Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013)

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1 Proceedigs of the 2d Iteratioal Coferece o Computer Sciece ad Electroics Egieerig (ICCE 20) GPU Implemetatio of the Electric Power System Model for Real-Time Simulatio of Electromagetic Trasiets Slawomir Cieslik Electrical Egieerig Istitute Uiversity of Techology ad Life Scieces Bydgoszcz, Polad slavcies@utp.edu.pl Abstract This paper presets decompositio method of mathematical model of electric power system ad implemetatio its i digital system with GPU. The electric power system model ca be used for real-time simulatio of electromagetic trasiets. The article presets the results of experimetal research of calculatio time for two differet digital systems usig both the CPU ad GPU. Keywords-electromagetic trasiets; real-time simulatio; parallel processig; GPU; thread computig; CUDA I. INTRODUCTION Simulatio of electromagetic trasiets i electric power systems is oe of the basic research methods. Complicated algorithms of digital cotrollers used i automatic regulatio of electric power systems ofte hiders reliable modelig of its electromagetic trasiets (especially if eural etworks ad geetic algorithms are used). Oe of the ways of performig a reliable simulatio of electric power systems with digital regulators is usig of a real-time electric power system simulator, which allows to coect a real cotroller (i which case there is o eed to defie its characteristics, as it is iteractig with the simulator). Such a simulator eeds to geerate results with miimal errors as well as promptly calculate the data i order to eable the commuicatio with the eviromet (i.e. the real cotroller). The mathematical model of the electric power system is a itegral part of the simulator. Oe of the ways of implemetig the mathematical models for the digital realtime power system simulator is their decompositio through various methods. Therefore, it is appropriate to research o mathematical models of electric power systems i the aspect of results accuracy ad calculatio time. This article describes the method of mathematical model decompositio based o the partitio of the modeled power system ito structural elemets represets as electric multipoles [,2]. The problem of improvig the accuracy of the results has bee discussed i author s previous work [], i which there is show a example of implemetig various algorithms for itegratio of differetial equatios for certai groups of structural elemets. The issue of calculatio time is also cosidered i this article. I order to shorteig solvig time i PC-based digital simulators GPUs are used [,,5]. The CUDA architecture eables a relatively easy implemetatio of the decomposed mathematical model ad parallel calculatios i separated blocks which are resposible for respective structural elemets. This article presets the results of a research o electromagetic trasiets i a exemplary electric power system with the use of two differet digital systems: CPU Itel Core i7 970 with GPU NVIDIA GE Force GTX 50 ad CPU Itel Core i7 20 QM with GPU NVIDIA GE Force GT 50M. II. THE METHOD OF MODEL DECOMPOSITON The method of mathematical modelig with the use of electric multipoles is oe of may methods [,7] of mathematical modelig of electric power systems, but it is oe of the few that allows the model decompositio, so that the calculatios are execute i parallel. The cocept of usig electric multipoles is successfully applied i mathematical modelig of complex electromechaical ad electric power systems. v 0S v k i k v S v 2k i 2k k v αs v βk i βk v ζs v ξk i ξk Figure. The equivalet scheme of the modeled electric power system. The electric power system show o Figure. is cosidered as a electric coectio of structural elemets (). Every k-th structural elemet k is represeted i geeral case by electric multipole with matrices: [ v v v v ] T v k = k 2k βk ξk, () - electric potetials of the k exteral odes, [ i i i i ] T i k = k 2k βk ξk, (2) - exteral brach currets of k.

