non-linear approaches for reducing large power systems

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Non-Linear Approaches for Reducing Large Power Systems X. Lei, B. Buchholz, 0. Ruhle, B. Kulicke, A. Menze

Abstract

Issues on the establishment of equivalent networks are becoming essential for the deregulated power market. This paper presents a comprehensive tool for network reduction of large power systems. Through integrating digerent methodologies into a simulation program, the dynamic equivalent can be established by adopting one common database. With a readily integrated modified Gauss-Newton algorithm, network reduction can be executed under the dynamic conditions either in the time domain or in the frequency domain in coping with non- linear nature of the system involved. Furthermore, a novel algorithm based on dynamic coherency approach implemented readily into the simulation program is also presented. This novel approach determines coherent generators on non-linear basis in the time domain using the cross correlation technique, taking dynamic characteristics of the system involved into consideration. Two case studies are presented in this paper. Each of the non-linear approaches presented is applied for one of the case studies as application example. The results achieved validate the functionality of the approaches presented.

1 Introduction

Deregulation and privatization in the power market are significantly changing relations among power gener- ation, transmission system and distribution. Utilities, especially transmission companies (Transco) are faced to be oriented in a competitive market environment not only to maximize their profits. In this new situation, detailed network data among competitive companies are becoming more and more confidential. Thus, exchanges of the network data among these companies could become dif- ficult. However, with enhanced stability requirements, dynamic behaviours of the interconnected power systems involved need to be carefully studied online and offline. This includes studies such as overall system stability, dynamic security assessment and coordinating system controls in a global manner, etc. All of these studies require knowledge of interconnected neighbour networks. Thus, issues on the establishment of equivalent networks for a large, multi-owner interconnected power system become essential.

On the other hand, in power-system analysis, it is also a common practice to represent the parts of a large interconnected power system by some form of a reduced order equivalent model. Due to limited size and capacity of tools adopted for studies, such as real-time- digital-simulator (RTDS) or electromagnetic transients program for DC-application (EMTDC), users are sometimes forced to reduce their networks in order to match the size and capacity of the tool used. Further- more, a reduced model can simplify network calcula- tion and save investigation time in some cases, e. g. for AC filter design. Depending on different applications, such as studies on investigation and verification of dynamic behaviour, fundamental frequency overvoltage studies and AC filter performance calculations, equiv-

alent models are established either by static or dynamic network reduction.

Research in network reductions of large power systems has been undertaken since three decades. Various algorithms and procedures have been addressed in litera- ture. Static network reductions have been established as a conventional technology. However, due to the rigorous requirements to handle the complexity of large interconnected power systems, only a few of the algorithms for dynamic network reductions have been available for engineering applications. As a result, dynamic network reductions are often done using heuristic methods.

The motivation of this paper is to provide a comprehensive tool for reducing the order of large and very large power systems. Thereafter, two methodologies of establishing equivalent are presented, which have been implemented into a simulation program: a Gauss-Newton approach and a dynamic coherency approach. The Gauss- Newton approach can be used for establishing static and dynamic equivalent, while the dynamic coherency approach that has been newly developed and differs from existing algorithms, is more efficient for establishing dynamic equivalent with an enhanced accuracy.

This paper is focused on establishing dynamic equivalents with a simulation program. Two case studies are demonstrated based on real power systems. The fist case study is to establish an equivalent model of a large power pool where the power generated is transmitted to a load center by means of HVDC. This equivalent model is established for RTDS studies by using the Gauss-Newton approach. The second application estab- lishes a dynamic equivalent of a large interconnected power system by means of the dynamic coherency approach. The results of two case studies validate the effi- ciency of the non-linear approaches presented as modules in the simulation program.

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2 Methods for Establishing Equivalents

Previously, various approaches for network reductions have been proposed [ 1-31. Some of those procedures have been employed in practice [4]. Others are under development. These proposals are typically based on the linearization of the system involved and not integrated into a simulation program. Thus, they require data converting from/to a simulation program for receiving the established operating point and for verifying the results. In addition, linear methods cannot properly cap- ture complex dynamics of the system, especially during critical faults (major disturbances). This could present difficulties for reducing the orders of the non-linear models in that the equivalent established to provide de- sired performance at a small signal condition might not guarantee acceptable performance in the event of major disturbances.

