The aim of this research is to develop a power system dynamic equivalents toolbox (PSDYNET) for digital type power system simulator. In dynamics study of large power

system, it is of the essence to represent the external system by means of dynamic equivalents with the intention of improve the solution speed and to reduce the problem into a solvable size

Kok, Boon Ching

Lo, Kwok Lun

Mustafa, Mohammad Wazir

Zin, Abdullah Asuhaimi Mohammad

Repository@USM

[ENE01] Development of power system dynamic equivalents toolbox for digital type power system simulator  Kok Boon Ching1, Abdullah Asuhaimi Mohammad Zin1, Mohammad Wazir Mustafa1, Lo Kwok Lun2  1Department of Electrical Power Engineering, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor Darul Takzim, Malaysia.  2Department of Electronic and Electrical Engineering, Royal College Building, 204 George Street, University of Strathclyde, Glassgow G1 1XW, United Kingdom.  Introduction  The aim of this research is to develop a power system dynamic equivalents toolbox (PSDYNET) for digital type power system simulator. In dynamics study of large power system, it is of the essence to represent the external system by means of dynamic equivalents with the intention of improve the solution speed and to reduce the problem into a solvable size. This is for the reason that detailed representation of large power system is restricted in digital simulation program such as EMTP, PSCAD/EMTDC and TNA. At the present time, TNB is still practicing to use the static equivalents for their system analysis and studies. As a result of this, their off-line simulation studies are presently limited to steady state analysis and electromagnetic transient studies only. PSDYNET has been developed in this research so as to expand the TNB works to transient and electromechanical simulation studies.  Accurate modelling of power system dynamic equivalent model at the boundary points is an important prerequisite for meaningful exploratory studies, analysis and designs of systems those involving power electronic applications such as HVDC transmission, Static VAR Compensators and FACTS. Generally, the power system structures will include internal system (study system) and external system (which is represented by dynamic equivalents system).  Efforts to find appropriate dynamic equivalents have been reported since three decades ago. In the common practice, the external system is normally replaced by one or more coherent groups of synchronous machines (Podmore, 1979). Other methods like modal analysis (Oliveira et al, 1988), energy function (Wang et al., 1997) and rotor angle based coherency analysis (Qi, 1994), etc. has been reported in the literature.  An analytical method is proposed to identify the dynamic equivalents network, i.e. by means of the parametric identification method. The method is based on the line flow function of the original power system. The external system will be reduced and replaced by dynamic equivalent generator models. The active power (P) is afterwards utilises to identify the dynamic equivalent generator parameters such as inertia constant, H, damping factor, D and synchronous and transient reactances (Xd, Xq, Xd’, Xq’), etc. The parameters identification process is optimized using the non-linear optimization algorithms such as Levenberg-Marquardt (LM) algorithm.   Towards the end of this research, a flexible and accurate dynamic equivalents toolbox is expected as the whole research output. The toolbox will consists of a range of power system programs such as power flow analysis program, time domain simulation program, and dynamic equivalents identification program.   Methodology and tool  The introducing of dynamic equivalents for large power system basically involves the reducing numbers of differential equations to be solved while preserving the most important dynamic characteristics of the external system. This alternative method introduces the order reduction process of electrical grid system and at the same time, it preserves only the frontier buses. Fictitious generators are then located at the frontier buses to represent the external system. Figure 1 shows the related interactive buses in a power system which is divided into study and external systems. The 4th Annual Seminar of National Science Fellowship 2004549             FIGURE 1 The interactive buses in a system.  