Power system dynamics has an important bearing on the satisfactory system operation. It is influenced by the dynamics of the system components such as generators, transmission lines, loads and other control equipment (HVDe and SVC controllers). The dynamic behaviour of power systems can be quite complex and a good understanding is essential for proper system planning and secure operation.
Simultaneous Solution
The system equations can be expressed as x = !(x, V) J{x, V) - [Y]V = 0
Implicit method of integration is used to discretize Eq (12.68) For example, using trapezoidal rule, we get
- h Fxk = Xk - Xk-l - "2[!{Xk, Vk) + !{Xk-l, Vk-d] = 0 (12.70) Combining this with Eq (12.69) and linearizing, we get
12 Simulation for Transient Stability Evaluation
425 m is the number of dynamic subsystems (number of gen.erators if no other dy- namic devices are considered), IN is the network Jacobian
Eq (12.71) can be solved as follows
[LN] = [J~] - Ck[A~rlBk -[L~]~Vk = Fh
In exact Newton's method, the Jacobian matrix [Jk] is updated at each iteration and step, while the VDHN method only updates the Jacobian when the system configuration changes or the iteration count exceeds a certain threshold Although the VDHN method exhibits slower convergence, it significantly reduces the number of computations required at each step.
Case Studies
The results of transient stability evaluation for three test systems are presented in this section The test systems selected are
(1) IEEE Transient Stability Test Systems [8]
(2) 10 generator, 39-bus New England Test System [9]
The 'STEPS' program, which stands for Structure Preserving Transient Energy based Program for Stability, is a specialized PC-based tool designed for simulating power systems and analyzing transient stability A key feature of this program is its ability to compute the structure-preserving energy function, which plays a crucial role in ensuring accurate stability assessments.
In the upcoming chapter, we will demonstrate that utilizing the scanning of energy functions is a more straightforward approach for assessing system transient stability than analyzing numerous swing curves Moreover, employing a structure-preserving transient energy function allows for direct stability evaluation through methods such as the potential energy boundary surface or the unstable equilibrium point (UEP) technique for identifying critical energy levels A detailed discussion on the application of energy function methods will be provided in Chapter 13.
In this benchmark test, all generators are modeled using classical models, while the loads are represented as constant impedances The study focuses on a three-phase fault occurring at bus #75, which is resolved by disconnecting the line between bus #75 and bus #9.
The simulation shows that the critical clearing time lies in the range (0.354-0.355 s) The swing curves (rotor angle with respect to Centre of Inertia) for all the
Figures 12.6 to 12.S illustrate 17 generators, highlighting results for both stable and unstable scenarios The instability is primarily due to generator 16, which is linked to bus #130, experiencing a loss of synchrony These findings align with the results presented in [S].
The generators are modeled using classical representations, while the loads are treated as constant impedances The analysis focuses on a three-phase fault occurring at bus #7, which is resolved by disconnecting the line between bus #7 and bus #6.
The simulation results indicate that the critical clearing time ranges from 0.105 to 0.109 seconds Figure 12.9 illustrates the swing curves of critical generators, highlighting both stable and unstable scenarios The observed instability arises from machines connected at buses 104 and 111 losing synchronization with the other machines These findings are consistent with previous results documented in [5].
The system data is taken from [10] The machines are represented by two-axis models (1.1) The generators are assumed to be provided with static exciters
12 Simulation for Transient Stability Evaluation 427
In this study, loads are simplified as constant impedances to facilitate analysis A three-phase fault at bus #14, which is resolved by disconnecting the line between bus #14 and bus #34, is the primary disturbance examined The critical clearing time for this fault is determined to be between 0.34 and 0.35 seconds However, when classical models are employed for the machines, this critical clearing time decreases to a range of 0.26 to 0.27 seconds.
The swing curves for the ten machines, represented by classical models, are illustrated in Figs 12.10 and 12.11, while Figs 12.12 and 12.13 depict the swing curves when generators are modeled in detail Additionally, the variations in field voltages for all generators in this detailed modeling scenario are presented in Figs 12.14 and 12.15.