2 Proceedigs of the 2d Iteratioal Coferece o Computer Sciece ad Electroics Egieerig (ICCE 20) The mathematical represetatio of each k has to be carried out i such a way that we ca provide a exteral equatio i the form as follows i k + Akvk + Bk = 0, () There is a square matrix A k with dimesios equal ξ ξ i the () equatio. Also we ca fid there the ξ - elemet matrix B k. The elemets of the metioed matrix are defied by structural elemet parameters ad iteral physical variables associated with the k. The exteral braches of structural elemets are coected at ( ζ +) odes of electric power system ( 0 S, S,, α S,, ζ S). The electric potetial of oe of them is zero ( v 0 S = 0), therefore for the remaiig ζ -th idepedet odes we ca defie the followig matrix A P A P ad B P. (8) = S j j T j = S jb j Mathematical modellig of electromagetic trasiets i electric power systems is doe accordig to the algorithm show i Figure 2. [ v v ] T α v v S = S S ζs, () called by electric potetial odes matrix of the electric power system. Relatioships betwee electric potetials of exteral odes of structural elemets ad electric potetials of odes of the electric power system are expressed usig the trasposed of a icidece matrix P k for each k-th elemet, as follows v T Pk vs k =, (5) P k matrices appearig i the (5) equatio are costat for a give. The umber of rows of the give matrix is equal to the umber of idepedet odes of the electric power system, ad the umber of colums is equal to the umber of exteral braches of structural elemet. The matrix elemet o the itersectio of the row ad the colum is equal to oe oly if the correspodig to the umber of the ode lie umber is coected to a brach of the multipole with a umber correspodig to the umber of the colum. I other cases, the elemets of this matrix are equal to zero. Usig the st Kirchhoff s law for all idepedet electric power system odes we receive the equatio P ji j = 0, () where the exteral brach currets of the structural elemets were iserted ad desigated from exteral equatios of multipoles (). Regardig to the equatio (5) received the followig odal equatio A S v S + B S = 0, (7) Figure 2. Mathematical modellig algorithm. The goal is that the tasks iside the loop show o Figure 2 should perform i parallel. There are examples of usig certai umerical methods of solvig systems of equatios usig the GPU i the literature [8,9]. I this article there is a particular emphasis the possibility of parallel tasks ad calculatios A.. A.8., which is associated with decomposig the mathematical model. where: 5

3 Proceedigs of the 2d Iteratioal Coferece o Computer Sciece ad Electroics Egieerig (ICCE 20) III. MATHEMATICAL MODEL OF -PHA STRUCTURAL ELEMENT Parallel umerical calculatios i the CUDA eviromet are doe with the best performace, oly if i each block ad each thread exactly the same computatioal tasks are implemeted, except isertig differet data. Decompositio mathematical model of a electric power system, which is to be implemeted i the CUDA architecture, apply the most cosistet three-phase model of the structural elemet, whose equivalet scheme is show i Figure. Also there is a list of the parameters of a mathematical model of this structural elemet i the Table I. v v 5 5 v i i 5 i e A R A L C A A i v e u LA B u CA R B L B C B i 2 2 v 2 e u LB C u CB R C L C C C i v u LC u CC Figure. The equivalet scheme of three-phase structural elemet. TABLE I. PARAMETERS OF THE MATHEMATICAL MODEL OF THE STRUCTURAL ELEMENT No Symbol Parameter descriptio E 0ζ ζ phase voltage offset e ζ(t) [V] 2 E mζλ ζ-harmoics voltage amplitude e ζ(t) [V] ψ eζ λ ζ-phase, λ-harmoics of voltage e ζ(t) [rad] f Frequecy () of voltage sources e ζ (t) [Hz] 5 R ζ ζ-th phase elemet resistace [Ω] L ζ ζ-th phase elemet iductace [H] 7 C ζ ζ-th phase elemet capacitace [F] () I the geeral case each phase source voltages frequecies are differet. The mathematical model of the structural elemet based o discrete mathematical models of iductace L ad capacity C was proposed i [0]. After use trapezoidal rule the followig depedecies for a a matrix values are derived A =, where a a a = diag( αa, αb, αc ), ad for the vector b B =, where b = [ βa βb βc ] T i - b exteral equatio (): β ( ζ = αζ eζ + ( t ) u ( t ) - - ( L ξ ξc ) i ( t ) u ( t )), ζ ζ where ξ = 0,5h. Capacitor voltages are equal ucζζe ulζζe LζζS ks E + + CζζS - ( t ) C ( i ( t ) + i ( t )) u ( t ) (0) + = ξ ζes k ES + kes + CζζE, () Iductor voltages are equal - ( t ) L ( i ( t ) i ( t )) u ( t ) IV. + = ξ kes + kes LζζE. (2) DESCRIPTION OF MODELED ELECTRIC POWER SYSTEM Figure 2. shows a 5 kv electric power system schematic diagram, meawhile Figure. is treated as the equivalet scheme. The system cosists of 25 (-25) structural elemets. 5 0 kv 5 kv L L5 L L2 L 7 Tr L7 L L L 2 L8 5 L5 9 L0 8 L9 7 L L Figure. Schematic diagram of modeled electric power system []. L2 - ( R + ξ L + ξ ) ζ = ζ ζ Cζ α, (9)

