电力系统故障恢复过程中临界参数值的数值计算 Numerical Computation of Critical Parameter Values for Fault Recovery in Power Systems
摘要：对于一个给定的电力系统扰动，确定导致所产生的轨迹精确地位于操作点的吸引区域的边界（在状态空间中）上的临界参数值是有用的。这些临界参数值在参数空间中形成了系统恢复到原始稳定工作点的参数值集和系统没有恢复到的参数值集之间的边界。摘要利用操作点吸引区域边界上存在的控制不稳定平衡点（CUEP），提出了一种数值计算临界参数值及其相关边界的算法。关键的想法是改变一个参数值，以使以CUEP为中心的球的轨迹所花费的时间最大化。这将驱动参数值到其临界值。该算法在一个测试案例中得到了演示，它被用于寻找网络中的临界惯性量，从而使系统能够从一个特定的故障中略微恢复。它还用于对临界参数值曲线的数值跟踪 Abstract—For a given power system disturbance, it is useful to determine the critical parameter values that cause the resulting trajectory to lie exactly on the boundary (in state space) of the region of attraction of the operating point. These critical parameter values form the boundary, in parameter space, be tween sets of parameter values for which the system recovers to its original stable operating point, and sets of parameter values for which it does not. The paper presents an algorithm for numerically computing critical parameter values and their associated boundaries in parameter space by exploiting the presence of a controlling unstable equilibrium point (CUEP) on the boundary of the operating point’s region of attraction. The key idea is to vary a parameter value in such a way as to maximize the time spent by the trajectory in a ball centred at the CUEP. This will drive parameter values to their critical values. The algorithm is demonstrated on a test case where it is used to fifind the critical amount of inertia in the network such that the system is marginally able to recover from a particular fault. It is also used to numerically trace a curve of critical parameter values given by varying the moments of inertia for a pair of generators. 指标术语——电力系统、大扰动稳定性、稳定边界。 Index Terms—Power systems, large disturbance stability, stability boundary. I. 近年来，由于需求的扩大、非传统、非线性负荷的广泛使用，以及不确定可再生发电的渗透，电力系统的运行越来越接近其极限。这些条件使电力系统更容易受到大的干扰，如传输故障。因此，电力系统操作人员能够有效、可靠地评估系统从潜在干扰中恢复的能力是很重要的，以便如果恢复不确定，就可以采取预防措施。电力系统的瞬态动力学的典型模型，在秒到几十秒的量级上，包括描述时间演化的微分方程，和表示收敛于代数流形[1]的非常快速的动力学的代数方程。该模型中的一个稳定的平衡点（SEP）代表了理想的工作行为。系统是否能够从一个特定的扰动中恢复到所需的SEP，这取决于系统的参数。关于参数变化对s的影响已有广泛的研究 I. INTRODUCTION In recent years, power systems have been operating closer to their limits as a result of expanding demand, widespread use of nontraditional, nonlinear loads such as those that employ power electronics, and increasing penetration of uncertain renewable generation. These conditions make power systems more vulnerable to large disturbances, such as transmission faults. Therefore, it is important for power system operators to be able to efficiently and reliably assess the ability of a system to recover from a potential disturbance so that, if recovery is uncertain, preventive action can be taken. Typical models for transient dynamics of power systems, on the order of seconds to tens of seconds, consist of a mixture of differential equations describing the time evolution, and algebraic equations representing very fast dynamics that converge to an algebraic manifold [1]. A stable equilibrium point (SEP) in this model represents desirable operating behaviour. Whether the system is able to recover to the desired SEP from a particular disturbance depends on the system parameters. There has been extensive study of the effect of parameter variation on the small disturbance (linear) response near the SEP. However, for more realistic industrystandard power system models the relationship between system
参数和大扰动（非线性）行为尚未得到令人满意的表征。在某种程度上，这是由于大扰动动力学中固有的非线性的挑战性，它不能像在局部稳定性的情况下那样安全地线性化。这项工作的目标是描述，对于一个特定的干扰，参数边界：参数空间边界的参数值的系统能够恢复原始SEP，和集的参数值无法恢复，如图1所示
parameters and large disturbance (nonlinear) behaviour has not been satisfactorily characterized. In part, this is due to the challenging nature of the nonlinearities inherent in large disturbance dynamics, which cannot be safely linearized away as in the case of local stability.
