( )

( ) ( ) ( )

( )( ) ( )

( )T T

tsT

x t Ax t w t

y t Cx t v t

w t Q SE w s v s

v t S R

(14)

where yn ny ,x are samples of output and state vectors. yn n

t tv ,w are static, zero average state and output noise vectors, consequently.

B. Identification Data In order to identify above model, we need to provide output samples. However, identification data should usually be provided in the following structure;

( ) ( ) ( 1) ( 1)

( ) ( 1) ( 2) ( )

T

T

f t y t y t y t k

p t y t y t y t k (15)

where f(t) and p(t) are future and past data set. k should be strictly bigger than n. It can be a guess. Therefore, this is not a restrictive condition. Now, we can formulate SSSI problem as below: “There are N samples of output vectors,

( ) (0) (1) ( 1)Y t y y y N , from a system of order n. Find matrices A, C, Q, R, S and n for the structure defined in (14).”

C. Block Hankel Matrices Stochastic subspace identification algorithms begin data processing by forming the following block Hankel Matrices.

,

(1) (2) ( )(2) (3) ( 1)

{ ( ) ( )}

( ) ( 1) (2 1)

Tk k

kk

H E f t p t

k k k

(16)

where ( ) { ( ) ( )}, 0,1, , , 2 1 ,Tl E y t l y l l L k L k n is correlation of future and past data. Actually, the algorithm uses statistical properties of output samples for further processing. The word Stochastic in the expression (Stochastic Subspace System Identification (SSSI)) may arise from this point.

D. System Order (n) SSSI uses the following Singular Value Decomposition (SVD) of Hankel matrix in order to provide system order;

,

00

Tsys sys

k k sys noise Tnoise noise

Tsys sys sys

VH U U

V

U V

(17)

We can detect noise singular values by detecting a big gap among the singular values of Hankel matrix. Thus, noise singular values can be neglected since they are very smaller than system singular values. Therefore, system order, n can be defined as

dim( )sysn (18) E. Estimation of State Space Matrices SVD of Hankel matrix can also provide us with an extended observability matrix;

1/2k sys sO U (19)

It can also present controllability matrix; 1/2 T

k sys sysC V (20) We can easily obtain system matrix in attention to controllability and extended observability matrices;

†1 ( 1: ,1: )

(1: ,1: )

k k y

k y

A O O p kn n

C O n n (21)

Investigating subspace system identification expresses the

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following advantages for such algorithms: SSI Algorithms are the only system identification

methods that can be easily and extensively applied to all MIMO and SISO systems.

Estimation of system order is one of the steps of SSI algorithms. This advantage reduces amount of time, cost and calculations.

SSI methods can handle big packages of data. Online operations of SSI methods are easier and can

easily be applied to MIMO systems. SSI methods use robust mathematical tools such as SVD,

LQ decomposition, least square and QR decomposition. They also don’t need nonlinear optimization.

Some SSI algorithms only use output data to identify a model. This is a considerable advantage.

In accordance to previous discussions, in order to estimate the proposed oscillatory instability index, we present the following algorithm which uses subspace system identification methods:

1. Measure angles, speeds, powers or electro-mechanical torques variations for each machine.

2. Estimate the whole system matrix, A, using above mentioned subspace method of system identification.

3. Calculate eigenvalues and eigenvectors of A. 4. Provide all the elements of H and Kd using (10) and

(11). 5. If dK is becoming zero, Hopf bifurcation or

oscillatory instability is going to occur. 6. Go back to stage 1. The proposed algorithm can be implemented online in

two ways; i) Implementing the algorithm using blocks of measured data which will be updated at the end of each algorithm run. 2) Updating data whenever a new sample data is provided. The first method is superior in economical view but the second method provides a more accurate and softer result. Using subspace identification methods in the proposed algorithm is rational since such algorithms are usually fast and non-parametric. Therefore, they are suitable for online identification and monitoring of large systems.

V. TESTS We apply the proposed index to several test power systems in order to assess its performance, advantages and possibility of its application as an index of oscillatory stability for power systems.

All of below simulations carried out using a computer with 2.5GHz CPU, 4GB RAM. We used MATLAB software to simulate test systems; a combination of SIMULINK models and m-files. Load models were constant power load. In each case, we increased the system total load by a constant step during time to the extent that the system became oscillatory instable.

We sampled several signals which are more likely related to electro-mechanical oscillations. These signals are usually

angles, angular speed, electrical torques or electrical powers of generators, and bus voltages. In our simulations, the variables were measured in per unit. Therefore, there is no difference between electrical torques and powers. Our algorithm requires generator angles and electric torques while computing (10). Thus, it is necessary to include these two kinds of signals in every measurement. Other signals are arbitrary. They may be helpful for system identification process. However, it should be noticed that large amount of identification data increases processing time of algorithm and the costs.