Differing from most existing methods, significant features of the approaches presented in this paper are:

- Equivalents are established on transient conditions with enhanced accuracy, in coping with non-linear natures of the system involved.

- Complete procedure of each two approaches is implemented in one simulation program, respectively, where power systems involved are modelled on non- linear basis.

In the following section, as basis of the procedures for establishing equivalent simulation program is brief- ly outlined fKst, then the two non-linear algorithms integrated within the program are presented.

2.1 Simulation Program

A complex non-linear model of a power system can be described in a set of differential-algebraic equations by assembling the models for each generator, load and other devices such as controls in the system, and con- necting them appropriately via the network algebraic equations. To describe such a non-linear power system, the simulation program NETOMAC (network torsion machine control) [5 -61 is adopted in which the system of differential equations are transformed into sets of algebraic equations solved by using the well known trap- ezoidal rule. This program has been widely used for the simulation of electromechanical and electromagnetic transient phenomena as well as steady-state behaviour of a power system. The program provides an instanta- neous value module similar to the program EMTDC /EMTP and a full stability module like the PSS/E program in the time domain. It provides also a frequency analysis module with the fast Fourier transform (FFT) and an eigenvalue analysis module in the frequency domain. In addition, it integrates an optimization/identification mode [7-81 for solving various optimization tasks and parameters identification problems. Recently a novel method for automatic dynamic network reduction has been integrated into the program. One of the significant benefits of the program is that such a wide field of applications uses only one surface and one database without data converting among the modules.

An additional advantage of this tool is its simplicity in the verification of results. Once the equivalent is established, the verification of the equivalent can be per- formed with the same program, where the original system model is readily available. With the help of various analysis modules available in the program, the verification can be executed in the time- and frequency domain, and it can also be executed with the eigenvalue module to compare the oscillation modes of the original and the reduced system.

2.2 Gauss-Newton Approach

The procedure of the Gauss-Newton approach proposed for establishing dynamic equivalent in this paper is executed in four steps:

- Identification of coherent generators.

- Aggregation of coherent generators.

- Construction of passive network.

- Determination of parameters of aggregated equivalent generators and corresponding equivalent controllers.

For identification of coherent generators the eigenvector approach described in [9] is adopted. Based on eigenvalue analysis, the closed-form solution of the linearized swing equations is obtained in terms of weighted eigenvector, from which the coherency indices are de- rived. These coherency indices are as measures of the maximum deviation in the relative rotor angle between two generators during the entire transient period and in- directly ascertain the coherency among the generators.

The passive network reduction is executed automatically with the simulation program by applying a method similar to that given in [lo], where the individual equivalent generators replace the generators in the each coherent group and the reduced passive network main- tains the balanced steady-state load-flow conditions.

After groups of coherent generators are identified, the generators in each coherent group are aggregated to form an equivalent generator in each individual coherent group. The equivalent inertia is the sum of the inertia of all generators in the group. The reactance, transient reactance and sub-transient reactance of each equivalent generator are obtained by paralleling the corresponding values of all generators in each coherent group.

In order to achieve enhanced accuracy of the aggregation, differing from existing algorithms, the algorithm proposed determine parameters of aggregated generators through a “least squares” fitting process by adopting the Gauss-Newton algorithm to be described in the last part of this section. Starting with values resulted by aggregating coherent generators as initial values, the final parameters of each equivalent generator are determined at the minimum of the target function in subsequent “least squares” fitting process. To obtain such parameters of equivalent generators (and their equivalent controllers) that can well represent system behaviours of the original system, the target function for the fitting process must be properly selected. Suppose an interconnected power system to be reduced consists of two sub-

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! Subsystem i j remaining j j unchanged j

Subsystem to be reduced

Original model

No changes I Subsystem I

Equivalent Reduction

Fig. 1. Establishing an equivalent model

systems as shown in Fig. 1: A subsystem remaining unchanged and a subsystem to be reduced.