The goal of this work is to characterize, for a particular disturbance, the parameter boundary: the boundary in parameter space between sets of parameter values for which the system is able to recover to the original SEP, and sets of parameter values for which it is not able to recover, as illustrated in Figure 1
先前的工作[2]已经成功地开发了有效的方法来数值计算两个自由参数的选择的参数边界，或在高维自由参数的参数边界上找到最近的点，对于动力学是由感应电机驱动的系统。最近，该算法在大量的动力系统中的应用得到了严格的理论证明，包括动力系统[3]。本文在一个具有四阶机器模型的同步机器网络上验证了该算法的有效性。该算法已经在感应电机[2]系统上演示，该系统表现出类似于梯度系统的动力学，同步机器的行为更像哈密顿系统。我们感兴趣的算法最适合于表现出强阻尼和最小振荡的系统。由于感应电机对其线性化贡献了真实的特征值，并且类似于一个梯度系统，它们为这些算法提供了一个更容易的测试用例。
Prior work [2] has been successful in developing efficient
methods for numerically computing this parameter boundary
for a choice of two free parameters, or finding the nearest
point on the parameter boundary in higher dimensions of free
parameters, for systems where the dynamics were driven by
induction motors. More recently, rigorous theoretical justifi-
cation has been provided for the use of this algorithm in a
large class of dynamical systems, including power systems [3].
This paper demonstrates the effectiveness of the algorithms
on a more realistic network of synchronous machines with
fourth order machine models. Whereas the algorithm has
already been demonstrated on a system of induction motors
[2] which exhibit dynamics similar to a gradient system,
synchronous machines behave more like Hamiltonian systems.
The algorithms of interest work best for systems which exhibit
strong damping and minimal oscillations. As induction motors
contribute real eigenvalues to their linearization and resemble
a gradient system, they represent an easier test case for these
algorithms.
s.相比之下，同步机器网络通常是弱阻尼，具有复杂的特征值，因此振荡，这是对这些算法的挑战。研究结果表明，所提出的算法可以克服这些挑战。在第三节和第四节介绍算法和主要结果之前，第二节提供了基本动力系统概念的一些动机。第五节和第六节给出了一个例子，第七节给出了结论。
In contrast, a network of synchronous machines is
typically weakly damped and possesses complex eigenvalues,
hence oscillations, which represents a challenge to these
algorithms. This work illustrates that the proposed algorithms
can overcome these challenges.
Some motivation for the underlying dynamical systems
concepts is provided in Section II before the presentation of
the algorithms and main results in Sections III and IV. An
example is presented and discussed in Sections V and VI, and
conclusions are provided in Section VII.
微光为了激励和提供所提出的算法和理论的直觉，我们首先考虑一个单机无限总线电力系统的简单说明性例子。这个系统
II. MOTIVATION
To motivate and provide intuition for the proposed algorithm and theory, we first consider the simple illustrative example of a single-machine infinite-bus power system. This system
图1。参数空间中分离参数的边界，使系统能够从其无法恢复的参数中恢复到其原来的稳定平衡点。只有两个平衡点，一个是稳定的，另一个是不稳定的。我们所考虑的扰动是发电机终端上的一个故障，它是通过将发电机产生的电力设置为零来模拟的。假设故障在有限的时间后清除，系统恢复到故障前的状态。设故障后初始条件（PFIC）表示故障清除时的系统状态。从扰动中恢复取决于PFIC是位于吸引区域的内部还是外部，我们今后将其称为SEP的稳定区域。当PFIC精确地位于稳定区域的边界，即稳定边界时，就会出现了恢复的边界情况。因此，其目标是找到PFIC位于扰动后稳定边界上的临界参数值
Figure 1. A boundary in parameter space separating parameters for which
the system is able to recover to its original stable equilibrium point from
parameters for which it is not able to recover.
