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\title{Nonlinear Estimation of External Power System Dynamic Equivalent Parameters}
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             \author[1]{Dr. R.  Gueddouche}

             \affil[1]{  LSEI, University of Science and Technology Houari Boumediene}

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\date{\small \em Received: 14 February 2012 Accepted: 2 March 2012 Published: 15 March 2012}

\maketitle


\begin{abstract}
        


Based on the concept of the external power system dynamic equivalent for the study system, this paper proposes a novel evolutionary method for the identification of the equivalent?s parameters, comparing the answers of the complete network and its equivalent following the small disturbances which emerged in the study system. The proposed method is demonstrated and compared with the original system using the 10 machines 39 buses New England test system. The comparison shows that the proposed approach can preserve all dynamic properties of the original network.

\end{abstract}


\keywords{Dynamic equivalents, Dynamic modeling, Genetic algorithm, Nonlinear identification, Multimachine system, Parameter identification, Power system, Struc}

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\let\tabcellsep& 	 	 		 \par
Introduction lectric power systems have long been perceived and exploited like national entities or regional areas. The interconnections between such zones being mainly used with the aim of help in case of failure in a nearby zone. In the new context of the electricity sector, this situation has changed since the interconnections are increasingly used in order to maximize the exchange of electricity through power exchanges recently introduced. This leads the transmission system operators (TSOs) to operate their systems increasingly close to their limits. In an individual way, each TSO is obliged to re-examine its way of doing the usual studies on its own network. In particular, a more refined modeling of the influence of neighboring systems on its own should be considered. Moreover, the process of extension of the synchronous area which continues today, led to a significant increase in the size of dynamical systems to study. Besides the difficulty posed by the size, new structural phenomena occur from this extension as the electromechanical oscillations of low frequency observed between generators remote network.\par
Many dynamic equivalence techniques have been developed over the years \hyperref[b0]{[1]}- \hyperref[b5]{[9]}. More research focused on coherency equivalents and modal equivalents nowadays, however, they need detailed data of the external system which may be difficult to get in the power market environment. Some estimation methods are based on available information from the boundaries nodes and do not require any knowledge detail of the external network. The classical estimation methods are mainly based on the linearization of the system around an operating point with theoretical constraints validity \hyperref[b6]{[10]} - \hyperref[b8]{[12]}. Consequently, they are limited by their validity in nonlinear practical applications, offline and online applications. This paper proposes a new evolutionary approach for estimating parameters to determine a dynamic equivalent model of an external power system from synchronized measurements of disturbance occurred in internal system, which obtained by PMUs (Phasor Measurement Units). This approach was applied to 10 machines 39 buses New England test system \hyperref[b9]{[13]}. Comparing the properties of the equivalent system with ones of the original system, the result shows that the reduced system can represent dynamic behaviors of the original system well, for any kind of disturbance. 
\section[{II.}]{II.} 
\section[{Dynamic Equivalents}]{Dynamic Equivalents}\par
The equivalent is represented by a model with unknown parameters. In operation, the power system is often perturbed by small random disturbances. The estimation of unknown parameters of the equivalent model is operated by comparing the measurements obtained from the real network, and the same measurements made on the equivalent network by minimizing some objective function. 
\section[{a) Dynamic equivalent model}]{a) Dynamic equivalent model}\par
The basic requirement of the dynamic equivalents is that the response of the equivalent system can fit the original system dynamically and approximately when faults happen in the internal system. In our equivalent system, equivalent generator represents dynamic effect of the external system on the internal system located at the boundary bus (Fig. \hyperref[fig_0]{1}). When there are multiple boundary buses connecting the external system with the internal system, multiple generators are adopted to represent dynamic influence of the external system on the internal system, with an equivalent generator located at each boundary bus \hyperref[b10]{[14]}. Parameters to be estimated are: ????,????,?????,??,??,???????, voltage regulator gain ?? ?? and its time constant ?? ?? . The parameters identification for nonlinear systems includes deterministic and evolutionary methods. Evolutionary methods offer greater ease of adjustment problems and other advantages over deterministic methods because they:\par
-Require only the calculation of the objective function, without this last being forced to be continuous or differentiable.\par
-Easily adaptable to multi-objective problems optimization, and complex systems with very important unknown number of parameter.\par
-Have great theoretical probability to find the global optimum.\par
-Offer great simplicity of implementation.\par