The sampling rates were chosen in accordance to the amount of data we would require as an input to the identification algorithm. Due to limitations on computer resources (CPU speed, RAM …), we cannot use very large amount of identification data. Therefore, the sampling rates cannot be too small. Another limit on sampling rates arises from the nature of test systems. In other words, the sampling rate should be chosen such that the frequencies of critical modes can be recovered through processing of sampled signals. Thus, in all of the following tests, the sampling rate is 0.001 second in order to provide the same simulation framework for all tests. We used 5 second sets of measured data for system identification. Therefore, data sets containing 5000 samples of signals are inputs to the algorithm in every iteration. This amount of identification data already fulfills identification and estimation issues for the three test systems.

Fig 2 shows a single line diagram of two-area test system. The system includes two areas which are connected by a weak line. The system is used for investigation of small signal analysis and inter-area oscillations in [9]. Two-area power system is proper for investigation of proposed index, since we have to extract some properties of power system from small signal variations.

We use computer simulation for investigation of small signal behavior of two-area power system. There are loads on 7th and 9th bus of system. In computer simulations, in order to insert disturbances into system, we increase the loads by a constant factor to the extent which instability occurs. Computer simulations show that the system has three basic modes; an inter-area mode and two local modes. Damping factors of other modes are more than 60 percent. Therefore, we can ignore their effects when we are interested in small signal analysis. We provide small signal disturbances in three ways; i) Increase of load in bus 7; Fig 3 illustrates effect of such a

disturbance on small signal analysis of system. In

Fig 2: Single Line Diagram of Two-Area System

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this case, Inter-area mode moves to the instability region faster than other modes.

ii) Increase of load in bus 9; Fig 4 illustrates effect of gradually increase of load 9 by a constant factor. In this case, local mode of the second area reaches the instability margin faster than other modes.

iii) Increase of both loads; in this case, we increase both of the loads in bus 7 and 9 by a constant factor. We conduct small signal analysis in each increase. Fig 5 illustrates the results. In this case, local mode of area one and inter-area mode affect instability, both together.

In order to find instability margin for each of the above cases, one can investigate Fig 6-8. They show variations of damping factor in accordance to total load variations. The

test system is instable when damping factor for at least one of its modes becomes non-positive.

Now, we try to identify stability margin using the proposed method. Therefore, we simulate two-area test system based on the mentioned three types of disturbances, and then we sample the proper identification signals for each type. We also add measurement noises to the output signals. Thus, we estimate the following two stability margin indices using measured data;

cEVI dDMI K

(22)

where DMI stands for Damping Matrix Index, and EVI stands for EigenValue Index. Fig 9-11 show the results. Almost both of indices have a linear behavior. The proposed index; DMI, shows almost the same behavior, but it has a different behavior in comparison to EVI.

Table 1 shows estimated marginal loads of each of indices. It expresses that proposed index, DMI, works better than EVI. It also expresses that subspace system identification algorithms have a good performance when they are used for estimation of instability indices. It is an attractive property of SSI algorithms in addition to their easy application. One should consider that we can monitor and predict the oscillatory instability of power system using the proposed index, application of few number of system signals and processing capabilities of SSI algorithms.

VI. CONCLUSIONS Comparison of proposed index with a well-known

instability index (Eigen Value Index; EVI) expresses a superior capability for the proposed index; it is very easy to apply it for monitoring and prediction of abnormalities of power systems since it only uses sampling of a few available signals and processing capabilities of well-known subspace system identification algorithms. It also has a fairly linear behavior without discontinuities with respect to system load increases. In comparison to EVI, the proposed index expresses a very successful online behavior when using SSI algorithms. The proposed index and associated identification methodology may be used as an effective online tool to help system operators for monitoring power system stability.

Fig 3: Effect of Load 7 on modes; Modes travels to right as load 7 increases

gradually.

Fig 4: Effect of Load 9 on modes; Modes travels to right as load 9 increases

gradually.

Fig 5: Effect of Loads 7 and 9 both together on modes; Modes travels to

right as the loads increase gradually.

Table 1: Oscillatory instability Prediction using different indices for two area test system.

Method Marginal Loading (MW)

Computer Analysis EVI DMI

Increase of L7 3900 3900 3900

Increase of L9 4250 4500 4300

Increase of L7 and L9 3275 3300 3250

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Fig 6: Effect of gradually increase of load 7 on damping factors.

Fig 7: Effect of gradually increase of load 9 on damping factors.

Fig 8: Effect of gradually increase of loads 7 and 9 on damping factors.

Fig 9: Effect of gradually increase of load 7 on different oscillatory

instability indices.

Fig 10: Effect of gradually increase of load 9 on different oscillatory

instability indices.

Fig 11: Effect of gradually increase of loads 7 and 9 on different

oscillatory instability indices.

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