has only two equilibria, one stable and one unstable. The
disturbance we consider is a fault on the terminals of the
generator, which is modelled by setting the electric power
produced by the generator to zero. The fault is assumed to
clear after a finite length of time, and the system returns to its
pre-fault conditions. Let the post-fault initial condition (PFIC) denote the system state when the fault clears. Recovery from the disturbance depends on whether the PFIC lies inside or outside the region of attraction, which we henceforth refer to as the stability region of the SEP. The bounding case for recovery occurs when the PFIC lies precisely on the boundary of the stability region, i.e. the stability boundary. Therefore,
the goal is to find critical parameter values for which the PFIC
lies on the post-disturbance stability boundary
考虑参数接近其临界值，系统恢复到原来的SEP的情况。随着故障清除时间的增加，轨迹逐渐进入不稳定的平衡点（UEP），如图2和3所示，直到清除时间变大，系统不再恢复到原来的SEP。请注意，虽然选择本例中的参数是为了不改变向量场，但在一般情况下，我们会考虑可以同时影响PFIC和向量场的参数
Consider the case where parameters are close to their critical
values and the system recovers to the original SEP. As the
fault clearing time is increased, the trajectory spends more
and more time in a neighbourhood of an unstable equilibrium
point (UEP), as depicted in Figures 2 and 3, until the clearing
time becomes so large that the system no longer recovers to the
original SEP. Note that although the parameter in this example
was chosen so as not to alter the vector field, in general we
will consider parameters which can influence both the PFIC
and the vector field
通过精确地考虑系统的UEP，对这一现象进行了直观的解释。因为它处于平衡状态，所以它（理论上）会在那里停留无限的时间。然而，如果它开始时非常接近UEP，但不是完全在UEP，通过流动的连续性，它将在UEP附近花费很长时间，然后发散到无穷大或SEP。因此，在参数边界上寻找参数值的想法是改变单个参数，使球在特定UEP周围的时间最大化，称为控制UEP（CUEP）
An intuitive explanation of this phenomenon follows by
considering the system exactly at its UEP. Because it is in equilibrium, it would (theoretically) remain there for infinite time. However, if it were started very close to, but not exactly at, the UEP, by continuity of the flow it would spend a very long time near the UEP before diverging to infinity or an SEP. Therefore, the idea for finding a parameter value on the parameter boundary is to vary a single parameter in such a way as to maximize the time spent in a ball around a
particular UEP, called the controlling UEP (CUEP), that lies
图2.控制不稳定平衡点附近不同故障清除时间（Tcl）下单机无限总线状态空间的轨迹。
Figure 2. The trajectories in state space of the single-machine infinite-bus
for different fault clearing times (Tcl) in the neighbourhood of the controlling
unstable equilibrium point.
图3。轨迹与控制的不稳定平衡点的距离作为时间的函数。在稳定的边界上。然后，可以使用延拓方法在两个自由参数维度中跟踪参数边界。此外，参数边界上的点可以用来初始化一个优化公式，在更高维的参数边界上找到最近的点。
Figure 3. The distance of the trajectory from the controlling unstable
equilibrium point as a function of time.
on the stability boundary. A continuation method can then be used to trace the parameter boundary in two free parameter dimensions. Furthermore, the point on the parameter boundary can be used to initialize an optimization formulation that finds the nearest point on the parameter boundary in higher dimensions.
