Model Order Reduction based on meta-heuristic optimization methods

Transcription

1 International Research Journal of Applied and Basic Sciences 3 Available online at ISSN 5-838X / Vol, 7 (): 5-3 Science Explorer Publications Model Order Reduction based on meta-heuristic optimization methods Mehdi Safaeian. Islamic Azad University, Hidaj Branch, Iran Corresponding Author m.safaeian@gmail.com ABSTRACT: Determining the stability region and recognizing the limits that exist in a large scale system such as in a power network are very significant. According to daily changes in these systems, such as changing power plan or network structure, it is necessary to use a dynamic method with instantaneous measurement for modeling. On the other hand, one of the biggest challenges in system identification is order selection. Due to the large number of parameters and daily changes in such system, system order permanently changes. In this paper a new method has been suggested. In this method we use metaheuristic algorithms, such as SA and SFL that are merged with other order selection methods to obtain more accurate results. Also, we use ERA/DC and Prony for extracting dynamic model. These Methods have been implemented on extracted data from simulated Iran power network Model in DIgSILENT software. Keywords: Optimization, Identification, Order Reduction, Prony, ERA/DC, PSO, Harmony Search, SA, SFL. INTRODUCTION Power network is a very large interconnected system. Stability analysis of this contributed system is very critical concerning economical and security issues. Due to dynamic structure of a power system, it is necessary to use a dynamic or an online model. Instantaneous measurement can be used to extract a new model to observe daily changes of system parameters and configuration. This results in calculating stability limits with more accuracy and reliable stability index (J. J. Sanchez-Gasca). Oscillatory instability has been known in power industry for a very long time, and several methods have been developed to prevent it. Such a condition may cause significant technical and financial damages to system itself and consumers. Now, according to identification algorithm, a suitable model will be extracted for analyzing dynamic behavior of power network (] K. Kimt, H. Schattlertt - J.J. Sanchez-Gasca). Model order selection is one of the important and challenging problems in system identification. Most of the time, system model and its details are not clear and accessible. Several methods for estimating model order have been proposed in previous studies. J. R. Smith and H. Ghasemi have proposed an approximate method. They suggested a primary order based on the number of output samples. This suggestion improves with singular value matrix of the system. Hassan Ghasemi uses singular value decomposition. Order can be selected whenever these values have a big drop. Also S. Lew and Goro Obinata proposed complicated methods for order reduction. These methods are complicated and need special identification algorithm to be implemented. In 9, meta- heuristic algorithms were used for these methods for the first time (Ebrahim Rahimpour- S.N. Sivanandam). They presented a new method for extracting model order. In these studies model order was one of the main optimization parameters. This new method we follow is a combination of optimization algorithm with identification methods for solving this problem. Unlike the other methods, it is very simple and can be used with every identification method. In spite of (Ebrahim Rahimpour- S.N. Sivanandam), due to the large number of parameters in power network, we use optimization algorithm for order selection and system model is obtained in an identification method separately. Recently, optimization algorithms are widely used in different sciences. The power of these algorithms in solving complicated mathematical problems with no exact answer or with time consuming and costly answer is obvious. Several optimization methods that we used included IPSO, Harmony search, SFL and SA randomly.

