Parallel Di erential-Algebraic Equation Solvers for Power System Transient Stability Analysis Transient Stability Analysis (TSA) 1].
Real-time or faster-than-real-time power system tran- Transient Stability Analysis is a compute-intensive sient stability simulations will have signi cant im- problem requiring the solution of nonlinear, discontin- pact on the future design and operations of both in- uous systems of DAEs for the purpose of simulating dividual electrical utility companies and large inter- such electrical power systems phenomena as network connected power systems. The analysis involves so- component and transmission line failures. Existing lution of extremely large systems of di erential and DAE solvers often use higher order implicit Runge- algebraic equations. Di erential-Algebraic Equation Kutta integration techniques or Backward Di eren- (DAE) solvers have been used to solve problems simi- tiation Formulas (BDF), techniques that yield more lar in nature to the transient stability analysis (TSA) accurate solutions or permit larger time-steps. DAE problem. This paper discusses the possibility of the solvers have been especially developed to handle dis- use of the existing DAE solvers to solve the transient continuities in functions, a condition encountered in stability analysis application. We also discuss our re- search in developing a scalable, parallel DAE solver for use by the power system community and in related This paper describes research on the use of existing DAE solvers to solve the power system TSA problem.
In section 2, a detailed description of the power sys- tem problem is given. In section 3, we discuss various existing DAE solvers and their potential for applica- tion to the TSA problem. Our ongoing research in the In engineering and science, many problems give rise benchmarking of various sequential solvers and the de- to smooth and/or discontinuous systems of di eren- velopment of parallel solvers is discussed in section 4.
tial equations coupled with nonlinear algebraic equa- Lastly, in section 5, our conclusions on the application tions. These coupled systems of di erential and al- of DAE solvers for TSA problem are discussed.
gebraic equations are commonly known as di eren- tial/algebraic equations (DAEs). Research in the eld of DAEs started about a decade ago and is still in its early stages. A number of engineering and sci- ence problems have been solved with the help of DAE solvers developed by 2, 3, 5, 10, 13]. But to this date, Transient Stability Analysis (TSA) examines the dy- there has been no investigation of the application of namic behavior of a power system for as much as sev- DAE solvers to power systems problems such as the eral seconds following a power- ow disturbance. The general nature of the transient stability problem is ex- This work has been supported in part by Niagara Mohawk Power Corporation,the New York State Scienceand Technology plained in 11] as follows: Consider a simple mechani- Foundation, the NSF under co-operative agreement No. CCR- cal analog where a number of masses, representing the 9120008, and ARPA under contract No. DABT63-91-K-0005.
generators in an electric power system, are suspended from a \network" consisting of elastic strings repre- senting the electric transmission lines. Now one of the strings is suddenly cut, corresponding to a sudden loss of a transmission line. Thereafter the masses will experience transient coupled motion, and the forces j and 0q are the time constants involved in the in the strings will uctuate. This sudden disturbance may cause one of the following two e ects: axis, -axis, -axis and -axis components of the ele- 1. The system will settle down to a new equilibrium ments described. Detailed explanation on the notation state, characterized by a new set of string forces involved in the described equations can be found in 1].
corresponding to the power voltages in the elec- There are multiple generators supplying power to 2. Because of the transient forces, one or more addi- the network and the generators are coupled through an tional strings will break, causing a weaker net- electrical power network. The power network can be work, resulting in a chain reaction of broken modeled by nonlinear algebraic equations as follows: strings and total collapse of the system. If the sys- tem has the strength to survive the disturbance and settle into a new steady state, it is referred where is the network current, is the network volt- to as \transient stable for the fault in question." age and is the network admittance. Apart from these two sets of equations, TSA involves equations The transient stability analysis problem can be de- for loads (which can be di erential or algebraic) and scribed by a system of DAEs where the generators are equations for the generator control (that are di eren- represented by di erential equations and the power tial). All these equations together form a system of system network interconnecting the generators is rep- DAEs that must be solved simultaneously.
resented by nonlinear algebraic equations. The reac- tion of a single generator to variations in its load can be modeled using the following Ordinary Di erential bility analysis DAEs are non-symmetric in nature.
They are of bordered-block-diagonal form wherein blocks of generator equations along the diagonal are coupled with the power system distribution network.