这些算法之所以能工作，是因为我们能够利用存在一个特殊UEP的性质，即当一个参数接近其临界值时，轨迹花费在该UEP附近的时间会发散到无穷大。在[3]中显示，这个特性对于电力系统是通用的。因此，这个特性允许我们将寻找临界参数值的问题转换为一个显式的优化问题，其中我们最大化了轨迹所花费的时间
These algorithms work because we are able to exploit the
property that there exists a special UEP such that the time the trajectory spends in a neighbourhood of this UEP diverges to infinity as a parameter approaches its critical value. It was
shown in [3] that this property is generic for power systems. Consequently, this property allows us to convert the problem of finding critical parameter values into an explicit optimization problem, where we maximize the time spent by the trajectory
在CUEP周围的一个球中，受代数约束。罗马数字 3理论电力系统的标准模型是微分方程（DAEs）的耦合集。设x∈Rn是动态状态，y∈Rm是代数状态，然后写成
in a ball around the CUEP, subject to algebraic constraints. III. THEORY A standard model for power systems is a coupled set of differential and algebraic equations (DAEs). Let x ∈ Rn be dynamic states, y ∈ Rm be algebraic states, and write
其中f： Rn+m→Rn和g： Rn+m→Rm都是C1光滑的。动态状态x包括发电机状态，如转子角度、频率和内部电压，而代数状态y包括母线电压的大小和角度。在合理的假设下，该系统可以被视为一个等效的常微分方程组（ODEs）。然后，针对ode开发的[3]理论可以用来证明CUEP的存在性及其所期望的性质
where f : Rn+m → Rn and g : Rn+m → Rm are both C1 smooth. The dynamic states x include generator states such as rotor angles, frequencies and internal voltages, whereas the algebraic states y include bus voltage magnitudes and angles. Under reasonable assumptions [4] this system of DAEs can be treated as an equivalent system of ordinary differential equations (ODEs). Then the theory in [3], which was developed for ODEs, can be applied to show the existence of a CUEP along with its desired properties
考虑对该系统的特定干扰，例如某条传输线路上的故障。设p为系统的任意参数，如故障持续时间或发电机惯性矩。注意，一般的PFIC，向量场，和代数方程将依赖于p。设x0(p)表示与参数值p对应的PFIC。我们设有以下系统：
Consider a particular disturbance to this system, such as a fault on a certain transmission line. Let p be any parameter of the system, such as fault duration or a generator moment of inertia. Note that in general the PFIC, the vector field, and the algebraic equations will depend on p. Let x0(p) denote the PFIC corresponding to parameter value p. We have the following system:
设φ为(3)-(5)的动态状态的流动，因此x (t) = φ（t、x0、p）为在向量场f（·、p）和代数方程g（·、p）下向前流动给出的状态。为了简写，我们写出φ（t，p）：= φ（t，x0(p)，p）来表示从初始条件x0(p)开始的系统在t时刻的动态状态的流动。设xe (p)是某个参数值p的一个平衡点。我们定义了xe (p)的稳定流形，记为Ws（xe(p)），由
Let φ be the flow of the dynamic states of (3)-(5), so that x(t) = φ(t, x0, p) is the state given by flowing x0 forward by time t under the vector field f(·, ·, p) and the algebraic equations g(·, ·, p). For shorthand we write φ(t, p) := φ(t, x0(p), p) to denote the flow of the dynamic states of the system at time t from initial condition x0(p). Let xe(p) be an equilibrium point for some parameter value p. We define the stable manifold of xe(p), denoted Ws(xe(p)), by
Ws (xe(p)) := {x ∈ Rn : limt→∞ φ(t, x, p) = xe (p)}.(6)xe(p)的稳定流形由状态空间中在正向时间内收敛于xe (p)的所有点组成。
Ws(xe(p)) := {x ∈ Rn : limt→∞ φ(t, x, p) = xe(p)}. (6) The stable manifold of xe(p) consists of all points in the state space which converge to xe(p) in forwards time.
