Power system is a complicated system. It is consist of several main parts which
can be arranged as following; generation, transmission, distribution and loads.
Each of these parts contains several complex structures and components (Kouzou
et al., 2010; Samimi and Golkar, 2012). The
need for controlling and providing continuous and secure operation of power
system causes many researchers to devote their investigations to such a large
nonlinear system (Radaideh, 2003; Belhadj
et al., 2007; Sabahi et al., 2008).
However, there are some documents which still reveal the need for more effective
methods to analysis and monitor the undesirable happenings of power systems
(Al-Odienat, 2006; Zribi and Rifai,
2006; Haidar et al., 2007). In order to get
rid of such problems, the parameters of power system should be identified using
online measurements of signals. In this case, the role of identification methods
becomes important. In this study, It is supposed to use Subspace System Identification
(SSI) methods for improving the power system operation.
Numerous investigators have worked on SSI methods. They used different SSI
algorithms for different applications (Katayama, 2005;
Keyvaani et al., 2010; Jamaali
et al., 2011). The first footsteps of SSI applications in power systems
may be seen in Kamwa et al. (1996). The study
provided low order model of large power system using N4SID algorithm of SSI.
Results of the study express that SSI based model is in lower orders, more optimized
and more suitable for controller design in comparison with classical system
identification and modeling.
A Heffron-Phillips model of synchronous generator was identified by Karrari
and Malik (2004) using subspace identification algorithms and online measurements.
According to Soliman et al. (2008), the parameters
of a Heffron-Phillips model of a synchronous generator was extracted from closed
loop data using SSI algorithms. It divides identification problem of a closed
loop system to two open loop identification and then it uses SSI algorithms
to identify each open loop transfer function. Using some mathematical processing
of provided transfer function, it provides a transfer function as generator
model. Wu and Malik (2006) discussed a model predictive
controller design for multi-machine power system using SSI algorithms. The design
uses a recursive subspace system identification algorithm in order to provide
a MIMO self-tuning adaptive controller; therefore it can be used for online
Zhou et al. (2006) mentions use of different types
of power system signals applicable to SSI algorithms. It uses such signals to
provide identification data. Modal analysis of power system was developed using
subspace system identification methods and provided data. Ghasemi
et al. (2006) also discussed modal analysis and oscillatory stability
study of power systems based on SSI methods. It provided a voltage stability
measure using identified critical modes of power system. Cai
et al. (2009) discusses a PSS using stochastic subspace system identification
approaches. It also mentions small signal analysis of power systems.
In this study, draw-backs of classical methods for power system analysis and
controller design are investigated. Moreover, Subspace System Identification
is used to extract beneficial properties of power system for analysis and controller
designs. A Linear Quadratic Gaussian (LQG) controller is designed based on the
information provided by subspace system identification methods.
SUBSPACE SYSTEM IDENTIFICATION (SSI) METHODS
The considered system is:
are samples of input, output, state vectors and
are static, zero average state noise and output noise vectors. Subspace system
identification problem can be formulated as below:
There are N samples of input vectors u = [u0 u1 u2
and output vectors y = [y0 y1 y2
from a system of order n. Find A, B, C, D, Q, R, S matrices and n for the structure
defined in Eq. 1.
There are two basic subspace system identification algorithm expressed in Table
1. They use the same measurements, same block Hankel matrices, different
types of projections, SVD of different matrices, the same method for extraction
of system order and different extended observability matrices. MOESP does not
need to estimate future states of system but N4SID provides future state vectors
by using a weighting matrix.
||Comparison of two basic SSI algorithms, MOESP (MIMO output-error
state space) algorithm and N4SID (numerical algorithm for subspace state
space system) algorithm
MOESP uses extended observability matrix to extract system matrices but N4SID
uses future states and through a least square problem estimates system matrices.
Table 1 expresses the following advantages for subspace system identification algorithms; SSI Algorithms are the only system identification methods that can easily and extensively be 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 dont need nonlinear optimization. Some SSI algorithms only use output data to identify a model. This is a considerable advantage.
Since, the algorithms expressed in Table 1 use exogenous
inputs, they are called deterministic subspace identification algorithms. Those
SSI algorithms that dont use exogenous inputs are stochastic. To full
fill the comparison, a stochastic SSI algorithm is provided in a stochastic
SSI algorithm uses output data and provide A and C matrices. They also provide
an innovation model in order to estimate future states.