The mode of instability remains consistent regardless of whether machines are represented by classical or detailed models, primarily due to the separation of machine 2, which decelerates while other machines accelerate Additionally, it is noted that groups of generators situated near one another or arranged in a radial system exhibit coherent behavior, as seen with generators 1 and 2.
3, 4 and 5, 6 and 7 form three coherent groups It is interesting to observe t.hat the variations of Efd within coherent groups are similar.
Dynamic Equivalents and Model Reduction 427
In large systems, simplifying the model order is essential to minimize complexity, especially due to computational constraints in online applications Additionally, the lack of available data can pose significant challenges in these scenarios.
In nonlinear simulations for stability evaluation, the focus is primarily on the effects of large disturbances within a designated area known as the study system The external system, while not directly relevant to stability analysis, influences the study system's response to internal disturbances, making it essential to represent it through dynamic equivalents Historically, Ward type equivalents, derived from distribution factors utilized in power flow studies, have been adapted for dynamic analysis This evolution has resulted in the development of two distinct types of dynamic equivalents.
Figure 12.6: Swing Curves - 17 Generator system
12 Simulation for Transient Stability Evaluation 429
Figure 12.7: Swing Curves - 17 Generator system
Figure 12.8: Swing Curves - 17 Generator system
12 Simulation for Transient Stability Evaluation 431
Figure 12.9: Swing Curves - 50 Generator system
Figure 12.10: Swing Curves - 10 Generator system
12 Simulation for Transient Stability Evaluation 433
Figure 12.11: Swing Curves - 10 Generator system
Time(s) (b) Unstable (t CI = 0.35 5) Figure 12.12: Swing Curves - 10 Generator system (Detailed Model)
12 Simulation for Transient Stability Evaluation 435
Figure 12.13: Swing Curves - 10 Generator system (Detailed Model)
Figure 12.14: Variation of EFD - 10 Generator system (Detailed Model)ã
12 StIIL'Ulation for Transient Stability Evaluation 437
Figure 12.15: Variation of EFD - 10 Generator system
The dynamic equivalents based on modal analysis involve a two stage procedure of
(a) construction of matrices which represent equivalents of the external system
(b) Interfacing these matrices with the transient stability simulation of the study system to simulate the complete system
The construction of the modal equivalent produces linear equations of the form
(12.76) (12.77) where VT and IT are terminal bus voltage and current injection at terminal bus respectively (expressed in rectangular coordinates)
The equations discussed are not typically applicable as models for physical devices Additionally, reducing models through modal analysis necessitates the calculation of eigenvalues and eigenvectors, a process that can be quite time-consuming.
Coherency based equivalents involve a two stage procedure of
(a) Identification of coherent groups in the external system
Dynamic aggregation involves combining a coherent group of generating units into a single equivalent unit that maintains the same speed, voltage, and total mechanical and electrical power during disturbances, as long as the units within the group remain coherent.
A coherent group of generating units consists of generators that oscillate at the same angular speed and maintain terminal voltages in a consistent complex ratio during disturbances Consequently, these units can be interconnected to a common bus, potentially utilizing a transformer that accommodates complex ratios, including phase shifts.
Coherency identification can be achieved through heuristics like electrical distance or by employing a simplified linearized power system model In the latter method, coherent groups are determined using linear simulations in response to specific disturbances The rationale for utilizing simplified models is grounded in certain foundational assumptions.