The main objective of the methodology is to identify the best parameters of the equivalent generators in the reduction order model which are then produce least error in dynamic response as compare to the full order model. An optimization algorithm is utilised together with minimization process in order to attain this matching.  Steady state preservation The first step in the network order reduction is to verify its steady state preservation. This is done by load flow study. The complex power flow that should be injected into the fictitious generators at the frontier buses is calculated through the load flow analysis. The bus power balancing equation is as follows: ,0∑∈=++Jkligiik PPP Ii∈∀                          (1) where J is the set of buses which linked to the ith frontier bus, I is the set of frontier buses, Pik is the active power flowing from ith bus to kth bus, Pgi is the generation active power at the ith bus, Pli is the load active power at the ith bus. The voltage level at the frontier buses will be adjusted to the values as calculated in the full order system load flow study.  Dynamic equivalent generator model Fourth order generator model with excitation system is employed to represent the equivalent generators model located at the frontier buses. The generator model is a two-axis with one field winding in d-axis and one damping winding in q-axis. The differential equations describing the synchronous generator written in its individual d-q rotor reference frame is expressed as follows (Galarza, 1996): 0ωωδ −= iidtd                                                 (2) Pgi1Equivalent Generator Model 1 Frontier bus i1 Bus j1 qididiqimiii IIXEPMdtd)([1 '' −−=ω  Pij1Pij2 Frontier bus i2 Bus j2 )]()( 0'' ωω −−+− iidiqiqidi DIIXE       (3) Pgi2 Equivalent Generator Model 2 Study System ])([1 '''0'qidididifdiidqi EIXXETdtdE −−−=         (4) External System])([1 '''0'qiqiqidiiqdi IXXETdtdE −+−=                 (5) whereSiniiiSHM ω2= , Sni is the machine MVA rating, δ is machine rotor angle, ωSi is machine synchronous speed, Pm is machine mechanical power, E’d and E’q are d-axis and q-axis voltages respectively, X and X’ are synchronous and transient reactances respectively, T is time con-stant in second, Ef is exciter field voltage, and H is the machine inertia. Equations (2) and (3) describe the rotor mechanical dynamics in the state variables δ and ω, while (4) and (5) represent the rotor electrical equations in the state variables  and . The machine model also involves algebraic equations for the stator transformed into the complex network reference frame and is written as follows: 'dE'qE0)sin( '' =+−−− qiqiSidiiiTidi XIRIVE θδ       (6) 0)cos( '' =+−−− didiSiqiiiTiqi XIRIVE θδ      (7)  Algorithm of parameters identification  The parameters identification will be done in time domain. The tuning of the parameters of equivalent generators is done so as to match the full model time response output. The objective of the time domain identification is, for the time interval t = 0 to t = T0, to fit the parameter, x to minimize the error function (Stankovic, 2003), ∫ −−= 00))()(())()(()(TfmrTfmrT dttytytytyxJ  (8) where yr(t) and yfm(t) are the output of the reduced and full models, respectively. The 4th Annual Seminar of National Science Fellowship 2004550Minimization of the squared difference of active power flows at the frontier lines is utilised for the identification. This condition is express as: ),(min xJ  22 )())(( eqikIi Kk Ii Kkoikikoik PPxfP −=−∑∑ ∑∑∈ ∈ ∈ ∈       (9) where K is the set of buses in the study system linked with frontier buses, x is the vector of parameters to be identified,  is the active power flow from ith to kth bus in the full order system,  is the active power flow from ith to kth bus in the equivalent system. oikPeqikik Pxf =)(The flow chart of the overview methodology is depicted in Figure 2. The frontier buses are those buses of the external system which are linking with the studied system.                   FIGURE 2 The flow diagram of parameters identification technique.  