An evolutionary approach based on genetic algorithms is presented in this document, to identify the dynamic equivalent model parameters, by minimizing the range of speed variation, electrical power variation and terminal voltage variation produced in all generators of the internal system between the original and the equivalent system. 
\section[{i. Error function}]{i. Error function}\par
The original system equations are generally written: \hyperref[b11]{(5)} Where: A= State matrix (4m× 4m). B = Control matrix (4m×2m). C = Output matrix (2m× 4m).\par
? Î?"?? Absolute angle rotor variation (rad. elec/sec).\par
? Î?"ð??"ð??" Angular velocity rotor variation (pu).\par
? Î?"????? Internal voltage variation (pu).\par
? Î?"??ð??"ð??"?? Internal excitation voltage variation (pu).\par
? ?????? Mechanical torque variation (pu).\par
? ?????? Order excitation systems variation (pu).\par
? Î?"???? Active power variation (pu).\par
? ?????? Change in terminal voltage generators (pu).\par
The equations describing the dynamic equivalent system are expressed similar to those of the original system (  {\ref 5}), but with a simplified structure and less parameters. \hyperref[b12]{(6)} are all functions of ??, the equivalent system parameter vector to be identified. Therefore, an error function may be defined by: \hyperref[b13]{(7)} Where Î?"?? represents the responses of the original system, which are directly measureable.\par
And Î?" (??), must be calculated by simulating the equivalent system with the same disturbance. 
\section[{ii. Objective function}]{ii. Objective function}\par
The idea is to minimize the error function \hyperref[b13]{(7)} between the measure and the model output for all machines belonging to the internal network. The mathematical model of the optimization problem is a multi-objective function and can be formulated as follows: \hyperref[b4]{(8)} Where:\par
By weighting the measurements by a weighting factor ????, our objective function becomes: \hyperref[b5]{(9)} Where: ?????????= number of machines belonging to the study system (internal system).\par
iii. Encoding and initial population\par
The encoding of individuals is an important parameter in population research methods. These are represented as a strings (chromosomes) containing characters or genes of a predetermined alphabet. There are different ways to code a solution. In our study, the individual is represented by eight parts of chromosome corresponding to the eight parameters to be estimated, each gene (parameter) is represented by its physical value, which means that, the real coding is adopted (fig. \hyperref[fig_3]{2}).\par
The GAs requires an initial population to start search process. Applied methods generate randomly a set of solutions belonging to the following area: \hyperref[b6]{(10)} Where:\par
The individuals' number in the initial population is chosen such that:  To form a group of (Local parents), in which each parent "i" satisfied the fitness function minimum: \hyperref[b9]{(13)} Where:\par
-A number of individual « ???????? » ;\par
To form a group of (Global parents), in which each parent "k" satisfied the fitness function minimum: \hyperref[b10]{(14)} Where:\par
The local parents group includes the favorable solutions for each type of measure that corresponds to each machine belonging to the internal system. However the global parents group includes the favorable solutions for each type of measure that corresponds to all the machines belonging to the internal system. The population called, "Parents population" is made up of the two groups (Global parents \& Local parents). 
\section[{b. Multi-parent recombination}]{b. Multi-parent recombination}\par
Recombination is used mainly in evolutionary strategies. Contrary to k points crossover operators, which exchange information between two parents, the recombination creates the descendants, by weighting many parents components'. We define three weighting operators: Where:\par
??h?? ? Integer belonging to \hyperref[b0]{[1,}\hyperref[b4]{8]}, which represents crossed chromosome order.\par
???? 1 ,???? 2 : Two individuals of local parents group.\par
We generate ???????? individuals for construct the new population (generation) from the two groups' "local parents" and "global parents", with the following steps: -we choose a random pair of individuals from the local parents group.\par
-a crossed chromosome is chosen randomly « chr ».\par
-a random real number, "h", is generated between 0 and 1, to select the weighting operator (Fig. \hyperref[fig_6]{4}). (15)\par
-we move to the next individual "i +1" when the number of weighting point "k" of the individual "i" is complete.\par
3. Accordance with the procedure for selection of parents, the best individuals evaluated by the fitness function \hyperref[b8]{(12)} between parents population and this new population, construct the new parents population and clear the old parents population for the next generation (elitist strategy).\par
III. 
\section[{Algorithm}]{Algorithm}\par
The complete algorithm of the proposed method is given below:\par
Step1 : Introduction of static and dynamic data of the complete system.\par
Step2 : Power flow calculation, linearization, disturbance choice and network dynamic simulation.\par
Step3 : Border node choice and equivalent system linear model construction from results of the power flow.\par
Step4 : Generating the initial population according to the procedure specified in subsection (III.B.3).\par
Step5 : Run equivalent system dynamic simulation and evaluate all individuals by fitness function \hyperref[b8]{(12)}.\par
Step6 : Construct the population of parents following the procedure described in paragraph (III.B.5.a), with equations \hyperref[b9]{(13,}\hyperref[b10]{14)}.\par