1 The coherent gr,oups of generators are independent of the size of the distur- bance Therefore, coherency can be determined by considering a linearizoo system model
12 Simulation for Transient Stability Evaluation 439
2 The coherent groups are independent of the amount of detail in the gener- ator model Therefore a classical synchronous machine can be considered for the identification of the coherent groups
1 B Stott, "Power system dynamic response calculations", Proc IEEE, Vol
2 H.W Dommel and N Sato, "Fast transient stability solutions", IEEE Trans Vol PAS-91, July/Aug 1972, pp 1643-1650
3 G.T Vuong and A Valette, "A complex Y-matrix algorithm for transient stability studies", IEEE Trans Vol PAS-105, No 12, Dec 1985, pp 3388-3394
4 EPRI Report, "Extenced transient-midterm stability package", Report EL-461O' Electric Power Research Institute, Palo Alto, CA, Jan 1987
5 W.F Tinney and J.W Walker, "Direct solutions of sparse network equa- tions by optimally ordered triangular factorization", Proc IEEE, Vol 55, Nov 1969, pp 1801-1809
6 J.M Undrill, "Structure in the computation of power-system nonlinear dynamical response", IEEE Trans Vol PAS-88, No.1, 1969, pp 1-6
7 T.S Parker and L.O Chua, Practical numerical algorithms for chaotic systems, Springer-Verlag, New York, 1989
8 IEEE Committee Report, "Transient stability test systems for direct sta- bility methods", IEEE Trans on Power Systems, Vol 7, Feb 1992, pp 37-43
9 M.A Pai, Energy function analysis for power system stability, Kluwer Academic Publishers, Boston 1989
10 Vijayan Immanuel, Application of structure preserving energy func- tions for stability evaluation of power systems with static var compensators, Ph.D Thesis submitted to Indian Institute of Science, Bangalore, August 1993
11 H.E Brown, R.B Shipley, D Coleman and R.E Neid Jr., "A study of I stability equivalents", IEEE Trans Vol PAS-88, No.3, 1969, pp 200-
12 J.M Undrill and A.E Turner, "Construction of power system electrome- chanical equivalents by modal analysis", IEEE Trans Vol PAS-90, Sept/Oct 1971, pp 2049-2059
13 J.M Undrill, J.A Casazza, E.M Gulachenski and L.K Kirchrnayer, "Elec- tromechanical equivalents for use in power system studies", IEEE Trans Vol PAS-90, Sept.jOct 1971, pp 2060-2071
14 S.T.Y Lee and F.C Schweppe, "Distance measures and coherency recogni- tion transient stability equivalents", IEEE Trans Vol PAS-82, Sept.jOct
15 R Podmore, "Identification of coherent generators for dynamic equiva- lents", IEEE Trans Vol PAS-97, No.4, 1978, pp 1344-1354
16 A.J Germond and R Podmore, "Dynamic aggregation of ~enerating unit models" , IEEE Trans Vol PAS-97, No.4, 1978, pp 1060?1O69
17 W W Price et al., "Testing of the modal dynamic equivalents technique", IEEE Trans Vol PAS-97, No.4, 1978, pp 1366-1372
18 A Bose, "Parallel processing in dynamic simulation of power sytems", Sadhana, Indian Academy of Sciences, Vol 18, part 5, 1993, pp 815-828
Application of Energy Functions for DIrect Stability Evaluation
I n t r o d u c t i o n
Evaluating transient stability through digital simulation involves solving nonlinear differential algebraic equations over several seconds following significant disturbances, like a three-phase fault This process can be computationally intensive, especially since transient stability depends not only on the system's configuration and loading conditions but also on the specific disturbances encountered Therefore, it is essential to consider multiple credible contingencies to accurately assess the transient stability of the system.
A fast and direct method for stability evaluation is essential, eliminating the need for complex differential-algebraic equation solutions For a two-machine or single-machine infinite bus (SMIB) system, the equal area criterion provides a straightforward stability assessment based solely on the system's state—angle and velocity—at the moment of fault clearing This approach can be extended to multimachine systems through the use of energy functions.
The application of energy functions in power systems dates back to 1947, when Magnusson first reported on the topic In 1958, Aylett introduced an energy integral criterion, further advancing the field Additionally, Lyapunov's function, which generalizes energy functions, has been explored in works by Gless, EI-Abiad, and Nagappan, contributing to the understanding and development of energy functions in power system analysis.
Liapunov's method was initially applied to power system stability, marking a significant advancement in the field Since then, numerous developments have been documented, particularly in the monographs by Pai Additionally, recent state-of-the-art papers provide a comprehensive overview of ongoing research efforts in this area.
Liapunov's method, while broadly applicable, demonstrates conservativeness in practical systems and faces challenges when extending to more complex models For instance, in classical generator models with constant impedance loads, the presence of transfer conductances complicates the application of Liapunov functions, often necessitating gross approximations Additionally, even under the assumption of a lossless transmission system, network reduction can lead to significant transfer conductances.