From the subsystem remaining unchanged looking into the subsystem to be reduced, state variables (Pi, Qi, U;, 0;) on the tie lines (i = 1, . . ., k) indicated in Fig. 1 can well represent behaviours of the external systems, e. g. behaviours of the subsystem to be reduced. Thus, these state variables on tie lines are selected to form the target function for the fitting process that is described as follows:

min f ( x ) ,

r=O

where the subscript “org” and “eq” denote state variables in the original system and the system with the equivalent, respectively, and x denotes the parameter vector to be determined by the fitting process. Due to the fact that the target function (1) considers transient performance over a pre-defined integration time, parameters of individual equivalent generators and controllers determined at its minimum can reflect non-linear natures of both original and equivalent models. Note that depending on requirements on equivalent, the target function can also contain Pi , Q; only, if power swings are a main concern of the equivalent.

To minimize the target function f ( x ) given in eq. (1 ), the “least-squares-error” technique is proposed based on a modified “Gauss-Newton” algorithm [8]. In the following, the basic approach of the algorithm is described:

subject to: xlow I x I xu,,,

wherexlowandx,,,(x E R“)are the parameterlowerand upper limits, Forg ; ( x ) denotes the i-th state variables (i = 1, . . . , m) of the original system which are supposed as the known values, Feq ; ( x ) denotes the i-th corresponding state variables (i = 1, . .., m) of the reduced system that can be represented by Taylor first order derivations of Feq ; ( x ) at xk:

Eq. (3) can be expressed in a compact form:

Feq ( x ) = Fq (xk ) + Flxk Ax, (4) with the vector Fq(x) and the Jacobian matrix F(”*rn) =FIX, at xk.The necessary condition for the optimal solution off(x) (see eq. (2)) is then given:

By inserting eq. (3) into eq. (2), the well-known Gauss Equation is deduced:

F ~ F A ~ = F~(F, , , -F, , ) . (6)

From eq. (6) , the optimal parameter set x can be determined iteratively by calculating:

xk+, = xk + ( F ~ F )-I F T( F~~~ - F ~ ~ ) (7)

to meet the convergence condition

f(Xk+l ) I f(x, 1. (8) The iterative minimization of eq. (2) continues until

pre-defined criteria are satisfied. To improve the convergence of the “least-square” algorithm described above, a special treatment has been proposed for dealing with the linear dependent components in the matrix, which ensures the convergence of the algorithm.

The fitting process for determining parameters of equivalent generators and corresponding controllers is illustrated in Fig. 2, where the values of the selected state variables of the original system are first calculated by the simulation program, and then stored in a common block of the program. Once the fitting process starts, the stored values are automatically fed into the target function to compare the values of the corresponding state variables of the equivalent model. By means of the fitting process described, the equivalent model can be well established with a required accuracy, in coping with non-linear natures of the system involved.

Suppose full generator models represented by Park equations are adopted in both of the original and equiv-

,_____________.___........... j Original system j

Measurement / ______________,_____________.

model I

A

IparametersI A I j

Simulation svstem

Equivalent

Fig. 2. Fitting process for establishing equivalent

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alent system models. Parameters of each equivalent generator to be determined by the fitting process include:

S, rated apparent power, H inertia, xd d-axis synchronous reactance, xi d-axis transient reactance, x i d-axis sub-transient reactance.

If any equivalent exciter and governor are also considered as equivalent controllers, parameters to be determined in equivalent controllers are mainly gains and selected time constants.

It is necessary to point out that for static network reductions where all generators are replaced by voltage sources, target functions can be varied from the standard form given in eq. (1) according to different purposes of the study. For example, for setting up an equivalent used for a fundamental frequency overvoltage study, the peak value of the voltage of the system is relevant. Thus, the target function can be described only considering the voltage term. In case of establishing equivalent for har- monics studies, frequency spectrum of the system involved is relevant. Then, the target function can be described in terms of the square-errors between the frequency spectrums of the original and the equivalent model. In this case, the corresponding procedure is executed in the frequency domain of the simulation program.