Step7 : While the number of generation has not reached the maximum number ?? ð??"ð??"???? :\par
Step7.1 : Generate the new population using described procedure in paragraph (III.B.5.b).\par
Step7.2 : Run equivalent system dynamic simulation and evaluate all individuals by fitness function \hyperref[b8]{(12)}.   
\section[{Application}]{Application}\par
To validate the proposed approach, developed algorithm was used to build the dynamic equivalent of transmission network IEEE New-England 39 bus \hyperref[b9]{[13]}. It represents a simplified New England transmission network (northeastern United States) that is part of real U.S. network. This network consists of 10 generators (PGtotal= 6.19 GW, QGtotal= 1.28 GVAr) and 39 bus with 19 load bus. The original system structure is as Fig. \hyperref[fig_9]{5}. new population (Fig. \hyperref[fig_7]{3}).  In Fig. \hyperref[fig_9]{5}, the shaded part is the internal system and the rest is the external system to be reduced. There are a boundary bus 16 and two tie lines 16-15 and 16-17 between the internal and the external systems. Fig. \hyperref[fig_10]{6} is the sketch of the equivalent system.  \hyperref[tab_1]{II}. These parameters were obtained after several tests with an appropriate adjustment. 
\section[{? > 0}]{? > 0} 
\section[{Table II : Control Parameter Values}]{Table II : Control Parameter Values}\par
To allow a better parameters estimation, different disturbances were applied in different parts of the internal system. Each proposed disturbance has a period of 10s. An event is generated each time to the first second of the simulation. The algorithm verifies the equivalents behavior in period of 10s, by comparing its response with the response of the original system. If the equivalent response exceeds some level relatively to the original systems response in a new disturbance, the equivalent generator parameters are recalculated and updated. Each solution obtained by applying the previous disturbances must be part of the initial population of the next estimation procedure. The estimated parameters of the equivalent generator are presented in Table \hyperref[tab_1]{III}. 
\section[{b) Validation}]{b) Validation}\par
A small perturbation represented by disconnecting the line 3-18 is applied to bus18 of the equivalent system and the original system respectively at time, t=2s and lasted for about 12 cycles. Time domain dynamic simulations were performed both on the two systems.  In order to verify the performance of the proposed dynamic equivalent approach in the case of the large disturbances, three-phase fault was applied to bus 30 of the equivalent system and the original system respectively at time t=2s, and the fault lasted for about 3 cycles. Time domain dynamic simulations were carried out on the original system and the equivalent system, and dynamic responses are shown in fig. \hyperref[fig_0]{10}    The simulation results of equivalent system with estimated parameters are encouraging and show the effectiveness of the developed algorithm and the ability of the estimated equivalent to reproduce the influence of the external system on internal system for small and severe disturbances.\par
V. 
\section[{Conclusion}]{Conclusion}\par
In this study, we proposed a nonlinear estimation method for parameters, based on an evolutionary algorithm. In order to obtain the external system dynamic equivalent, several dynamic simulation disturbances were applied to increase the accuracy of the model estimated by the developed algorithm. The proposed approach does not require data of external system. This approach requires only configuration information, settings and operating status of the internal system. Dynamic simulation was performed on both the original system and the equivalent system under different operating conditions, and the results show that the obtained equivalent system can represent the main dynamic characteristics of the original system well. Thus, the approach proposed is proved to be feasible and has potential for tackling the complex practical application in power system.  
\section[{Global Journal of Researches in Engineering}]{Global Journal of Researches in Engineering}\begin{figure}[htbp]
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\caption{\label{fig_0}Fig. 1 .}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Nonlinear}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Fig. 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{13}\includegraphics[]{image-7.png}
\caption{\label{fig_5}1 .Fig. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Fig. 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-9.png}
\caption{\label{fig_7}Step7. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{23}\includegraphics[]{image-10.png}
\caption{\label{fig_8}Nonlinear 2 ?? 3}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5}\includegraphics[]{image-11.png}
\caption{\label{fig_9}Fig. 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6}\includegraphics[]{image-12.png}
\caption{\label{fig_10}Fig. 6 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-13.png}
\caption{\label{fig_11}}\end{figure}
      \begin{figure}[htbp]
\noindent\textbf{.} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_0}Table . I}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{I} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_1}Table I :}\end{figure}
 			\footnote{F © 2012 Global Journals Inc. (US) 2012 July ð??" =} 			\footnote{F © 2012 Global Journals Inc. (US) 2012 July} 			\footnote{© 2012 Global Journals Inc. (US) 2012 July} 			\footnote{F © 2012 Global Journals Inc. (US) 2012 July Fig. 8 : Angle of the machine 10} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\end{document}