设xs (p)表示某个参数值p下的电力系统的SEP。然后，Ws（xs(p)）是SEP xs (p)的稳定区域。设∂S表示一个集合S⊂Rn [5]的拓扑边界。那么∂Ws（xs(p)）是稳定的边界。注意，当且仅当x0（p∗）∈∂Ws（xs（p∗））时，p∗是一个临界参数值。[3]的工作，建立在先前的理论关于分解∂Ws（xs（p∗））的稳定流形包含[6]，表明在实际假设下，这样一个临界参数值p∗存在和x0（p∗）包含在稳定的流形
Let xs(p) denote the SEP of the power system for some parameter value p. Then Ws(xs(p)) is the stability region of the SEP xs(p). Let ∂S denote the topological boundary of a set S ⊂ Rn [5]. Then ∂Ws(xs(p)) is the stability boundary. Note that p∗ is a critical parameter value if and only if x0(p∗) ∈ ∂Ws(xs(p∗ )). The work in [3], which builds on
prior theory regarding a decomposition of ∂Ws(xs(p∗ )) into a union of stable manifolds of the UEPs it contains [6], shows that under practical assumptions, such a critical parameter
value p∗ exists and x0(p∗) is contained in the stable manifold
在稳定边界上的一些UEP xe（p∗）。我们称xe（p∗）为CUEP。由于x0（p∗）∈Ws（xe（p∗）），流φ（t，x0（p∗），p∗）收敛于xe（p∗），所以它在xe（p∗）周围的一个球上花费无限的时间。在[3]中显示，x0(p)的轨迹在球中绕xe (p)所花费的时间在p中是连续的，并且当p接近其临界值p∗时，它发散到无穷大。这激发了第四节的算法通过最大化CUEP周围的球的时间来寻找临界参数值p∗。
of some UEP xe(p∗) which is in the stability boundary. We call xe(p∗) the CUEP. Since x0(p∗) ∈ Ws(xe(p∗)), the flow φ(t, x0(P∗), p∗) converges to xe(p∗), so it spends an infinite amount of time in a ball around xe(p∗). It is shown in [3] that the time the trajectory of x0(p) spends in the ball around xe(p) is continuous in p and that it diverges to infinity as p approaches its critical value p∗. This motivates the algorithm of Section IV for finding a critical parameter value p∗ by maximizing the time in a ball around the CUEP.
增值算法上一节提出的理论支持了一种算法，该算法通过最大化围绕CUEP的球的轨迹所花费的时间来计算临界参数值。为了本讨论的目的，我们假设CUEP已经被其他一些方法发现了，如[7]，[8]。
IV. ALGORITHM
The theory presented in the previous section underpins an algorithm which computes critical parameter values by maximizing the time spent by a trajectory in a ball around the CUEP. For the purposes of this discussion, it is assumed that the CUEP has already been found by some other method, such as those of [7], [8]
该算法的第一步是找到与在球内花费固定目标时间的系统对应的参数值。设ˆx表示CUEP，r表示球周围的半径，τ表示球内的目标时间，φ（t，p）表示时间t时的参数值p的系统状态。另外，设t1，t2表示球通过系统轨迹的边界的初始和最终交叉，设x1，x2分别表示系统在时间t1，t2时的动态状态。同样，设y1、y2分别表示t1、t2时的代数状态。让z = [x1 y1 t1 x2 y2 t2 p]|，这些条件可以用代数的方法来描述
The first step of the algorithm is to find the parameter value corresponding to the system spending a fixed target time inside the ball. Let ˆx denote the CUEP, r the radius of the ball around it, τ the target time inside the ball, and φ(t, p) the system state at time t for parameter value p. Furthermore, let t1, t2 denote the initial and final crossings of the boundary of the ball by the system trajectory, and let x1, x2 denote the system dynamic states at times t1, t2, respectively. Similarly, let y1, y2 denote the system algebraic states at times t1, t2, respectively. Letting z = [x1 y1 t1 x2 y2 t2 p]| , these conditions can be described algebraically by