The integration of the modified Gauss-Newton algorithm provides a flexible formulation of target functions. According to different formulations of the target function, equivalents can be established to satisfy spe- cific requirements. To achieve acceptable equivalents by using this module, however, users may need some experiences in setting up reasonable target functions. Once the target function is established, the determination of parameters of the equivalent can be automatically executed within the simulation program.

2.3 Dynamic Coherency Approach

A novel algorithm for establishing dynamic equivalents has been newly developed based on dynamic coherency approach. This approach performs in the time domain and does not involve any linearization of the system. The most significant features of this algorithm are :

- By analyzing characteristic similarities in waveforms of individual generators on transient conditions with the help of the cross correlation technique, coherent generators are statistically determined with enhanced accuracy.

- By exciting the system involved with time-dependent voltage-white-noise having a uniform random feature, dynamic characteristics of generators are identified independently from excitation types and locations.

The implementation of this algorithm into the simulation program provides an effective means for automatically establishing dynamic equivalents of a large power system. In this paper, the basic ideas of the algo-

UI 14- !,Voltage white noise

Fig. 3. Schematic interconnected power system

rithm are outlined. A detailed description with theoreti- cal fundamentals is discussed in a subsequent paper.

To illustrate the procedure for establishing dynamic equivalents using the algorithm proposed, an interconnected system is given schematically in Fig. 3. This system consists of two subsystems connected via three tie lines. The subsystem I will remain unreduced, while the subsystem I1 is to be reduced.

The procedure for establishing dynamic equivalents is executed in the four steps, almost the same as the Gauss-Newton approach discussed in the last section:

- identification of coherent generator, - aggregation of coherent generator, - construction of passive network and - determination of parameters of corresponding equiv-

alent controllers.

Before the procedure starting, following pre-definitions are made: - Tie lines between the subsystems remaining un-

changed and to be reduced (e. g. tie line AD, BE and CF in Fig. 3) where time-dependent voltage-white- noises are injected.

- Node (e.g. node “N” in Fig. 3) that remains in the equivalent established.

- Lowest correction coefficient at which coherent generators are determined or number of generators that will remain in the equivalent established.

According to above pre-definitions, the full model of the system involved is established with the simulation program first. Then, the reduction procedure runs automatically until the equivalent is established. After establishing a load-flow condition that defines the operating point of the power system involved, the full model of the system is divided into two subsystem models automatically as shown in Fig. 3: - one (subsystem I) remaining unchanged and - the other (subsystem 11) to be reduced.

Thereafter, time-dependent voltage-white-noises are injected into the subsystem I1 through each tie line (e. g. AD, BE and CF) as disturbance with uniform random feature to excite the subsystem 11. In the following four steps to be executed are described:

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2.3.1 Identification of Coherent Generators

Generator responses to disturbance injected are simulated over a time period. The excited speed deviations of individual generators in the subsystem I1 are taken as characteristic value for the correlation analysis. To iden- tify coherent generators, similarities of waveforms among generators over time are analyzed by means of the correlation coefficient pij which represents the similarity between the two time-dependent speed of genera- tori andj, mi([) and q ( t ) [ l l].

Suppose the total energy E of time-dependent sig- nals ~ ( t ) over time is calculated as:

E[Ioi( t )12] = Ilco(r)12dt = const, -m

(9)

By definition of the correlation coefficient pij:

E [ W ; ( ~ ) ~ j ( f ) ] (10)

= Jm' and as following from the Schwarz inequality

with K = const, the similarity of the waveforms between the two time-dependent speeds can be identified as in terms of the correlation coefficient:

p.. 'I = 0 oi(t) and mi([) are uncorrelated when E [ o i ( t ) o j ( t ) ] = 0,

0 < lpVl < 1 wj(r) and wj(r) are correlated when E[w;(t)o,(t)] < K ,

IPijl= 1 mi(t) and oj(t) are coherent when E [ o j ( t ) o j ( t ) ] = K.