Recent developments in this field have been a tmed at solving the prob- lems of
1 Accurate determination of critical clearing time for a given fault
2 Application to detailed generator and load models
3 The inclusion of controllers such as excitation, HVDC and SVC
4 Applications to on-line dynamic security 'assessment
The advancement of Structure Preserving Energy Functions (SPEF) has enabled the integration of load models and network-based controller models, such as HVDC and SVC, into power system analysis Previously, loads were simplified to constant impedances, and the network was reduced to include only the internal buses of generators represented by classical models However, this approach is inadequate, as the reduced network admittance matrix contains transfer conductances, making it impossible to strictly define energy functions for such systems.
Mathematical Formulation
Consider that a power system is described by nonlinear differential equations x = fr(x), to < t < tF (13.1)
At time tF, a fault is assumed to occur, with the simplification that tF = 0 Prior to this fault, the system is considered to be in equilibrium During the fault, the system can be represented by the equation x = fF{X), where tc denotes the moment the fault is cleared Following the fault, the system is described by the equation x = f(x) for the time interval tc < t < ∞.
The postfault dynamics is generally different than the prefault dynamics since the clearing of the fault is usually accompanied by disconnection of line{s)
13 Application of Energy Functions for Direct Stability Evaluation 443 x • s
Figure 13.1: Definition of critical clearing time
To evaluate transient stability in a post-fault system, it is essential to determine if the initial state lies within the stability region surrounding the stable equilibrium point (SEP) If the initial state, denoted as x(t c), is within the attraction region A(x s) of the SEP (x s), then the system is considered stable.
There are two steps in this evaluation i) characterization of the boundary of stability region, 8A(s) ii) computation of post-fault initial state x(t c ) and comparison with 8A(s)
The fault-on trajectory crosses the stability boundary at the critical time, t = tcr If the clearing time, tc, is less than tcr, the system remains transiently stable, making tcr the critical clearing time for a specific fault This critical time is influenced by X(tF), the initial condition of the faulted trajectory, which is equivalent to the system's stable equilibrium point prior to the fault.
The region of attraction is defined by the stable manifolds MS(Xi) associated with unstable equilibrium points (type 1) located on the boundary Notably, a type 1 unstable equilibrium point (UEP) possesses a one-dimensional unstable manifold MU(Xi).
It can be shown that [10), under some conditions,
(13.4) where E1 is the set of type 1 equilibrium points
Stable and unstable manifolds of a hyperbolic equilibrium point, Xi, are characterized by the eigenvalues of the system's Jacobian, which must reside exclusively in the right half-plane (RHP) or the left half-plane (LHP).
MS(Xi) = {x I 4>(3), t) + Xi as t + co}
W'(Xi) = {X I 4>(x, t) + Xi as t + -co} where 4>(x, t) is a solution curve (trajectory) starting from X at t = 0 Thus,
The stability region is defined by
The sufficient conditions that guarantee the result of Eq (13.4) are [10]
1 All the equilibrium points on the stability boundary are hyperbolic
2 The intersection of MS(Xi) and MU(xj) satisfies the transversality condi- tion for all the equilibrium points Xi, Xj on the stability boundary
3 There exists a C 1 function (continuous with continuous derivatives) W :
(i) W(4)(x, tằ ~ ° at X Â E where E is the set of equilibrium points The equality is satisfied only at some points on the trajectory and not over a finite time interval
(ii) W(4)(x, tằ is bounded implies 4>(x, t) is bounded W is termed as the energy function
Assumptions 1 and 2 represent fundamental characteristics of dynamical systems, applicable to nearly all such systems The transversality condition ensures that at each intersection point of the two manifolds, their tangent spaces collectively span the tangent space Rn.
Condition (i) in assumption (3) ensures that every trajectory either diverges to infinity or converges to an equilibrium point, thereby ruling out the existence of bounded oscillatory trajectories such as limit cycles or chaotic motions As a result of this condition, the energy function on the stable manifold of a unique equilibrium point (UEP) Xi attains its minimum value at Xi.
The region of attraction is unbounded if the system does not have a source (EP with all the eigenvalues of the Jacobian lying in the RHP) If the
13 Application of Energy Functions for Direct Stability Evaluatiof& 445 stability region is bounded, the energy function achieves a local maximum at a source on the stability boundary 8A(x s }
The transversality condition is violated in saddle-to-saddle connections, also known as heteroclinic orbits, as illustrated in Fig 13.2, where a trajectory links two Unstable Equilibrium Points (UEPs) Additionally, a homoclinic orbit exists, which is a trajectory that connects a UEP to itself.