通过求解F (z) = 0，得到了所期望的条件。(8)一个解决方案意味着||x1ˆx||2=||x2ˆx||2=r，因此x1，x2在CUEP周围的球的边界上。此外，φ（t1，p）= x1和同样的t2确保了t1，t2是轨迹与球的边界相交的初始时间和最后的时间。此外，τ = t2 t1强制执行了第一个和最后一个交点之间的时间约束。最后，g（x1，y1）= g（x2，y2）= 0确保y1，y2是与动态状态x1，x2对应的代数状态
The desired conditions are obtained by solving, F(z) = 0. (8) A solution implies that ||x1 ˆx||2 = ||x2 ˆx||2 = r, so x1, x2 are on the boundary of the ball around the CUEP. Also, φ(t1, p) = x1 and likewise for t2 ensures that t1, t2 are the initial and final times at which the trajectory intersects the boundary of the ball. Furthermore, τ = t2 t1 enforces the time constraint between the first and final intersections. Finally, g(x1, y1) = g(x2, y2) = 0 ensures that y1, y2 are the algebraic states corresponding to the dynamic states x1, x2
这个方程组可以解决使用纽顿拉夫森，它需要相应的雅可比矩阵，J (z) =∂F(z)∂z=I 0 f|1 0 0 0∂φ（t1，p）∂p∂∂g∂1∂y|1∂∂|100000000000000丆I 0 f|2∂φ（t2，p）∂p0 0 0 ∂∂x|2∂∂y|2000000（x2ˆx）|0 0 0 0 0 1 0 0 ䷖1 0 (9)f|1：=f（x1，y1）和f|2 ：= f（x2，y2）。为了形成这个雅可比矩阵，需要计算偏导数∂φ（t1，p）∂p和∂φ（t2，p）∂p。它们可以从与流φ（t，x0）相关联的轨迹灵敏度∂φ（t，x0）∂x0中获得。这是通过引入p作为满足平凡微分方程˙p=0的状态变量来实现的。[9]的结果表明，轨迹灵敏度可以有效地与标称系统动力学进行数值积分。牛顿-拉夫森的更新有通常的形式
This system of equations can be solved using NewtonRaphson, which requires the corresponding Jacobian matrix, J(z) = ∂F(z)
∂z = I 0 f|1 0 0 0 ∂φ(t1,p)
∂p
∂g
∂x |1
∂g
∂y |1 0 0 0 0 0
(x1 ˆx)| 0 0 0 0 0 0
0 0 0 丆I 0 f|2
∂φ(t2,p)
∂p
0 0 0 ∂g
∂x |2
∂g
∂y |2 0 0
0 0 0 (x2 ˆx)| 0 0 0
0 0 1 0 0 ䷖1 0
where f|1 := f(x1, y1) and f|2 := f(x2, y2). To form this Jacobian matrix, it is necessary to compute the partial derivatives ∂φ(t1,p) ∂p and ∂φ(t2,p) ∂p . They can be obtained from the trajectory sensitivities ∂φ(t,x0) ∂x0 associated with the flow φ(t, x0). This is achieved by introducing p as a state variable that satisfies the trivial differential equation ˙p = 0. It is shown in [9] that trajectory sensitivities can be efficiently numerically integrated along with the nominal system dynamics. The Newton-Raphson update has the usual form
其中，k是迭代次数。为了提高计算效率，该系统通常采用J（zk）的LU或QR分解来求解。这种牛顿-拉夫森算法的结果是一个参数值p，它驱动系统在CUEP周围的球内花费一个目标时间τ。
where k is the iteration number. For purposes of computational efficiency, this system is usually solved using either LU or QR decomposition of J(zk). The result of this Newton-Raphson algorithm is a parameter value p which drives the system to spend a target time τ inside the ball around the CUEP.