Thus, coherently oscillating generators have a correlation coefficient of + 1, while generators oscillating in phase opposition share a correlation coefficient - 1. In other cases, the correlation coefficient varies between + 1 and - 1. Experience shows that generators having a correlation coefficient larger than + 0.8 can be identified as a cluster of coherent generators that can provide suf- ficiently accuracy in establishing equivalent generators. Using the correlation analysis, the number of the equiv-

Fig. 4. Identification of coherent generators (correlation 1 - 2 = 0.93; correlation 1 - 3 = 0.0; correlation 3 -4 = -0.5)

alent generators is determined automatically in the procedure. As shown in Fig. 4, generators 1 and 2 are identified as acluster of coherent generators with a high value (+0.93) of the correlation coefficient, and generators 3 and 4 are not coherent generators due to a negative value of the correlation coefficient.

2.3.2 Aggregation of Coherent Generators

Generators being identified as members of a cluster are aggregated to form an equivalent generator that is connected to a single node (e. g. node "N" in Fig. 3). The equivalent generator must have similar behaviours as those of the original group of generators with respect to voltage and frequency variations. This can be achieved by calculating the impedance of the direct and quadrate axes for certain oscillation frequencies and by connect- ing them in parallel. The impedance of each generator to be aggregated is determined for following oscillation frequencies that are relevant for stability studies: Z" = R" + j X"

Z'=R'+ jX'

Z = R + jX

with 50 Hz (s = 1) for transient impedance;

with 0.5 Hz (s = 0.01) for sub-transient impedance;

with 0.001 Hz (s $1) for synchronous impedance.

The equivalent admittance of the aggregated generator in each coherent group is determined by (each impedance of individual generators in the group is connected in parallel):

Ye; = q"+ Y;'+ . . . + Y;: Ye; = u,' + Y; + . . . + Y;, Yeq =u , +Y2 + . . * + Y * .

(12)

From eq. (12), each equivalent circuit data of the generator such as xd, xh and x: are easily calculated. The equivalent inertial He, is calculated with the sum of the inertial with respect to the equivalent sum of their apparent power:

where inertial Hi is normalized by the apparent power of the i-th generator, and the power injections P,, + j Qeq of each generator are added to give an accumulated power injection:

This technique for determining the parameters of aggregated generators ensures that the sub-transient, transient and steady-state short circuit behaviours of the equivalent generators are mostly similar to those of the original group of coherent generators.

2.3.3 Construction of Passive Network

The passive network is constructed in such a way that the short circuit and load-flow behaviours of the

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original system remain unchanged. The algorithm adopted for constructing passive network is the same as that used in the Gauss-Newton approach. For establishing equivalent generators without changing established load-flow condition, an “ideal” transformer with a zero- impedance and a complex rating is connected to the node (e. g. node “N” in Fig. 3), where an equivalent generator is placed. By inserting “ideal” transformers, possible phase shifts of individual generator terminal voltages can be compensated.

2.3.4 Determination of Parameters of Corresponding Equivalent Controllers

The algorithm can distinguish between natural generator oscillations and controller influences. This is re- alized by modelling the system with and without considering controllers. In case of considering controllers, the Gauss-Newton algorithm described in the last section is used for determining parameters of the equivalent controllers. IEEE standard controllers or user-defined models can be adopted for equipping equivalent generators.

The dynamic coherency procedure for establishing dynamic equivalents runs based on non-linear models of the power system involved in the time-domain simulation. In comparison with algorithms based on linearized systems that are essentially only for small signal analysis, non-linear behaviour of the original system, especially during critical faults (major disturbances) can be captured by the corresponding equivalent.

3 Case Studies

3.1 Case Study I

The power generated by hydropower stations in an Asian country will be fed into a power pool and then transmitted to the load center through HVDC transmission links. In addition to an existing HVDC link, two new f 500-kV bipole HVDC systems, each with a rated capacity of 3 000 MW, will be installed.

