Identification of a continuous time nonlinear state space model for the external power system dynamic equivalent neural networks.pdf

identifi cation of a continuous time nonlinear state space model for the external power system dynamic equivalent by neural networks hamed shakouri g a hamid reza radmaneshb aindustrial engineering faculty university of tehran tehran iran bengineering faculty shahed university tehran iran a r t i c l ei n f o article history received 14 june 2008 received in revised form 5 march 2009 accepted 16 march 2009 keywords dynamic equivalent artifi cial neural networks nonlinear identifi cation numerical differentiation a b s t r a c t based on the concept of the external power system dynamic equivalent obtained for the study system in this paper a reduced order artifi cial neural network is proposed to construct a model for the external part the mastermind behind the proposed method is to identify the external part as a dynamic algebraic ann and this separation between dynamic equations in the state space and algebraic equations is useful to solve the prediction problem moreover using similarity transformations the state space model can be simplifi ed such that all the nonlinearities are embedded in the algebraic part since usually the study sys tem equations are available in the continuous time domain the external part is converted to the contin uous time domain by a novel method to obtain this model the system should be excited fi rst by a sort of random disturbances and then data measured on the boundary nodes is used to identify the model iden tifi cation process is accomplished by training the proposed network which can be used to predict behav ior of the external system with a high degree of accuracy such an equivalent has wide applications for dynamic stability studies 2009 elsevier ltd all rights reserved 1 introduction modeling as the fi rst step of any system analysis is an impor tant task in scientifi c studies usually the model is obtained either using physical laws or through an identifi cation process 1 2 therefore a variety of linear and nonlinear models are developed to get in use for power system studies and many computer pack ages are implemented as well 3 5 a set of low order linear models is still used in some dynamic studies 6 8 as well as many nonlinear models 9 however for vastly interconnected networks such as the european and the mid dle east networks even low order models lead to large system matrices causing memory managing and or system analysis diffi culties therefore in company with system expansion utilizing dy namic equivalents has been spread manifestly a review through different former approaches on dynamic equivalents can be found in 10 on the other hand due to the lack of complete system data and or frequently variations of the parameters while system oper ations the importance of estimation methods is revealed notice ably especially on line model correction aids for employing adaptive controllers power system stabilizers pss or transient stability assessment 11 14 the capability of such methods have become serious rival of the old conventional methods e g the coherency 15 16 and the modal 17 approaches the equivalent estimation methods have spread because it can be estimated based on data measured only on the boundary nodes between the study system and the external system 18 20 this way with out any need to information from the external system estimation process tries to estimate a reduced order linear model which is re placed for the external part evidently estimation methods can be used in presence of perfect data of the network as well to compute the equivalent by simulation and or model order reduction sophisticated techniques have become interesting subject to researchers for solving identifi cation problems since 90s for example to obtain dynamic equivalent of an external subsystem an optimization problem has been solved by genetic algorithm ga likewise levenberg marguardt algorithm 21 neural net works is the most prevalent method between these techniques be cause of its high inherent ability for modeling nonlinear systems including power system dynamic equivalents 22 27 some researchers have modeled the external part as a nonlinear arx narx 22 23 since simultaneous solution of the algebraic and dynamic equations in the state space model is more conve nient it would be better rearranged the narx model in a state space form to avoid complexity of solution caused by nonlinear terms and transformation to model external part as a state space model 26 27 introduce two anns a bottleneck ann is used by the former to extract states of the reduced order equivalent and a recurrent ann is embedded in an ordinary differential 0142 0615 see front matter 2009 elsevier ltd all rights reserved doi 10 1016 j ijepes 2009 03 016 corresponding author e mail addresses hshakouri ut ac ir h shakouri g radmanesh shahed ac ir h r radmanesh electrical power and energy systems 31 2009 334 344 contents lists available at sciencedirect electrical power and energy systems journal homepage equations odes solver by the latter the method involves diffi culty of calculation for the two anns which model the exter nal part it should be mentioned that these previous works 22 23 26 27 have no emphasis on implementation of a complete prediction it is important to solve the dynamic algebraic equiva lents dae of the study system and the external part simulta neously considering this point it can be deduced that the methods proposed by aleksandar et al 26 27 have some sort of defi ciency from this point of view because the interactions be tween the study system and the external part has been modeled one way not simultaneously moreover estimation of the external part state variables by the bottleneck sounds meaningless and so improper however this paper intends to construct a state space model without the aforementioned shortcomings to explain the problem in summary we have to mention again that an external