下一步是改变一个参数，以最大化围绕CUEP的球中的轨迹所花费的时间。这是通过现在将τ作为(7)中的一个变量来实现的，并给出zcont = [x1 y1 t1 x2 y2 t2 p τ ]|。然后zcont和约束F（zcont）= 0定义了一条曲线，因为zcont中比F（zcont）= 0中的方程的数量多一个变量。随着τ的增加，可以使用一种类似于[10]概念的延续方法来跟踪解决方案
The next step is to vary a parameter so as to maximize the time spent by the trajectory in the ball around the CUEP. This is accomplished by now treating τ as a variable in (7), giving zcont = [x1 y1 t1 x2 y2 t2 p τ ]| . Then zcont, together with the constraints F(zcont) = 0, defines a curve since there is one more variable in zcont than the number of equations in F(zcont) = 0. A continuation method, similar in concept to those in [10], can be employed to track the solution as τ is increased
这种延拓方法的关键思想是沿着由一组代数方程定义的曲线迭代地跟踪一组点序列，如图4所示。在每一步中，预测都是通过沿着与曲线的切线移动来进行预测的。然后通过投影回曲线上来修正预测的点。这个过程会重复进行，直到曲线完成。对曲线的单位切向量η可以通过让来找到
The key idea of this continuation method is to iteratively
trace a sequence of points along a curve defined by a set of
algebraic equations, as illustrated in Figure 4. At each step a
prediction is made by moving along the tangent to the curve.
Then the predicted point is corrected by projecting back onto
the curve. This process repeats until the curve is complete.
The unit tangent vector η to the curve can be found by letting
然后η满足Jcont（zcont）η = 0（12）||η||22 = 1。（13）对Jcont（zcont）|进行QR分解，并设η为，中的最后一列
Then η satisfies Jcont(zcont)η = 0 (12) ||η||22 = 1. (13)The solution is given by taking a QR decomposition of Jcont(zcont)| , and setting η equal to the final column of the
图4.一种沿着由一组代数方程式定义的曲线迭代跟踪点序列的延拓方法。在每一步中，预测都是通过沿着与曲线的切线移动来进行预测的。然后将预测的点投影回曲线上。这个过程重复，直到跟踪曲线。正交矩阵q。从曲线上的第j个点开始，下一个预测点是，zp = zj cont + κη（14），其中κ为步长。对于校正步骤，这涉及到投影回曲线，我们要求zj+1丆p与η正交，所以η|（zj+1丆丆p）=0。（15）从（14）开始，这意味着
这个校正步骤是通过设置，Fproj（zj+1）=F（zj+1）η|（zj+1−−）κ（17）和相应的雅可比矩阵，−（zj+1）=−（zj+1）η|来完成的。（18）然后，我们可以利用标准的牛顿-拉夫森更新得到Fproj（zj+1 cont）= 0的解。该方法改变参数值p，以便向τ最大化的方向移动。因此，p被驱动到其临界值。一旦找到一个临界参数值，就可以将一对参数作为自由变量处理，并计算其参数边界。将τ从上述延续过程结束时固定到其值。让z绑定= [z| p1 p2]|作为一对参数p1，p2。然后F（zbound）= 0描述了参数空间中任意靠近期望参数边界的曲线。使用如上一步所述的延拓方法，可以逐步追踪这条曲线，以得到所需的参数边界。v.模型
第四节中提出的算法应用于一个循环排列的三个同步机器的测试用例。该机器由双轴四阶同步机器模型描述，完整的细节由
The algorithms presented in Section IV were applied to a test case of three synchronous machines arranged in a loop. The machines are described by two-axis fourth-order synchronous machine models, with complete details given by
在[1]中表示的方程式（6.110）-（6.113）。这三台机器都有恒定的场电压、恒定的机械扭矩和机械阻尼Dω，其中D是一个参数，隐含地表示由调速器引起的阻尼。发电机的参数见表一，三台发电机的惯性矩分别为0.077 pu、0.105 pu和0.084 pu。选择恒定场电压，确保终端母线总线电压均为1.05 pu。发电机有功功率分别用P1 = 0.4 pu、P2 = 0.5 pu、P3 = 0.9 pu进行初始化。假设线路电阻为零，电抗分别为X13 = 1.0000 pu、X12 = 1.2500 pu、X23 = 0.8333 pu