Figure 13.2: An example where transversality condition is not satisfied
The energy function can be used to approximate the stability region The bounded region
W(X} ~ Wcr (13.6) is an estimate of the stability region if W cr (termed as critical energy) is suitably chosen There are two ways in which W cr can be selected These are
The minimum value of W, evaluated at all Unstable Equilibrium Points (UEPs) on the stability boundary, is designated as W cr, with the corresponding UEP referred to as the critical (or closest) UEP This approach tends to underestimate the stability region, as it primarily focuses on predicting the exit point p of the faulted system trajectory at the stability boundary The exit point is located on the stable manifold of a UEP known as the controlling UEP, XC, which is in proximity to the faulted system trajectory Therefore, an accurate estimation of the stability boundary is essential for effective analysis.
(B) W cr = W(x c ), Xc is the controlling UEP (13.8)
This is a two step procedure given below
Step 1 Compute the controlling UEP lying on the boundary 8A(x s ) Calculate
Figure 13.3: Critical and controlling UEP
Step 2 At the instant of clearing of thp fault, compute W(t e ) If W(t e ) ~ Wcr, the system is transiently stable When W(t e ) = Wcr, the clearing time is said to be critical
The issues that are important in the application of energy functions for direct stability evaluation are
(i) Formulation of an appropriate energy function
(iii) Approximate determination of W (t e) without having to integrate the faulted
These issues are discussed in the following sections The equivalence of equal area criterion to the criterion in terms of energy function is shown in the next section.
Energy Function Analysis of a Single Machine System 446
A synchronous machine connected to an infinite bus is illustrated in Figure 13.4, where it is modeled as a voltage source (Eg) behind a reactance (xg) The equivalent circuit representation is depicted in Figure 13.5.
13 Application of Energy Functions for Direct Stability Evaluation 447
The swing equation for the machine is given by x g p e
Figure 13.5: Equivalent circuit for system in Fig 13.4
The electric power output of the machine can be derived from the equiv- alent circuit of Fig 13.5 as
Substituting Eq (13.10) in (13.9), we have
It is to be noted that B is negative (due to inductive susceptance)
Multiplying both sides of Eq (13.11) by (~) and integrating, we have
The L.H.S of (13.12) can be expressed as the sum of kinetic energy Wk and potential energy Wp given by
Wk+Wp - W{x) = constant = kl (13.15) where xt = [6 d6] at
There is no loss of generality in expressing energies with reference to a SEP x S , such that
Thus Eq (13.15) can be revised to
~M( ~!) 2 +BEgEb[COSo _ COS as] - GEgEb[sinO - sino s]
The constant k2 is determined from the initial condition W(t = tel
1 The energy function, W (x) is defined for the postfault system The initial condition for this corresponds to t = te where te is the fault clearing time
2 The initial energy (k2) at t = t e, is determined from the integration of the faulted system equation
Equivalence with Equal Area Criterion
The criterion for stability using an energy function is given by
The application of energy junctions is crucial for directly evaluating stability in power systems, represented by the equation WeT = W(xu) = W p (8u), where Xu serves as the controlling UEP Figure 13.6 illustrates the power angle curves for three scenarios: pre-fault, faulted, and post-fault systems, with the assumption that G = 0 for simplification in the curve representation.
From Fig 13.6 it is easy to see that the area (A2 + A 3) is given by
At the time of fault clearing, the kinetic energy is given by h Oc, ,
01 where ,8r is the initial (prefault), angle p
Figure 13.6: Power angle curves The potential energy at the time of clearing is given by h Oc
Using Eq (13.18) to (13.20) in the stability criterion gives
(13.21) Thus, the stability criterion using energy function is equivalent to the equal area criterion for stability
In the previous analysis, damping effects were overlooked, resulting in a constant energy function throughout the post-fault trajectory, which is equal to the energy acquired during the fault.