dynamic equivalent is useful when we concentrate on a study system which is entirely known and only effects of the external system on the study system is of importance and needed that is while the external system model is unknown and or of high order dynamics as far as the dynamics analysis of the study sys tem is concerned it would be necessary to include dynamical im pacts of the external part on the study system as a design requirement this paper is dealing with such a problem and pro poses a new method to identify a model for the external part as a dynamic algebraic neural network 2 numerical differentiation as an introductory tool we should fi rst mention the numerical differentiation method used in this paper introduced by li 28 a 2n 1 point centered difference formula of order 2n to approx imate the fi rst derivative of a function f x at the middle point x0is determined by f0 x0 1 h x n j n d2n 1 0 jf xj o2n 0 h2n 1 where d2n 1 0 j 1 j 1 n 2 j n j n j j 1 2 n 2 and d2n 1 0 0 0 thus the symmetric 7 point approximation for the fi rst derivative can be written as f0 x0 1 60 f 3 9f 2 45f 1 45f1 9f2 f3 o h6 3 both the backward and the forward difference formulas are avail able in 28 however it should be mentioned that usually centered difference formula has a better performance than the other formulas 3 the power system model 3 1 dynamic equations to simulate a multi machine power system the state equations are derived from 29 each generator can be represented by the following two axis model t0doi de0qi dt e0qi xdi x0di idi efdi 4 t0qoi de0di dt e0di xqi x0qi idi 5 ddi dt xi xs 6 2hi xs dxi dt tmi e0diidi e0qiiqi x0qi x0di idiiqi d xi xs 7 the excitation system type is assumed to be ieee type i tei defdi dt kei sei efdi efdi vri 8 tfi drfi dt rfi kfi tfi efdi 9 tai dvri dt vri kairfi kaikfi tfi efdi kai vrefi vi 10 3 2 algebraic equations the network solution method for this problem consists primar ily of solving for the bus voltages and injection currents of the re duced system along the direct and the quadrature axes 30 this method is based on the basic network equations of the system by kron reduction 9 i b yredv 11 where i and v are vectors of the complex bus voltages and the bus injection currents at the machine buses only supposec cij is a diagonal matrix angel shift matrix such that ci i ej di p 2 i 1 2 m 12 where m is number of the machines hence the direct and the quadrature axis voltage and current relationships are defi ned by v c vd jvq 13 i c id jiq 14 now the matrices a b c and d are defi ned as diagonal matrices with the diagonal elements defi ned as follows ai i rs r2 s x0dx0q 15 bi i x0q r2 s x0dx0q 16 ci i x0d r2 s x0dx0q 17 di i rs r2 s x0dx0q 18 in addition two matrices namely g0and b0 are defi ned by g0 re c 1byredc 19 b0 im c 1byredc 20 this way vdand vqcan be obtained using the regular matrix meth ods given below vd vq g0 a b0 b b0 cg0 d 1 ae0d be0q ce0d de0q 21 once vdand vqare known idand iqcan be calculated using 22 and 23 id g0vd b0vq 22 iq b0vd g0vq 23 h shakouri g h r radmanesh electrical power and energy systems 31 2009 334 344335 4 the concept of dynamic equivalent to understand the concept of dynamic equivalent it is relevant to start our study with the simplest case the frontier boundary nodes are infi nite buses fig 1 magnitude and phase of voltages on the frontier nodes vfn are constant without dynamics conversely when the frontier nodes cannot be assumed as infi nite buses both the magnitude and phase of voltage on the frontier node involve dynamical terms the voltage at the frontier nodes can be considered as a function of injection currents and the exter nal part state variables see fig 2 the notion explained above can be achieved analytically as well differential algebraic equations daes in part iii can be repre sented symbolically as follows 29 x fo x id q v u 24 id q h x v 25 0 go x id q v 26 where u is a vector of disturbances 25 is related to the algebraic generator stator equations and 26 is related to the algebraic net work equations now 24 26 should be rearranged to represent the study system and the external part of system equations sepa rately to do so the study system equations are written as follows xstudy fs xstudy ifn vfn u 27 ifn hs xstudy vfn 28 moreover the external part can be represented as xext fe xext ifn vfn 29 vfn he xext ifn 30 it is supposed that the sources of disturbances u are located only in the study system it should be mentioned that although the study system and the external part of system equations are represented separately but the algebraic eqs 28 and 30 are a set of the equations that must be solved simultaneously see fig 3 eqs 27 30 are arranged such that both magnitude and phase of the on frontier nodes currents ifn are treated as inputs study system fn v fn i fig 1 the study system connected to an infi nite bus study system fn i fnextefn ixhv fig 2 the study system connected to the external part of system fig 3 the whole system equations green box in the left hand dynamic equations blue box in the right hand algebraic equations 336h shakouri g h r radmanesh electrical power and energy systems 31 2009 334 344 to the model and both magnitude and phase of the frontier nodes voltages vfn are outputs for the algebraic equations of the exter nal part there are other options for choosing input output data set let us consider the following sets of variables vfn vfnifn ifn 31 vfn vfnsfn sfn 32 ifn ifnsfn sfn 33 where j j means the magnitude denotes the phase angle and sfn is the complex power measurements in the interconnecting frontier nodes for each set we can choose two of four waveforms as the in put set and two remaining waveforms as the output set eqs 31 33 are in polar form and it is possible to be rewritten in d q components form as well since integration of the frequencies to obtain the angles hides the fl uctuations caused by disturbances there should be a refer ence to compute differences between the angles to do so we need to choose one of the