One of the important issues is to establish AC equivalent systems for the power pool where the sending station of the HVDC link is planned. The equivalents to be established both for steady-state and dynamic modelling are mainly used for the following purposes:

- ACDC simulation studies. - Fundamental frequency over-voltage and switching

studies. - AC filter performance calculations.

In Fig. 5, the 500-kV grid of the power pool contain- ing about 100 generators is schematically illustrated. In the study, the 220-kV system was also taken into account. The rectifier station of the HVDC link is planned at “B” close to the hydropower plant. To investigate the impact between the power pool AC-system and the HVDC link, the AC equivalent shown in Fig. 5 was established nearby the rectifier station at “B”.

To establish such an equivalent, the procedure described above has been applied, taking all controllers of generators into account. The identification of coherent

Fig. 5. Dynamic reduction of a large power system

generators and passive network reduction were executed on initial steady state of the system. For determining parameters of aggregated generators and corresponding equivalent controllers, the fitting process ran on transient conditions of the system. To excite the system, a 100-ms three-phase fault on station “B” is pre-selected for the fitting process. A main concern of this case study is to investigate power behaviours around the rectifier station “B”. To ensure that the established equivalent represents the swing characteristics of the original system with an enhanced accuracy, the target function eq. (1) only con- tains active and reactive powers on the tie lines. As shown in Fig. 6, starting from the same steady-state condition,

Original system . . . . . . . Equivalent model

i u 0.5

MW

-2000

c) 0 , ,, , 1 I I 1 I I

h4w -lo00

-2000- tYVL I 1

0 0.5 1 1.5 2 2.5 3 s 3.5 t 4

Fig. 6. Behaviour of voltages and active powers a) U =f(t) for three-phase fault on station “B” b) P =f(t), line “A”-“B”, for three-phase fault on station “B” c) P =f(t), line “B”-“CI’, for three-phase fault on station “B”

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p.u.

t 1.OoJ U

0.75

1.25 Property Initial model Equivalent Machines 400 49 + 28

............ Nodes 2000 120 Branches 6 700 530

Tab. 1. UCTE/CENTREL dynamic simulation model

.............. i

- Original system ....... Equivalent model

Fig. 7. Voltage curves in case of sudden load rejection (sudden outage of HVDC on station “B”)

similar dynamic behaviours of the equivalent established are illustrated in comparison with the original system.

The equivalent established should be valid for relevant operating conditions. Thus, it is necessary to verify the equivalent achieved for a set of selected fault conditions. In accordance with the main concern of the case study presented, a load rejection (HVDC sudden out) on station “B” was considered as an example in verifying the equivalent established. Fig. 7 shows the voltage curves on station “B” after a sudden outage of the HVDC, which demonstrate a good consistency between both original system and equivalent model established.

3.2 Case Study I1

In order to test the quality of the reduction tool in practice, a data record describing the grid of UCTE /CENTREL was reduced. Calculations were executed both with the reduced and the full system model. The results are compared in the following chapters. Although a high reduction ratio was achieved, the computation results obtained with the reduced network are similar to the results of the transient stability calculations with the or- igin network.

The original model used as basis for building the equivalent consists of 4OOgenerators and describes thenet- work of the UCTE/CENTREL including the Preussen-E- lektra grid. In addition, a neighbouring grid is also interconnected. The different parts of the grid investigated are schematically illustrated in Fig. 8.

The original model is separated into two subsystems: one subsystem remains unchanged including the area under investigation, i. e. the northern part of Germany and

Fig, 8. Schematic illustration of the grid investigated

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Fig. 9. Relative deviation of fault clearing time

the western part of Denmark. These parts are named “model of the subsystem remaining unchanged” and “neighbouring grid” hereinafter. The other subsystem of the model named “equivalent” was reduced. Both areas are interconnected via seven original interconnection lines.

The complete extensive model was reduced using the automatic procedure for dynamic network reduction in the simulation program. Using this tool the number of generators was reduced to 49 equivalent generators. 28 generators are modelled in detail within the model of subsystem remaining unchanged and the neighbouring grid. Tab. 1 gives some characteristic figures about the model. Due to the fact that the common model will be used for transient stability investigations, a relatively high reduction ratio was chosen.