APPLICATION OF SSI METHODS FOR SMALL SIGNAL ANALYSIS OF POWER SYSTEMS
Power system is generally a nonlinear system. Therefore, one should follow the following stages to achieve small signal properties of a power system; (i) Finding the details of all included elements (Generator constants, Transformer and line parameters,
), (ii) Finding nonlinear model of power system using constant, parameters and theoretical relations of variables for different power system elements, (iii) Solving a load flow problem in order to provide an operating point, (iv) Linearization of nonlinear model using the provided operating point and (v) Application of modern small signal methods to provide small signal properties.
Providing an operating point, a nonlinear modeling and linearizing the model are all tough works in application, especially when the system is large. There is always a big gap between the analysis done on a piece of paper and the system behavior. Such a method is not applicable for monitoring of power system and this is a considerable draw-back for a scientific method.
Classical identification methods are useful in many applications. When using a classical system identification method, the biggest difficulty origins from Single-Input/Single-Output (SISO) structure of such methods. Classical system identification methods may fall into whirlpool of over parameterization. Coping with such problems is itself a new problem.
Our suggestion for overcoming such problems is to use Subspace System Identification (SSI) methods. SSI methods are good solution for Multi-Input/Multi-Output (MIMO) systems. They can be considered as the bridge for passing over the gap between real world system and theoretical analysis. The next section investigates SSI methods to glorify their useful advantages for small signal analysis of power systems.
The SSI advantages expressed in previous section can be used to overcome the difficulties with classical small signal analysis of power systems. The above five steps can be reduced to the following three steps Using SSI methods; (i) Measuring input/output signals of power system, (ii) Identification of a linear model for power system using SSI algorithms and (iii)Application of modern small signal methods to provide small signal properties.
As it can be seen, the four first steps vanished and two other steps replaced them. The fifth step left with no change. Therefore, one can provide small signal analysis of power systems in an easier and faster way.
Signal measuring is the starting point of system identification. The most effective
inputs must be used since measured signals should have enough persistent excitement.
In attention to differential equations of a single machine power system (Kundur
and Balu, 1998), mechanical torque and field voltage are proper inputs.
Suppose that input vector u and output vector y of a power system have been measured. As an identification problem, goal is to find small signal properties of power system (Modes, Damping Ratios, Oscillation Frequencies, Participation Factors) using several samples of u and y.
It is announced that having N samples of input/output vectors and utilizing a subspace system identification algorithm, one can identify the following state space linear model:
One can find system modes and as a result damping factors and damping frequencies by digging matrix A. But the state vector x of model is not that of real power system obtained using analytical methods, since the state vector x is not unique. Therefore, mode in state participation factors can't be utilized using identified A.
In order to cope with such a problem, it is proposed to use modal canonical realization of Eq. 3. Using T as a similarity transform matrix, one can provide the following modal canonical realization:
Generally, Λ is in Jordan and block diagonal structure. Mode in state participation factor (pki) is defined as:
where, akk is the diagonal element of system matrix. Since, in Eq. 4, the system matrix is diagonal with modes as its diagonal elements:
Therefore, modal canonical realization can maximize (100%) mode in state participation factor of model. In order to clarify the point, suppose that u is zero and z0 is initial condition vector of modal canonical realization. Therefore:
Therefore, the only participated mode in state zi is γi, so the participation factor of mode γi in state zi is 100% and each mode is mapped to a state.
Considering above point and output equation of Eq. 3, one can write:
Therefore, output yk is affected by mode γi and mode in output participation factor (pki) is proposed as:
In order to provide participation factors, one may need z0 which can be provided through following relation:
x0 is the initial condition vector of identified state space model which is also provided by SSI algorithms.
Some investigators (Hashlamoun et al., 2009)
discuss another kind of participation called state in mode participation factor.
In most of literatures state in mode and mode in state participation factors
are the same and they have been used interchangeably, however, there is a discussion
on some differences by Hashlamoun et al. (2009).
Single machine three bus system: Power system shown in Fig.
1 is a three bus single machine power system with no control and exciter.
The parameters of the system are those used in (Kundur and
Balu, 1998). It is supposed to extract all small signal properties of system
using SSI Algorithms and the methods illustrated in previous sections.
Identification process should be provided with measured input/output signals. Computer simulations have been conducted using the system shown in Fig. 1 in order to measure input/output data. It is recommended to use mechanical torque as input and rotor speed or its angle as output signals since the extraction of small signal properties of generator angle and speed is desired to be achieved. Mechanical power can be used as input signal since in a per-unit system, torque and power are the same.