2 If damping were to be considered then
W{t) =.W{t e ) -1: n(~~) dt which implies dW = _n(d5)2, dt dt (13.22)
As j is not identically zero along the post fault trajectory, Eq (13.22) satisfies the condition 3{i) defined in the previous section
In most cases, a single machine system has only one unique equilibrium point (UEP) on the stability boundary For a lossless system, where G equals zero, this equilibrium point is defined by the equation 5'/), = π - 58, typically aligning with the trajectory of the faulted system The potential energy along this fault-on trajectory attains a local maximum at this equilibrium point.
Notably, the potential energy during fault-on conditions can be tracked to compute Wer, even without prior knowledge of the 5'/)" value, by capturing its maximum value, although this approach may not be applicable to multimachine systems due to the fault-on trajectory not passing through the controlling Unstable Equilibrium Point (UEP).
In a single machine system, kinetic energy reaches zero at the moments when potential energy peaks along the post-fault trajectory Additionally, in a system devoid of damping, oscillations persist without decay, leading to a continuous exchange between potential and kinetic energies.
The controlling Unstable Equilibrium Point (UEP) does not always align with the faulted trajectory In a single machine system with fault resistance, it is theoretically possible for the machine to decelerate during a fault and approach the UEP, represented as 5~ = -(7r + 58) However, with minimal damping (n ~ 0), this UEP does not serve as the controlling UEP Analysis reveals that while the system may be stable during the initial swing, it becomes unstable during the reverse swing, as illustrated in typical swing curves This instability arises from the relationship Wp{5~) > Wp{5'/),) and underscores the importance of accurately identifying the controlling Potential Energy Point (PEP) to determine the correct value of the system's energy.
13 Application of Energy Functions for Direct Stability Evaluation 451
Figure 13.7: (a) Equilibrium points iri-a single machine system
Structure Preserving Energy Function
The extension of energy function, defined for a single machine system in the previous section, to multimachine power systems is complicated by the following factors
The presence of transfer conductance in multimachine systems raises concerns about the existence of global energy functions This is significant because transfer conductance can lead to unstable limit cycles around the saddle equilibrium point (SEP), ultimately diminishing the stability and region of attraction of the system.
The path dependent energy component due to transfer conductances can t
Figure 13.8: A swing curve be utilized in the numerical computation but has no theoretical validity
2 The number of UEPs on the stability boundary increase considerably with the system size and the computation of the controlling UEP (or critical energy) can be a problem
3 The inclusion of voltage dependent load models and detailed generator models with A VR need to be considered to increase the scope of application of energy functions for practical systems
The first and third factors listed above, can be, accounted fpr if structure- preserving model of power systems is used in the formulation of energy functions
The incorporation of detailed load models, including sve and HVDe controllers, addresses the challenges posed by significant transfer conductances in reduced network models Conversely, transmission losses in EHV transmission networks are minimal and can often be disregarded By maintaining load buses and utilizing voltage-dependent load models, it can be demonstrated that energy functions are present under specific assumptions.
1 The transmission network is lossless
2 The synchronous machines are represented by classical model
3 The active power load at any bus is constant (independent of the voltage)
The initial assumptions can be adjusted to incorporate a two-axis model featuring flux decay and a damper winding Additionally, the first and last assumptions may be modified by introducing path-dependent components in the energy functions.
The application of energy functions for direct stability evaluation offers a numerically assessable method that achieves good accuracy, particularly when path-dependent terms are minimal compared to other factors.
In a nonlinear voltage-dependent load scenario within an n-bus system featuring m machines, it is assumed that damping effects are negligible This leads to a conservative system characterized by the absence of energy dissipation, in addition to previously outlined assumptions.
The motion of ith machine with respect to the COl (Centre of Inertia) reference frame is described by the differential equations
- Pmi - Pei - M~ Mã Peol where m m
MT = L Mi, Peol = L(Pmi - Pei) i=l i=l
From the definition of the COl variables given by
In this load model, both active and reactive powers are treated as arbitrary functions of the corresponding bus voltages, allowing the system load equations to be expressed accordingly.
Pzi fpi(Vi) Qli = fqi(Vi)
The power Bow equations for the lossless system, are given below
The active power injected into the network at bus i is gli + g2i, i = 1, 2 m }
The reactive power injection at bus i is where
Ei Vi sin{