4 Proceedigs of the 2d Iteratioal Coferece o Computer Sciece ad Electroics Egieerig (ICCE 20) v v 2 v 2 v v 7 v v 0 v v v 7 5 v v 5 v 2 v 8 v 9 v v v v v v 5 v 2 v 7 v 8 v v 22 v 20 v 2 v 2 v 2 v 25 7 v 2 v27 v v v v v Figure 5. Equivalet scheme of modeled electric power system []. I this case, most of the structural elemets of the model correspods to exactly oe elemet of the real system, e.g. 9 correspods to the amout the lie marked as L7. There are also several cases i which the structural elemet of the model correspods to the group of elemets of the real system, for example 25 cotais electric lie segmet marked as L, a power switch ad a capacitors bak. The partitio of the electric power system structure (decompositio) for structural elemets is a very importat problem i terms of calculatio time (The problem of improvig the accuracy of the results has bee discussed i author s previous work []). O the equivalet scheme, each structural elemet is represeted as a electric six-pole, as show i Figure. V. EXPERIMENTAL RESULTS AND CONCLUSIONS Figures -8 shows results (computatio times) of experimetal computatioal tests carried out o two digital systems. I 2-d block system of equatios is solved, ad i -st block A.-A.8 tasks are executed as showed o algorithm (Figure 2). Figure 7. Parallel computatio total executio times measured o two digital systems. The size of computatioal task is preseted o X-axis, (50,8) meas that task cosists of 50-variables i the model of electric power system solved o each itegratio step ad 8-odes of electric power system case show o Figure. Next four cases apply to more complex computatioal problems: (7,), (92,50), (2,59) ad (228,2). Figure 8. -st block arallel computatio executio times measured o two digital systems. Please pay attetio that parallel computatio time of -st block is approximately costat (it's true oly if structural elemets umber is less tha GPU-cores umber). Described results showed, that computatio time o GPU-used system with more cores (8) was loger tha o less cores (9). The reaso is 27% higher system clock frequecy i 2 d case ( MHz vs. 058 MHz). Figure. Sequetial computatio total executio times measured o two digital systems. REFERENCES [] S. Cieslik, Decompositio of Mathematcal Models of Power Systems for Real-Time Digital Simulators i Aspect of Calculatio Accuracy, Ryek Eergii, vol. (98), Feb. 202, pp. 7-5, ISSN [2] S. Cieslik, Digital Simulators as a Assessmet Tool of the Impact of Distributed Geeratio o Power Grid Ifrastructure, Electrical Review, vol No. 8, Aug. 200, pp , ISSN

5 Proceedigs of the 2d Iteratioal Coferece o Computer Sciece ad Electroics Egieerig (ICCE 20) [] M. Drechy, Possibility of Parallel Computig Applicatio i Power Egieerig, Ryek Eergii, vol. (0), Aug. 202, pp. -70, ISSN [] L. Murray, GPU Acceleratio of Ruge-Kutta Itegrators, IEEE Trasactioa o Parallel ad Distributed Systems, vol. 2. No., Ja. 202, pp. 9-0, /2. [5] J.K. Debath, Wai-Keug Fug, A.M. Gole ad S. Filizadeh, Simulatio of Large-Scale Electrical Power Networks o Graphics Processig Uits, Proc. IEEE Electrical Power ad Eergy Coferece, 20, pp , /. [] P. Zag, J.R. Martí ad H.W. Dommel, Network Partitioig for Real-Time power System Simulatio, Iteratioal Coferece o Power Systems Trasiets (IPST 05), Motreal, Caada, Jue 2005, Paper No. IPST [7] J.R. Martí, L.R. Liares, J. Calviño ad H.W. Dommel, OVNI: A Object Approach to Real-Time Power Simulators, Proc. of the Iteratioal Coferece o Power System Techology, Beijig, Chia, Aug. 998, pp. -5. [8] G. Amador ad A. Gomes, Liear Solvers for Stable Fluids: GPU vs CPU, Proc. of the 7 th Ecotro Portugeês de Computação Gráfica (EPCG 09), 2009, pp [9] H. Courtecuisse ad J. Allard, Parallel Dese Gauss-Seidel Algorithm o May-Core Processors, Proc. of th IEEE Iteratioal Coferece o High Performace Computig ad Commuicatios, Jue 2009, pp [0] H.W. Dommel, Digital computer solutio of electromagetic trasiets i sigle- ad multiphase etworks, IEEE Trasactios o Power Apparatus ad Systems, Vol. PAS-88, 99, pp