The admittance matrix involved is extremely large, complex and sparse. Research is being conducted 9] to reorder this matrix into block-diagonal-bordered form in order to exploit the structure for parallelism.
tions involved in the transient stability analysis solu- tion is extremely large. Consider an example of 20 dif- ferential equations to describe each generator and two equations to describe the complex voltage/current at each network bus. Then an interconnected power sys- The equations described above roughly describe the tem with 2000 buses and 300 generators could generate \structure" of the di erential equations involved in a sparse, unsymmetric system of 10,000 nonlinear al- the transient stability analysis DAEs. The notation gebraic equations that must be solved simultaneously.
used in the above equations is as follows: For regional power systems, the number could be ve is the magnitude of source voltage of the gen- of a system is the minimal number of analytical di er- entiations needed such that the di erential equations e and m are the electrical and mechanical of the DAE can be transformed by algebraic manip- ulations into an explicit ODE system. The transient stability problem leads to an index-one or higher-index Euler method and can handle sti ODEs. This DAE system with the possibility of the ODEs involved code has been successfully used in the solution of problems arising in combustion modeling where there are frequent discontinuities in time. Be- cause of its ability to handle sti di erential equa- tions of the TSA are complex in nature (the voltages tions and frequent discontinuities, LIMEX is an and currents involved are both complex). These equa- excellent choice for the power system TSA solu- tions are also nonlinear. Research in developing a solver that can be used directly to solve a system of complex, nonlinear algebraic equations has been lim- ited and leaves the eld wide open for further research.
lation technique for the solution of DAEs.
(IRK) method of order ve. Research on solution consist of many individual power systems, it is highly of sti ODEs has focused a great deal on Implicit desirable that the DAE solver used for the transient Runge-Kutta methods. RADAU5 is designed to stability analysis be scalable. Also, the power net- solve index one, two, and three systems. IRK works can vary in size depending on the region under methods have a de nite advantage over multistep methods such as BDFs methods when the DAEs exhibit frequent discontinuities. Because of their one-step nature, IRK methods can be started at a higher order after every discontinuity, whereas multistep methods must be restarted, usually at a low order, after every discontinuity. This fact The power system community has been solving the helps the IRK methods to be more e cient. Also, transient stability problem approximately as a decou- with IRK methods, it is possible to construct high pled system of di erential and algebraic equations. As order A-stable or nearly A-stable IRK formulas, explained in 2], this method of reduction of large which is important while solving sti ODEs with DAE system to a system of explicit ODEs not only eigenvalues lying close to the imaginary axis. The destroys the sparsity of the system but also prevents generator ODEs in the transient stability analy- the exploitation of the system structure (symmetry of sis may be of this type. Since the TSA DAEs can the Jacobian). Several sequential DAE solvers exist be of index higher than one, RADAU5, with its that use various numerical techniques for their solu- ability to handle higher index systems makes an tion. Prominent methods among the existing solvers excellent choice for the transient stability analysis tiation technique. It uses a variable step-size, variable-order xed-leading-coe cient implemen- tation of BDFs formulas to advance solution from Backward Di erentiation Formulas (BDF) meth- one time step to the next. DASSL can solve su er no order reduction for index one systems.
Also, BDFs methods achieve the same order of Our research is concentrated on analyzing the follow- convergence for this class of DAEs as they do for ODEs. DASSL has a robust order selection strat- egy for DAEs with eigenvalues close to the imag- inary axis in the complex plane. DASSL can also discussed in 2], one-step methods such as ex- be used to solve semi-explicit index-two systems trapolation techniques have an inherent advan- of DAEs. DASSL by itself cannot handle discon- tage over BDFs methods when applied to DAEs tinuities, but a variation of DASSL, DASRT, has with frequent discontinuities. LIMEX is a code root nding capabilities to locate discontinuities developed for semi-explicit index one DAEs. It when they are su ciently large that DASSL can- implements an extrapolation of the semi-implicit not integrate through without intervention. Re- search is being conducted at the Northeast Paral- best suited for the TSA problem. We wil also present lel Architectures Center (NPAC) 10] to solve the a comparative review of their speed and performance.
The test data being used is the standard IEEE 118- bus data but we will also benchmark larger systems at coe cient implementation of BDFs formulas to solve linearly implicit system of DAEs. The xed-coe cient methods can be implemented e - ciently for smooth problems, but su er from inef- ciency and possible instability for problems re- CDASSL (Concurrent DASSL) 13] is the rst attempt quiring frequent step-size adjustments. LSODI that has been made at parallelizing these DAE solvers.