The objective function of the optimization problem solved by the Levenberg-Marquardt (LM) Algorithm can be expressed as (More,1977): [ ] [ )()()( xrxrxJ T= ]                                      (10) where                                (11) )()( xfPxr o −=is the criterion function. The minimize of J(x) can be achieved by differentiate equation (10) and equate to zero,  must satisfy the nonlinear equation, xˆ[ ] [ 0)ˆ()ˆ(2)(ˆ=−−==xfPxFdxxdJ oTxx]          (12) where xxdxxdfxFˆ)()ˆ(==  is Jacobian matrix. One of the methods to solve the equation (12) is based on Taylor series approximation of f(x) around a nominal point, xo, ]ˆ)[()()( ooo xxxFxfxf −+=                        (13) Substituting equation (13) into equation (12) yields, )]()[(]ˆ)][()([ oooToooT xfPxFxxxFxF −=−          (14) System Data Load flow and dynamic data )()()]()([ 1 qqTqqqT xrxFxxFxF =∆ +           (15) Solve iteratively xq+1 + xq + ∆xq+1, where r(xq) = Po-f(xq) is the residual in qth iteration. The iterations of equations (14) and (15) are continued until  approaches a minimum value. According to Levenberg-Marquardt Algorithm (More, 1977), equations (14) and (15) may be solved by adding a positive number to the diagonal of the matrix  for fear that of oscillatory behaviour in convergence and/or ill-conditioning of the matrix. So, equation (15) becomes, )ˆ(xJ)()( qTq xFxF)()(])()([ 1 qqTqqqT xrxFxDxFxF =∆+ +α      (16) where D is a diagonal matrix and the constant α  0. > Toolbox – PSDYNET program  The Graphic User Interface (GUI) layout for the Power System Dynamic Equivalents Toolbox (PSDYNET) is shown in Figure 3. It has been developed in Matlab environment for the purpose of constructing the power system dynamic equivalents network. Principally, it consists of three major programs – power flow analysis program, time domain simulations program, and dynamic equivalent identification Steady State Preservation ∑ =++ ligiij PPP 0  Transient Analysis Calculate power flowat frontier line, Poij. Estimation Xo = [ H, D, Xd, X'd, Xq, X'q ] Transient Analysis As full order model.  Calculate power flow, Peqij. Minimization|| ∇ J(x)|| < ε EndOptimization Algorithm Xq = [ Hiq, Diq, Xdiq, X'diq, Xqiq, X'qiq ] No Reduced order model Yes Full order model The 4th Annual Seminar of National Science Fellowship 2004551program. Other utility tools that available with PSDYNET are static report generation (for power flow) and plotting page (for time domain imulations).  s  FIGURE 3 The layout of PSDYNET program. imulation results and discussion fault by removing it at t = 2.1.    S Test system model  For the purpose of evaluating the method developed in this paper for dynamic equivalents identification, a detailed 39-bus IEEE system containing 10 generator units and 46 lines is used. This system represents the New England area 345-kV bulk transmission network. Machine 10 at bus 39 is an equivalent power source representing parts of the U.S.-Canadian interconnection system. Machine 2 is chosen as a reference (slack bus) for the other nine machines. A one-line diagram of the system is depicted in Figure 4. The system is separated into a study system and an external system as shown. The study system contains generating units 2 and 3 while the other generating units are belongs to the external system. Thus, there are three frontier buses (3, 15, and 39) and three frontier lines (3-4, 15-14, and 39-9). Figure 5 shows an equivalent network including three fictitious generating units and equivalent loads at bus 3, 15, and 39. The evaluation procedure is performed by applying a three-phase fault on bus 4 at simulation time, t = 2.0 and clearing the   FIGURE 4  The 39-bus New England system.   FIGURE 5  The equivalent network.  Results and discussion  Table 1 exhibits the estimated parameters for the fictitious generating units, under the consideration aforementioned (LM algorithm).  TABLE 1 The estimated equivalent parameters.  Gen Xd X'd T'do Xq X'q T'qo H D Eq1 0.371 0.128 7.5 0.344 0.141 1.67 6.7 0 Eq2 0.369 0.112 7.6 0.339 0.184 1.73 6.3 0 Eq3 0.392 0.136 7.9 0.355 0.132 1.84 7.6 0 G GG2 3 39935411 13 14 15 8 31 12 6 32 7Eq 2 Eq 1 Eq 3 1 8 910 765 4 2 3303912253728 26 2927 3818 17 316 2115 954 3624352220 34 2319 3310 1113148 3112 632 7External System Study System The 4th Annual Seminar of National Science Fellowship 20045520123456780 2 4 6 8 10 12 14 16 18 20Time (s)Rotor Angle (rad.)full_31 equiv_31 FIGURE 6 Rotor angle of machine 31.  0123456780 2 4 6 8 10 12 14 16 18 20Time (s)Rotor Angle (rad.)full_32 equiv_32 FIGURE 7 Rotor angle (δ) of machine 32.  