Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

with the state matrix. However, for DAE systems, computing the state matrix requires factorizing the algebraic Jacobian matrix gy . To avoid this step, it is also possible to study the complete Jacobian matrix of the DAE system, which has the advantage of being sparser that the state matrix. Finally, the power flow Jacobian matrix is useful for assessing the sensitivities between reactive powers and bus voltages.

Domain type: The typical eigenvalue analysis uses the matrix as is. This leads to the eigenvalues in the S-domain. Using the Z-domain transformation leads to a change of coordinates so that all eigenvalues with negative real part fall inside a unitary circle, while positive real part eigenvalues fall outside the unitary circle. The Z-domain transformation requires the factorization of the state matrix but can be e cient if used in conjunction with a method that computes only a reduced set of eigenvalues since the module of unstable eigenvalues can be easily estimated.

Participation factors: Participation factors are computed using right and left eigenvector matrices and allow defining the participation of each system state variable to each system mode. This information is relevant for synthesizing control systems and for defining sensitivities.

Using (8.3), the DAE system (8.2) can be rewritten as:

that is formally the same problem as (8.1).

In the discussion above there are two relevant issues:

1.The initial value problems (8.1) or (8.4) do not generally have an explicit solution due to the nonlinearity of f . In other words, to find an analytical expression for the trajectory ϕ(x, t) is generally impossible.

2.The implicit function theorem guarantees the existence of g˜ if gy is not singular. However, to find an analytical expression for (8.3) is generally not possible due to the nonlinearity of g.

To overcome these issues, the only solution is to use a numerical integration method. The numerical solution approximates x(t) using a series of discrete values ti of the independent variable t:

or, with a compact notation:

There is a huge variety of numerical integration methods. The most intuitive one is the explicit forward Euler’s method. At a generic time ti+1, the scheme of the explicit Euler’s method is:

where Δt is the step length that can be fixed or varied from step to step. Although it worked well for computing planet orbits, the explicit Euler’s method can be improved in terms of both accuracy and numerical stability. A small selection of methods that yield an improvement over the basic Euler’s scheme is presented in the following Section 8.3.

Assuming that a suitable numerical integration method is used, the importance of time domain simulations for power system analysis is twofold:

1.Assessing the electro-magnetic behavior of power system devices. Relevant studies are transients following line switching operations, symmetrical and asymmetrical faults or imbalanced conditions and power electronics converters [345]. This approach is called electro-magnetic transient analysis.

2.Assessing the electro-mechanical response of power system networks following a large disturbance such as line outages or short circuits. This approach is called transient stability analysis.

Since the time scales of electro-magnetic and electro-mechanical transients di er at least one order of magnitude (see Figure 1.6 of Chapter 1), the two studies have followed separate development and are generally tackled by different specific software tools. Only in recent years it has been recognized the need for interfacing Electro-Magnetic Transient (EMT) programs and Transient Stability (TS) ones [151]. Although this book focuses on transient stability analysis, HVDC systems and the increasing penetration of renewable energy sources and energy storage systems, most of which in dc (e.g., photovoltaic and fuel cells and battery energy storage systems) lead to the need of interfacing several small dc systems with a large ac network. Thus, Section 8.5 briefly discusses the EMT approach.

The main goal of transient stability analysis is to determine the e ects of large disturbances on the dynamic response of a given power system. This problem is conceptually di erent from the small-signal stability analysis. In fact, the system before and after the disturbance generally shows a stable equilibrium point. In other words, preand post-disturbance equilibria exist and all the eigenvalues of the state matrix have negative real part. The issue is that the trajectory of the system following the disturbance is unstable and never reaches the stable post-disturbance equilibrium point.

Thus, the object of transient stability analysis is to determine whether the system trajectory is stable or not. A variety of mathematical and engineering books have been written on this fascinating topic. In this context, it is relevant to summarize the most important solutions that have been proposed.

1.The first solution is to solve the numerical integration and observe the response of the power system. If the time domain simulation diverges, the system is unstable, otherwise it is stable. This approach has the advantage of being “exact”. The accuracy only depends on the numerical integration method and on the system model. On the other hand, the numerical integration is computationally demanding, especially taking into account that one has to solve a simulation for each contingency. Although modern computers can solve the numerical integration for real size power systems very quickly, the computational burden of a full contingency analysis is still high.

2.The second solution is based on the Lyapunov’s direct method. This method attempts to infer the stability of the ODE system (8.1) by building a function ϑ(x, x0) : Rnx → R called Lyapunov’s function able to “measure” the stability of the system. The total energy of the system is a good Lyapunov function and thus the Lyapunov’s function is called Transient Energy Function (TEF) in most publications on transient stability. The main advantage of the Lyapunov’s direct method is that the large disturbance stability of a multi-variable system is reduced to the study of a scalar function. Thus no numerical integration is needed. The main drawbacks are:

a.There is no general systematic method to define the Lyapunov’s function ϑ(x, x0). As a matter of fact, the TEF can be easily computed only for comparatively simple systems.

b.The stability region of the TEF has to be computed to provide the stability measure. This can be a di cult task for large systems.

c.For systems with losses (e.g., for the totality of real systems) the stability test provided by the TEF is only su cient, not necessary.

d.An hypothesis of the Lyapunov’s direct method is that the structure of the system must not change after a given initial instant ti. This can be a limiting hypothesis in case one wants to study the e ect of corrective actions (e.g., fast valving of synchronous machine turbines).

e.The multi-swing instability phenomenon cannot be taken into account. The multi-swing instability consists in a system that loses synchronism after the first oscillation following a large disturbance. The origin of this phenomenon is intrinsic of the nonlinear system and it has been conjectured to be caused by an unbounded chaotic motion [173].