For evaluating the quality of the reduced network in transient stability investigation, the maximum admissible fault clearing time for three-phase short circuits on the seven coupling nodes in the detailed model, connect- ing both the neighbouring network and the equivalent, were simulated. The node names are given in Fig. 8. These calculations have been done using the initial complete model and the interconnected equivalent model. Both results were compared. The relative deviation of

the fault clearing time (fct) &, is illustrated in Fig. 9 for all seven coupling nodes (Nnode). Note that the relative deviation of the fault clearing time is defined as follows

where and are the maximal admissible fault clearing time by using each of original and equivalent model. From Fig. 9, it is obvious that the maximum deviation for the fault clearing time is -4.0%, i. e. the max. admissible fault

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Gen4: f

a) MW, Mvar

Gen 1: t P,Q o

Gen 2: p,

t l O o o W 1 p p Q o Gen 3:

20001 I I I I

10007L I I I

clearing time computed using the equivalent is 4.0% lower than the fault clearing time calculated using the original model.

For verification of the results a three phase short circuit at a node in the network part modelled in detail with a fault clearing time of 150 ms were calculated with both the initial complete model and the interconnected equivalent data set. The Figures below show the active and reactive power output of the generators in the detailed part of the model. The results for the initial model are given below in Fig. 10a, whereas the results for the total equivalent are shown in Fig. lob.

4 Conclusions

Reducing the order of large power systems became an important issue in the new deregulated market environment. A comprehensive tool for network establishing dynamic equivalents of large power systems has been presented in this paper. l k o non-linear approaches for establishing equivalents have been integrated into the simulation program NETOMAC: modified Gauss- Newton approach and dynamic coherency approach.

Most important features of these approaches are as follows: - determining equivalent generators and their equiva-

lent controllers on transient conditions with enhanced accuracy, in coping with non-linear natures of the system involved,

- flexible solutions for establishing equivalents to meet special requirements,

- complete procedure for establishing equivalent within one simulation program without any data conver- sion.

In this paper, two case studies based on two real power systems were demonstrated by applying each of the non-linear procedures. The results achieved validate the functionality of the procedures proposed.

The prerequisite for reliable networkreduction is the availability of advanced tools and skilled engineers with experiences on system dynamics. The implementation of the non-linear algorithms into the simulation program provides an effective means for establishing equivalents of large and very large power systems in deregulated market environments.

5 List of Symbols and Abbreviations

5.1 Symbols

target function active and reactive power voltage amplitude and -angle vector of state variable vector of parameter to be identified impedance, reactance, resistance and admittance inertial of a generator apparent power of a generator reactance of d-axle of a generator reactance of q-axle of a generator lower and upper limits of parameters energy of time-dependent speed of generators over the time constant value speed of a generator correlation coefficient relative deviation of the fault clearing time maximal admissible fault clearing time no. of the coupling node generator 1 ,2 ,3 and 4 partial differential operator deviation operator dimension of a vector rated value element index of a vector function of time sun-transient and transient value element or vector in equivalent and in original model matrix inversion

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(.IT transpose (‘)k iteration index

(a), element index of a vector

5.2 Abbreviations

Transco transmission company AC alternating Current DC direct current RTDS real-time digital simulator EMTDC electromagnetic transient program for di-

rec t current NETOMAC network torsion machine control EMTP electromagnetic transient program PSSE power system software/engineering FFT fast Fourier transform IEEE Institute of Electrical and Electronic En-

gineers HVDC high-voltage direct current UCTE Union for the Coordination of Transmis-

sion of Electricity CENTREL Regional group of the Central European

Electricity Companies

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The Authors Xianzhang Lei (1958) received the BS degree from Zhejiang University, China, and his MSc and PhD degrees in electrical engineering from the Technical Uni- versity of Berlin, Germany, in 1982, 1987 and 1992, respectively. From 1987 to 1993 he worked as a research fellow in the Dept. of Electrical Engineering at the Technical University ofBerlin. After that, he was as Post-doctor fellow at the Yale University, USA. Since 1994 he

has been a senior manager in the Power Transmission and Dis- tribution Group at Siemens in Erlangen, Germany. He is junc- tion-professor at several universities in the People’s Republic of China. (Siemens AG Power Transmission and Distribution (PTD EM IT), Phone: +499 13 1770467, Fax: +499 13 177 04 44, E-mail: [email protected])