solves DAEs of index zero but if the coe cient More recently, attempts have also been made to de- matrix is singular, then it solves the resulting in- velop a data-parallel and message-passing versions of dex one DAE system. It can solve systems of sti DASPK 12] on the CM5. Our research is examin- and non-sti ODEs as well. LSODI has been suc- ing the scalability of the various DAE solution tech- cessfully used to solve systems arising from PDE niques. We are concentrating on selecting the most problems. Another version of LSODI, LSODA, suitable solver for the power system application, and switches automatically between sti (BDF) and creating a scalable library for an application based on non-sti (Adams) methods. Because of its abil- it. The parallel version of this DAE solver is being de- ity to handle sti ODEs and frequent discontinu- veloped for the use by the power system community, ities, LSODI presents a viable alternative to other and in related applications such as the chemical plant solvers discussed for the power system TSA prob- simulations (e.g., 13]) and the electrical circuit sim- ulations 2]. We are using the Multicomputer Toolbox 15] for the development of the parallel solver. This software is currently based on the portable message- nique of order four. Rosenbrock methods have a big advantage over IRK methods in that they MPI 4] next year and hence our programs will run completely avoid non-linear systems of equations, on all reasonable message-passing multicomputers.
at the same time providing the advantages of ac- curate solutions with sti di erential equations.
RODAS can solve DAEs that are expressible in semi-explicit form and that are of index one. Be- cause of the lack of data about the application of We have discussed the power system transient stability RODAS to any signi cant problem, and because analysis problem (TSA). We also presented a review of of the varied advantages of other solvers discussed some of the existing DAE solvers that we consider best above, our research will not involve RODAS, even suited for this problem. Benchmarking of sequential though it has the qualities of being a good con- DAE solvers and the development of a parallel, scal- able DAE solver are in progress. This research is being closely coordinated with parallel sparse matrix solver 9] research being performed at the Northeast Parallel Architectures Center (NPAC) at Syracuse University.
Currently at NPAC, we are involved in the following two issues: Transient Stability Analysis Benchmarks, We wish to thank David Koester and Alvin Leung, of NPAC, for their help with this research.
We are setting up transient stability analysis bench- marks with interfaces to di erent sequential solvers 1] P. M. Anderson and A. A. Fouad. Power Sys- like LIMEX, LSODI, RADAU5 and DASSL. This is tem Control and Stability. Iowa State University to test our empirical conclusion that these solvers are 2] K. E. Brenan, S. L. Campbell, and L. R. Petzold.
14] Anthony Skjellum. The Design and Evolution of The Numerical Solution of Initial Value Problems Zipcode. Parallel Computing, 1993. (Invited Pa- in Di erential-Algebraic Equations. Elsevier Sci- 15] Anthony Skjellum, Alvin P. Leung, Steven G.
3] P. Deu ard, E. Hairer, and J. Zugck. One Step Smith, Robert D. Falgout, Charles H. Still, and and Extrapolation Methods for Di erential- Alge- Chuck H. Baldwin. The Multicomputer Toolbox{ braic Systems. Numer. Math., 51, pages 501{516, First-generation Scalable Libraries. Technical Re- port 930805, Mississippi State University, Com- 4] Message Passing Interface Forum. Document for puter Science Dept., June 18 1993. Submitted a standard message-passing interface. Technical to HICSS{27: Minitrack on Tools and Languages Report Technical Report No. CS-93-214, Univer- for Transportable Parallel Applications.
5] E. Hairer and G. Wanner. Solving Ordinary Dif- ferential Equatios II | Sti and Di erential- Algebraic Problems. Springer-Verlag, 1991.
6] A. C. Hindmarsh. Large ordinary di erential equation systems and software. IEEE Control 7] A. C. Hindmarsh. ODEPACK, a systematized collection of ODE solvers. In R. S. Stepleman et al., editors, Scienti c Computing, pages 55{64.
8] A. C. Hindmarsh. The ODEPACK solvers. In R. C. Aiken, editor, Sti Computation, pages 9] David Koester, Sanjay Ranka, and Geo rey Fox.
Parallel Block-Diagonal-Bordered Sparse Linear Solvers for Electrical Power System Applications.
Scalable Parallel Libraries Conference, 1993.
10] Alvin Leung and Anthony Skjellum. Concurrent Dassl: A Second Generation DAE Solver. Scal- able Parallel Libraries Conference, 1993.
11] Iris Mack. Block Implicit One-Step Methods for Solving Smooth and Discontinuous Systems of Di erential/Algebraic Equations. PhD thesis, Applied Mathematics, Harvard University, May 12] Linda Petzold. Solving Large-Scale Di erential- Algebraic Systems via DASPK on the CM5. Scal- able Parallel Libraries Conference, 1993. (Invited 13] Anthony Skjellum. Concurrent Dynamic Simu- lation: Multicomputer Algorithms Research Ap- plied to Ordinary Di erential-Algebraic Process Systems in Chemical Engineering. PhD the- sis, Chemical Engineering, California Institute of


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