It is worth observing that all the drawbacks of the TEF are intrinsic of the Lyapunov’s direct method and thus can be hardly solved, unless one invents some new theory able to overcome mathematical issues. On the other hand, the main drawback of the numerical time integration is only to be time consuming. Thanks to the enhancement of micro-processors, the computational burden is more and more an irrelevant constraint. For this reason, in this chapter, only numerical integration methods are described. However, the Lyapunov’s direct method is so intriguing that it is di cult to resist to the temptation of trying to solve mathematical issues. As a matter of fact, for over three decades there have been attempts to provide suitable procedures for building the Lyapunov’s function [20, 57, 98, 228, 229, 257, 328].

For the sake of example, consider the OMIB system described in Section 7.1 of Chapter 7. The OMIB system can be modelled as a two-order ODE:

where all parameters are defined in Section 7.1. The TEF for this system is the sum of the kinetic and potential energy, EK and EP , respectively, of the synchronous machine, as follows:

where (δ0, ω0) = (0.5236, 1) is the stable equilibrium point discussed in Section 7.1. A three-phase fault occurs at t0 = 0.5 s at one end of the two parallel

Fig. 8.2 Time domain analysis for the OMIB system

stable equilibrium point. In our case, it is relevant to define the stability region of the post-fault condition. Figure 8.3 shows the potential energy EP (δ) of the post-fault OMIB system (e.g., for xeq = 0.6 pu). The minimum of EP (δ)

˜ ˜

corresponds to the stable equilibrium point (δ0, 1), where δ0 = asin(0.6) rad.

The stability region is bounded by the maxima of EP (δ), that occur for the

− − − ˜ unstable equilibrium points ( π δm, 1) and (δm, 1), where δm = π δ0.

Since EP 1 = EP (δm) < EP (−π − δm) = EP 2, the binding limit is EP 1. For loss-less systems, the Lyapunov’s direct method states that:

1. If the total system energy at t0 + tc is E ≤ EP 1, then the system is stable.

2.If the total system energy at t0 + tc is greater than E > EP 1, then the system is unstable.

For loss-less systems, the Lyapunov’s direct method is both necessary and su cient.

A simple yet powerful way to apply the Lyapunov’s direct method is the equal area criterion (EAC) that is shown in Figure 8.4. The areas in the plane (δ, p) are energies. In particular, the accelerating area Aa = pm(δc −δ0) is the energy increase during the fault. In fact, during the fault the electrical power pe generated by the synchronous machine is zero (the voltage at the short-

Fig. 8.3 Post-fault potential energy of the OMIB system

compensating the accelerating area Aa. The critical angle δc satisfies the condition Aa = Ad. In particular one has:

Aa + Ac = Ad + Ac

+δm

δc = 1.1759 rad

where δ0 = 0.5235 rad, δm = 2.4981 rad and p˜maxe = 1.6667 pu. The value of the critical angle δc confirms the results of the numerical integration.

The following final remarks are relevant.

1. If a damping is included in the machine equations, the Lyapunov’s direct method is conclusive only if the energy at t0 + tc satisfies E ≤ EP 1. If E > EP 1, the Lyapunov direct method is inconclusive or, in mathematical terms, only provides a su cient stability condition (see Figure 8.5).

2.The critical clearing time is big with respect to typical protection intervention times. Modern protections can detect and open a transmission line in about 4 four cycles (80 ms at 50 Hz) which is well lower then typical values of critical clearing times (about 200 ms). Thus, only in case primary protections fail to clear the fault, there can be a real risk of transient instability. As a consequence, transient instability has become a relatively rare event in the last decades.

Fig. 8.4 Equal area criterion for the OMIB system

For the reasons above, the interest in TEF has somewhat decreased in recent years. Unfortunately other instability phenomena, included but not limited to voltage and frequency instability, are still a major system operator concern. Numerical integration can tackle any kind of instability phenomena, not just the loss of synchronism of synchronous machines. Thus, numerical integration is and will likely always be the workhorse of any power system stability analysis. The challenge is how to improve e ciency and/or accuracy of numerical methods.

8.2Power System Model

As discussed in Chapter 1, the power system model used for time domain analysis is a set of nonlinear di erential algebraic equations (DAE) with discrete variables:

0 = g(x, y, η, u, t)

If discrete variables u are substituted for if-then rules, (8.12) becomes an hybrid dynamical system, e.g., a collection of continuous DAE, one per each

Fig. 8.5 Time domain analysis for the OMIB system with damping. The clearing time is tc = 0.26 s for all simulations

discrete variable change [131].1 Thus, one can assume that the problem to be solved for each set of discrete variables is a problem similar to (8.2).

In (8.12), di erential equations f depends on machine, regulator and load models. On the other hand, the form of algebraic equations in (8.12) can be twofold, namely (i) current-injection model and (ii) power-injection model. These two models are described in the following subsections.

8.2.1Current-Injection Model

The classical and most common model of algebraic equations for transient stability analysis is the current-injection one [298]. According to this model, the algebraic variables are bus voltage phasors v¯ and the algebraic equations express the current injections at network buses:

0 = i(x, v¯) − Y (x)v¯

1The notation “hybrid system” has not to be confused with hybrid transient simulator that is used in the literature on time domain integration of power systems for indicating tools that integrate together EMT and TS analyses [151].