0.99811.0021.0041.0061.0081.010 2 4 6 8 10 12 14 16 18 20Time (s)Rotor Speed (p.u.)full_31 equiv_31 FIGURE 8 Rotor speed (ω) of machine 31.  0.99811.0021.0041.0061.0081.010 2 4 6 8 10 12 14 16 18 20Time (s)Rotor Speed (p.u.)full_32 equiv_32 FIGURE 9 Rotor speed (ω) of machine 32.  5.15.125.145.165.185.25.220 2 4 6 8 10 12 14 16 18 20Time (s)Mechanical Power (p.u.)full_31 equiv_31 FIGURE 10 Mechanical power of machine 31.  6.426.446.466.486.56.526.540 2 4 6 8 10 12 14 16 18 20Time (s)Mechanical Power (p.u.)full_32 equiv_32 FIGURE 11 Mechanical power of machine 32.  00.20.40.60.811.20 1 2 3 4 5Time (s)Voltage (p.u.)full_31 equiv_31 FIGURE 12 Voltage at generator bus 31.  00.511.522.533.544.50 2 4 6 8 10 12 14 16 18 20Time (s)Active Power (p.u.)full_10_to_11 equiv_10_to_11 FIGURE 13 Active power flows at the line 10-11.   The 4th Annual Seminar of National Science Fellowship 2004553 The quality of the estimation was confirmed by transient stability study. Figures 6 to 13 depict the sample of the corresponding dynamic responses, comparing the behaviour of the full and reduced system after the three-phase fault. It is observed from the simulation results that the plotted dynamic behaviours are satisfactorily close to the original full system. Besides simplified the system modelling works, another main benefit that can be gained from the reduction process is to save the CPU run time. Table 2 gives the overall simulation per-formance of both the full and the reduced system which are running on PC platform with Pentium IV, 2.0 GHz processor and 256MB RAM.  TABLE 2 CPU run time for power flow analysis and time domain simulation  Analysis Time in sec. (Full) Time in sec. (Reduced) Power flow 0.341 0.180 Time domain simulation 19.339 12.688  Conclusions  The formulation of a new alternative methodology is proposed to obtain the power system dynamic equivalents which preserve satisfactorily the set of electromechanical modes associated to generators of the studied system. External system is replaced with fictitious generating units and its equivalent loads whose consequent parameters are included in the solution. Such formulation is posed as non-linear optimization problem that can be solved by a range of methods. In this paper, the Levenberg-Marquardt algorithm has been selected to solve the problem due to its robustness in convergence matter. For further study the problem, biological inspired algorithm such as Genetic Algorithm (GA) and social behavior inspired algorithm such as Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) are suggested for the parameter identification procedures.  Acknowledgements The authors are gratefully acknowledged the Ministry of Science, Technology and Innovation (MOSTI), Malaysia for given the fully financial support in this research through the National Science Fellowship awarded to Kok Boon Ching. Besides, the contribution of Tenaga Nasional Berhad Research (TNBR) Sdn. Bhd. in providing the research materials and valuable information is also highly appreciated.  References Galarza, R.J., (1996). Power system dynamic equivalencing, New York: Rensselaer Poly-technic Institute.  More, J.J., (1977). The Levenberg-Marquardt algorithm: implementation and theory. Journal of Numerical analysis, Springer 630: 105.  Oliveira, S.E.M., and Queiroz, J.F. (1988).  Modal dynamic equivalent for electric power systems, Part I: Theory. IEEE Trans. on Power System 3: 1723-1730.  Podmore, R., (1979). A comprehensive program for computing coherency-based dynamic equiva-lents. Proc. IEEE Conf. on Power Industry Computer Applications:  298-306.  Qi, Z.Z., (1994). Coherency based dynamic equiva-lence of power system, Scotland: University of Strathclyde.  Wang, L.,  Klein, M.,  Yirga, S., and Kundur, P., (1997). Dynamic reduction of large power systems for stability studies. IEEE Trans. on Power Systems 12: 889-895.  Stankovic, A.M., Saric, A.T., Milosevic, M., (2003). Identification of nonparametric dynamic power system equivalents with artificial neural networks. IEEE Trans. on Power Systems 18:1478-1486. The 4th Annual Seminar of National Science Fellowship 2004554

Development of power system dynamic equivalents toolbox for digital type power system simulator

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