generator power angle deviations as a refer ence for angle deviations it is apparent in fig 4 that the power an gles simulation results for wscc test system seem fl oat and these fl oat waveforms do not provide appropriate data for identifi cation process instead the waveforms in fig 5 are obtained by choosing power angle deviation 3 as a reference for the angle deviations now these waveforms appear as persistently exciting 1 and appropriate data for identifi cation process to carry out this trans formation we should convertxstoxnin 6 where n is the num ber of chosen generators as the reference for power angles hence for generator n 6 is eliminated and the degree of the differential equations reduces 5 the neural network structure in order to model daes of the external part a dynamic alge braic neural network is constructed which is illustrated in fig 6 layers 1 and 3 are composed of neurons with nonlinear sigmoid activation function and layers 2 and 4 are composed of neurons with linear activation function the fi rst part of the dynamic equiv alent layers 1 and 2 models the eq 29 in difference equation forms eq 30 is consistent at all times and it can be written at either t k or t k 1 since the targets of the ann are the pre dicted waveforms the second part of the network layers 2 and 4 is related to algebraic terms at k 1 from fig 6 it is obvious that the phases of the frontier nodes voltages and currents are con sidered as the input data and the magnitude of voltages and cur rents on the nodes are considered as the output data for the system according to the similarity transformation concept we know that realization of state space equations is not unique therefore we can rearrange the daes so that all of the nonlinearities are embedded in the algebraic part if the state space equations were strictly proper it would be possible to embed all of the nonlinear ities in the dynamic part however the state space equations of the dynamic equivalent represented by eqs 27 30 have a proper form therefore if there is not any nonlinear term in the algebraic part the inputs to the algebraic part do not perceive the nonlinear layer s in their paths to the outputs fig 7 illustrates three struc tures that can realize daes let us call the external inputs to the dynamic equations group g1 and the external inputs to the fig 4 power angles obtained from simulation of wscc fig 5 power angles by choosing power angle deviation 3 as a reference for angle deviations fig 6 the initial structure of the reduced order dynamic algebraic equivalent dea the number of the neurons of each layer is given under the number of each layer of the nn blocks these are used for the case study defi ned in section 7 h shakouri g h r radmanesh electrical power and energy systems 31 2009 334 344337 algebraic equations group g2 in fig 7a g1 paths through two nonlinear layers the dynamic equations and the algebraic equa tions bold green line but g2 paths through only one nonlinear layer the algebraic equations thin blue line in fig 7b nonlinear ity is concentrated only in the algebraic equations hence both g1 and g2 face to one nonlinear layer thin blue line similarly in fig 7 nonlinear layers in the paths from the inputs to the outputs of three realizations of daes fig 8 the simplifi ed structure of the reduced order dynamic algebraic equivalent dea the number of the neurons of each layer is given under the number of each layer of the nn blocks these are used for the case study defi ned in section 7 338h shakouri g h r radmanesh electrical power and energy systems 31 2009 334 344 fig 7c g1 faces to one nonlinear layer the dynamic equations thin blue line but g2 does not faces to any nonlinear layer dashed red line from fig 7 it is apparent that the structure c cannot be a perfect structure because there is no nonlinear term affecting the external inputs to the algebraic part it is worthy to mention that the structure b is more effi cient than a because each nonlinear unit in the algebraic part is shared between the two groups of the inputs inputs to the dynamic part and inputs to the algebraic part in their paths to the outputs furthermore complexity of the structure b is less than a due to the less num ber of nonlinear layers the simplifi ed version of the structure shown in fig 6 is illustrated in fig 8 since the bias connection for layer 1 is useless it is pruned from the ann 6 the continuous time state space model the state space model illustrated in fig 8 is a discontinuous discrete time model but the study system equations are usually given in continuous time domain a discontinuous time state space model can be represented as in fig 9 hence for the ith state var iable we can write xi k 1 fi xi k ui k 34 where xi k and ui k are called effective state variables and effec tive inputs at t k to construct the ith state variable at t k 1 eq 34 can be written in continuous time domain as xi t fi xi t ui t 35 a relatively accurate approximate of the fi rst derivative of xi t i e xi t can be determined by 3 therefore we can train the neu ral network in fig 10 the target of which is xi t i 1 2 ns where nsis the number of state variables considered for the equiv alent system the effective equivalent states xi t which indeed are artifi cial states and so dimensionless are xi t xext t 36 and the inputs are ui t vt3it3 vt3 it3 37 7 the test system simulation studies are conducted on the wscc system which is a nine bus power system composed of three generators and three loads the system is illustrated in fig 11 where specifi cations of the system are given 29 generator 3 is as assumed to be the study system and node 3 is the frontier node connecting the study system to the external part fig 10 the neural network input output model to approximate fi fig 9 the discontinuous discrete time state space model fig 11 the wscc 9 bus test system h shakouri g h r radmanesh electrical power and energy systems 31 2009 334 344339 the disturbances take place on mechanical torque and reference voltage of