Bernd Buchholz (1948) received his MS and PhD at the power engineering institute in Moscow/Russia in 1973 and 1976, respectively. After that, he was assigned project leader and later director of the R&D department at the Insti- tute of Power Supply until 1990. In 1990, he joined Siemens and took over the head of the R&D department of the division “Protection and Substation Control Systems”. Since February

2000, he is president of the Network Analysis and Consulting Group in the Siemens Power Transmission and Distribution Group. (Siemens AG, Dept. EV SENC, Paul-Gossen-Str. 100, 91052 Erlangen/Germany, Phone: +4991317-34443, Fax: +49 91 3 1 7-34445, E-mail: [email protected])

Olaf Ruhle was born in Germany in 1965. He received the Dip].-Ing. and the PhD (Dr.-Ing.) degree in electrical engineering from the Technical Uni- versity of Berlin in 1990 and 1994 respectively. Since 1993 he is a member of Power Transmission and Distribu- tion Group and the system planning department at Siemens in Erlangen, Ger- many. He is working in the area of simulation of dynamic phenomena in

electrical power systems. The majority of activities cover working on power system stability, dynamics of multimachine systems, control, optimization and identification problems in electrical power systems. He is responsible for the NETO- MAC / NETCAD / NEVA / NETOMAC Realtime program support, sale and training worldwide. (Siemens AG, Power Transmission and Distribution, Services, Network Analysis and Consulting, PTD SE NC5, Paul-Gossen-Str. 100, D- 91052 Erlangen, Germany, Phone: +49 9 13 1 732982, Fax: +49 91 3 1 735 159, E-Mail: [email protected])

Bernd Kulicke (1944) received the MSc degree in Electrical Engineering from the Technical University of Ber- lin/Germany and the Doctor degree in

[ 101 Tinny, W. F.; Bright, J. M.: Adaptive reduction for power Power Engineering from the University flow equivalent. IEEE Trans. on Power Syst., PWRS-2 of Darmstadtl Germany in 1970 and (1987) no. 3, pp. 351 - 360 1975, respectively. From 1970 to 1983

[ 1 I ] Papoulis, A.: Probability, Random Variables and Sto- he was with the Siemens Company, chastic Process. 2-nd ed. New YorkKJSA: McGraw-Hill, working in the High Voltage and Power 1984 ~~ ~ Engineering department. He is respon-

sible for the development of the NET- OMAC program and was mainly involved in performing Manuscript received on October 10,2000

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system studies including electromechanical and electromagnetic transients and stability problems. In 1984 he was ap- pointed a Professor and Director of the Institute of Electrical Power Engineering at the Technical University of Berlin. He is a member of the IEEE Power Engineering Society. (Tech- nische Universitat Berlin, Inst. f. Elektr. Energietechnik, Ein- steinufer 11 , D-10587 Berlin/ Germany, Phone: +49 30/314 23390, Fax: +49 30/3 14 21 142, E-Mail: [email protected] berlin.de)

Andreas Menze (1969) studied electrical engineering at the University of Hannover/Germany and earned his Dip1.-Ing. degree in 1996. From 1996 to 2000 he was engineer and project manager with PreussenElektra. Since 2000 he is with the system analysis division of E.ON Netz GmbH. His main fields of interests are system stability and dynamic interactions. (E.ON Netz GmbH, Regionalzentrum Nord, Vor

dem Nordwald 14, D-31275 Lehrte/Gerrnany, Phone: +49 5132 88 2041, Fax: +49 5132 88 2276, E-mail: andreas. menze@ eon-energie.c om)

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non-linear approaches for reducing large power systems

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