In order to have enough persistence excitation in input signals, one may add a white noise to input signals. To provide more realistic operating conditions, one may add a white noise to output signals, as well. Effect of noises will be investigated later.
300 samples of input/output data acquired through a 30 sec simulation. Using the SSI algorithms presented in Table 1, some linear models were identified and results are presented in Table 2. It is clear that to investigate performance of noises, the noise average cannot be manipulated because the operating point may vary which is not applicable in this study. Each noise variance was altered separately in order to see its effect.
In Table 3, one can see that an increase in input noise variance may lead to a better model from the view of FPE measure but one should be conservative when estimation of small signal properties is under consideration. Actually, a large increase in input noise variance may alter the operating point or its absorption area and may lead to instability.
Comparison of SSIM4 and SSIM5 with SSIM2 in Table 2, one
can see that output noise has no effect on subspace system identification. The
point is something related to application of consistence linear algebra tools
in SSI. Left eigenvectors of a wide matrix are not sensitive to additive white
noise considerably (Katayama, 2005). Therefore, the identification
is not sensitive to output noise.
|| Single Machine 3 bus power system
|| A stochastic subspace system identification algorithm
|| Voltage regulator model
Since, applied input noise is too weak, the identification process has no effect on normal operating conditions of power system.
State estimation and LQG controller design: The single inertia model
of a turbine generator connected to an infinite bus is represented by an 11th
order nonlinear state space set of equations. Its state vector is
are the rotor load angle relative to the infinite bus-bar and its differential,
are the electromagnetic states, ve and vf are voltage
regulator states defined in Fig. 2 and Ap and Tm
(pu torque) are governor states defined in Fig. 3.
||Small signal analysis of power system using ssi algorithms;
abriviations are: classical model (CM), subspace system identified model
(SSIM), subspace system identification algorithm (SSIA) and final prediction
|| Governor states
||Performance of subspace based LQG control (solid) of nonlinear
power system and the response without LQG (dashed)
The voltage regulator model, for a very fast excitation system, is a simplified representation of an IEEE Type AC4 aimed to give the transient performance. Field current can be forced up or down but cannot reverse. The governor model is a 2 time-constant approximation and represents main and interceptor values being governed in parallel to give a fast response. It is assumed that opening and closing speeds are the same which is not typical of present practice, opening being comparatively slow.
To validate the proposed approach to state estimation and controller design for nonlinear power systems, some simulations have been arranged using matlab7/simulink software. We used the nonlinear power system model described in above. It means that the power system is a turbine generator connected to an infinite bus through a line. The sampled signals were used to identify a linear model of power system. Then the state space matrices of the model were applied to design an LQG. The LQG can be redesign in any time instances which is more suitable.
Figure 4 and 5 show the load angle response
of system following a 3-phase fault of 100 msec, with two controller designs
compared with performance without supplementary control. The fault is assumed
to be at the high voltage terminals of the generator transformer when the generator
is at 0.8 pu power, unity power factor and the tie-line impedance is assumed
to be unchanged. Load angle to the infinite bus and field voltage are measured
and used as a basis of estimation.
|| Performance comparison of analytic LQG (dashed) and subspace
based LQG (solid)
An illustration of the performance of a subspace based LQG controller of nonlinear
power system is shown in Fig. 5. The LQG performance is very
good. An 11th order linear model of nonlinear power system has been used to
design a usual LQG controller. The performance of subspace based LQG and analytical
LQG have been compared in Fig. 5. Both LQG controllers are
the same in all common parameters. It is clear that the proposed SSI based LQG
controller can successfully cope with damping of the second swing while there
is a little increase in the first swing.
In this study, it is shown that system identification can be very helpful for analysis and controller design of power systems. Moreover, different models can be identified for different applications based on sampled signals of power systems. The pitfalls of analytic methods for power systems can also be avoided using models identified by system identification tools.
The study proposed that subspace system identification is a useful tool for small disturbance analysis and controller design of power systems. In this case, a state space multi-input/multi-output model of system was identified using the sampled data and subspace system identification algorithms. Furthermore, extraction of modes and their participation factors were easily investigated using SSI methods. However, it was a need to apply some modification in SSI algorithms. Additionally, It was shown that linear controller design based on the online system identification was very easy to implement.