2 Intl. Res. J. Appl. Basic. Sci. Vol., 7 (), 5-3, 3 Simulation results show that this new idea leads to more accurate model. Another idea that we use in this research is merging order reduction methods with our optimization algorithm. One of the main problems in convergence of an optimization algorithm is initial values. We solved this problem with classical order reduction methods. This idea is very efficient and reasonable. We use Prony and ERA/DC as identification methods which have been combined with different optimization algorithms. These methods have been implemented on a 7 order linear simple system and extracted data from Iran power network model in DIgSILENT. In next section we have a brief review optimization algorithm and our idea will be explained. At the end, we will have the simulation results and conclusion. REVIEW OF OPTIMIZATION METHOD We have a brief review of optimization methods here Particle Swarm Optimization (PSO) The PSO is inspired by the social behavior of a flock of migrating birds trying to reach an unknown destination. In PSO, each solution is a bird in the flock and is referred to as a particle. A particle is analogous to a chromosome (population member) in GAs. As opposed to GAs, the evolutionary process in the PSO does not create new birds from parent ones. Rather, the birds in the population only evolve their social behavior and accordingly their movement towards a destination. Physically, this mimics a flock of birds that communicate together as they fly. Each bird looks in a specific direction, and then when communicating together, they identify the bird that is in the best location. Accordingly, each bird speeds towards the best bird using a velocity that depends on its current position. Each bird, then, investigates the search space from its new local position, and the process repeats until the flock reaches a desired destination. It is important to note that the process involves both social interaction and intelligence so that birds learn from their own experience (local search) and also from the experience of others around them (global search). Harmony Research Algorithm (HS) Harmony search (HS) algorithm is based on natural musical performance processes that occur when a musician searches for a better state of harmony, such as during jazz improvisation. The engineers seek for a global solution as determined by an objective function, just like the musicians seek to find musically pleasing harmony as determined by an aesthetic. In music improvisation, each player sounds any pitch within the possible range, together making one harmony vector. If all the pitches make a good solution, that experience is stored in each variable s memory, and the possibility to make a good solution is also increased next time. HS algorithm includes a number of optimization operators, such as the harmony memory (HM), the harmony memory size (HMS, number of solution vectors in harmony memory), the harmony memory considering rate (HMCR), and the pitch adjusting rate (PAR). In the HS algorithm, the harmony memory (HM) stores the feasible vectors, which are all in the feasible space. The harmony memory size determines how many vectors it stores. A new vector is generated by selecting the components of different vectors randomly in the harmony memory. Shuffle Frog Leaping Algorithm (SFL) The SFL algorithm, in essence, combines the benefits of the genetic-based MAs and the social behaviorbased PSO algorithms. In the SFL, the population consists of a set of frogs (solutions) that is partitioned into subsets referred to as memeplexes. The different memeplexes are considered as different cultures of frogs, each performing a local search. Within each memeplex, the individual frogs hold ideas, that can be influenced by the ideas of other frogs, and evolve through a process of memetic evolution. After a defined number of memetic evolution steps, ideas are passed among memeplexes in a shuffling process. The local search and the shuffling processes continue until defined convergence criteria are satisfied. Simulated Annealing (SA) Simulated annealing is a stochastic search technique in which a randomly generated potential solution, N, to a problem is compared to an existing solution, O. The probability of N being accepted for investigation depends on the proximity of N to O. If N is accepted, its suitability as a solution is evaluated according to a swap probability function and it may be chosen to replace O. Both the acceptance and swap probability functions depend on a temperature parameter T, which reduces in value as the algorithm proceeds. 6

3 Intl. Res. J. Appl. Basic. Sci. Vol., 7 (), 5-3, 3 SIMULATION RESULTS In this part we compare new identification methods. Simulation has two different parts. In the first part we implement methods on a simple 7 order system and in the second part these algorithms were implemented on measured data from Iran Power network in DIgSILENT software. Finally, we compare and analyze the simulation results. We should notice that fitness function is based on difference between real and simulated output: f(n) = (y y) However, we try to maximize fitness function with reducing differences between real and simulated output. Identification of a simple system The first simulation was implemented on simple 7 order system whose impulse response and Eigen-value are shown below G(s) = s + s + 5s + s + 5s + 6s + 7s + 8 Impulse Response of G(s) Eigen Value of G(s) Figure. Simulation results are presented in the table below. We tried to use unique parameters for optimization methods SA T=, Temp. Damp=. 9 SFL Mem. No.=, Mem. Itr.= PSO C=, C=.5 Speed Damp.=.9 Table. HS HMCR=.9, PAR=.5, bw= Itr. No. Particle No. Other Parameters In the algorithms, HS, SLF and SA, the order of system was the same as real system. In figure -,-, real and simulated output and Eigen-values are shown. 7

4 Intl. Res. J. Appl. Basic. Sci. Vol., 7 (), 5-3, Real Output Estimated Output Real EigenValues Estimated EigenValues Figure. In HS algorithm selecting, iteration number greater than has no effect on convergences. The change in other parameters like HMCR and PAR result in different optimization speeds. SFL also has good result and always locks in order 7. There is no difference with changing frog numbers (, 5, 3, ). SA, unlike other methods, has unique particle for search. In addition, it has good optimizing speed. We can change its parameters for obtaining different convergence speed. PSO has special results. System order with this method is 6, instead of 7. But simulated output shows its accuracy. The approximate system model with reduced order is one of the goals of identification methods. Comparing dominant Eigen-values and output shows that this reduced model has acceptable output. We can see output of this system below: 6 5 Real Output Estimated Output

5 Intl. Res. J. Appl. Basic. Sci. Vol., 7 (), 5-3, 3.5 Real EigenValues Estimated EigenValues Figure3. Introduced parameters in table- were the best ones. Particle numbers or iteration number has a neglectable effect on convergence result, but C and C have important role on convergence. Changing them can result in reducing speed of convergence exit from of boundaries. Identification with Iran network data In the second part, we use prony method for extracting dominant mode of Iran power network. This model was implemented in DIgSILENT and a programmed short circuit happened, and then several signals were measured. The main goal is stability analysis of system and for that we should approximate dominant mode that was extracted from the measured data. Also, we could obtain stability index. For example, in figure there was reactive power of Montazer-Ghaem power plane according to a short circuit in network (p.u):.8 MontazerGhaem Active Power Figure. For more accuracy, we can use multiple output from several points. Because of high order of system and sampled data, we only used one output. Also, we should notice that the setting of part one was used for optimization algorithms that are shown in table below. SA T=, Temp. Damp=.9 SFL Mem. No.=, Mem. Ite.= Table. PSO C=, C=.5 Speed Damp.=.9 HS HMCR=.9, PAR=.5, bw= Iteration No, Particle No. Other Parameters In the next table, optimization algorithms are compared concerning the obtained order, fitness function and the number of dominant modes. 9

6 Intl. Res. J. Appl. Basic. Sci. Vol., 7 (), 5-3, 3 (SA) 5 7. (SLF).7 (PSO) Table3. (HS).7 System order Fitness Function No. of Dominant mode (less than %) It is shown that the algorithm has the same result approximately. In figure, real and simulated output are compared. In spite of differences that exist in system order, the number of dominant modes is the same. So, the results of this new method are acceptable MontazerGhaem Active Power Real y PSO y SLF & HS y SA y Figure5. CONCLUSION The main idea of this paper was obtaining system order with optimization algorithm. We used these algorithms and mixed them with identification algorithms for using the advantage of them. In identification methods, system order had the main role on output results. We used optimization algorithms for approximating system order. The simulation results showed this new method had good results in extracting system order and approximating the dominant mode. REFRENCES Eusuff MM, Lansey KE.3.Optimization of water distribution network design using the shuffled frog leaping algorithm, J Water Resour Plan Manage 3;9(3): 5. Geem ZW, Kim JH, Loganatan GV..A New Heuristic Algorithm: Harmony Search, Simulation 76, 6-68, Ghasemi H, Ca nizares C, Moshref A. 6.Oscillatory Stability Limit Prediction Using Stochastic Subspace Identification, Power Systems, IEEE Transactions, Volume, Issue,May Ghasemi H, Canizares CA, Reeve J..Prediction of Instability Points Using System Identification, Conf. on Bulk Power System Dynamics and Control-VI, August Goro Oa, Brian D, Anderson O..Model Reduction for Control System Design, Springer-Verlag, London, Hauer JF, Demeure CJ, Shcarf LL. 99.Initial Results in Prony Analysis of power System Response Signals, IEEE Trans. on Power Systems, Vol. 5, No., February Kang SL, Geem ZW.. A new structural optimization method based on the harmony search algorithm, Journal of Computers and Structures,, 8, Kennedy J, Eberhart R.995. Particle swarm optimization, IEEE International Conference on Neural Networks, Perth, Australia, 995, pp Kennedy J, Mendes R..Population structure and particle swarm performance, Proceedings of the IEEE Congress on Evolutionary Computation,, pp Kimt K, Schattlertt H, Venkatasubramanianttt V, Zaborszkytt J, Hirsch P. 997.Methods for Calculating Oscillations in Large Power Systems, IEEE Transactions on Power Systems, Vol., No., November Kirkpatrick S, Gelatt CD Jr, Vecchi MP.983.Optimization by simulated annealing, Science, New series, Vol., No. 598, pp , Lew S, Juanj JN, Longman RW. 993.Comparison of Several System Identification Methods For flexible Structures, Journal of Sound and Vibration, Vol.67, No.3, 3

7 Intl. Res. J. Appl. Basic. Sci. Vol., 7 (), 5-3, 3 Liong SY, Atiquzzaman Md.. Optimal design of water distribution network using shuffled complex evolution, J Inst Eng, Singapore ;():93 7. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E Equation of state calculations by fast computing machines, J. Chem. Phys. Vol., No. 6,pp. 87-9, Rahimpour E, Rashtchi V, Shahrouzi H..Applying artificial optimization methods for transformer model reduction of lumped parameter models, Electric Power Systems Research, 8 () 8 Elsevier Safaeian M, Karrari M, Shafiee M, Malik OP. 7.New Oscillatory Instability Assessment Method with Instantaneous Measurement, IFAC Workshop ICPS'7, Cluj-Napoca, Romania Sanchez-Gasca JJ, Chow JH. 997.Computation Of Power System Low-Order Models From Time Domain Simulations Using a Hankel Matrix, IEEE Transactions on Power Systems, Vol., No., November Sanchez-Gasca JJ, Energy GE..Identification of Power System Low Order Linear Models Using the ERA/OBS Method, Power Systems Conference and Exposition IEEE, vol., October Sivanandam SN, Deepa SN.9. A Comparative Study Using Genetic Algorithm and Particle Swarm Optimization for Lower Order System Modeling, International Journal of the Computer, the Internet and Management Vol. 7. No.3 (September - December, 9) pp -. Smith JR, Fatehi F, Woods CS, Hauer JF, Trudnowski DJ. 993.Transfer Function Identification in Power System Applications, IEEE Trans. on Power Systems, Vol. 8, No. 3, August Trudnowski DJ, Johnson JM, Hauer JF. 999.Making Prony Analysis More Accurate Using Multiple Signals, IEEE Trans. on Power Systems, Vol., No., February 3