equations (6.110)-(6.113) in [1]. All three machines have constant field voltage, constant mechanical torque, and mechanical damping Dω where D is a parameter implicitly representing the damping due to the governor. The parameters of the generators are provided in Table I. The moments of inertia for the three generators are 0.077 pu, 0.105 pu, and 0.084 pu, respectively. The constant field voltages were chosen to ensure that the terminal bus voltages are all at 1.05 pu in steady state. The generator active powers were initialized with P1 = 0.4 pu, P2 = 0.5 pu, and P3 = 0.9 pu. The line resistances are assumed to be zero, and the reactances are X13 = 1.0000 pu, X12 = 1.2500 pu and X23 = 0.8333 pu
所考虑的干扰是发电机一号终端发生的故障。故障建模为开关，恒定分流电抗，Xfault = 0.001 pu。在这个例子中，三个发电机之间的主要区别是它们的惯性矩。因此，故障位于发电机一号，因为它的惯性最低，因此最容易发生。在整个示例中都使用了1.0秒的故障持续时间。虽然这是一个不切实际的长故障清除时间，但它是说明算法特性的一个有用的选择。
The disturbance considered is a fault applied at the terminals
of generator one. The fault is modelled as a switched, constant
shunt reactance with Xfault = 0.001 pu. The main difference
between the three generators in this example is their moments
of inertia. Accordingly, the fault was located at generator one
because it has the lowest inertia, and hence is most vulnerable
to the fault. A fault duration time of 1.0 sec has been
used throughout the example. Whilst this is an unrealistically
long fault clearing time, it is a useful choice for illustrating
algorithm characteristics.
发电机惯性矩是该系统的重要参数，因为它们在故障恢复中起着重要的作用。我们引入了一个比例因子，它同时乘以所有三个同步机器的惯性矩。较低的比例因子值导致系统的惯性减少，从而降低了系统从故障中恢复的能力。这个测试用例的CUEP是通过两步过程确定的。首先，采用延拓法确定了四种潮流解。然后，对于每一个这些解决方案，都确定了所有可能的机器状态的选择。一旦知道了这些平衡，就可以直接通过观察系统进入不稳定状态时的轨迹来确定CUEP
The generator moments of inertia are parameters of interest
in this system as they play an important role in recovery
from the fault. We introduce a scaling factor which multiplies
the moments of inertia of all three synchronous machines
simultaneously. Lower values of the scaling factor lead to less inertia in the system, which reduces its ability to recover from faults. The CUEP for this test case was identified through a two step process. First, a continuation method was used to identify four power flow solutions. Then, for each of those solutions, all possible choices of machine states were determined. Once those equilibria were known, it was straightforward to determine the CUEP by observing the trajectory as the system was driven to instability
vi。结果第一步是驱动轨迹在球内绕CUEP停留固定长度的时间τ。如图5所示，它描述了从轨迹上的一个点到CUEP的距离作为时间的函数。给定一个半径为r=为10.5的球，使用(8)确定惯性矩比例因子，使轨迹在球内停留的目标时间τ=为6.5秒。接下来，增加球中的时间τ，以驱动惯性矩缩放因子到其临界值。这有效地将故障后的初始条件转移到故障后条件上
VI. RESULTS
The first step is to drive the trajectory to spend a fixed length of time τ inside the ball around the CUEP. This is illustrated in Figure 5, which depicts the distance from a point on the trajectory to the CUEP as a function of time. Given a ball of radius r = 10.5, the moment of inertia scaling factor was determined using (8) such that the trajectory spent the target time τ = 6.5 sec inside the ball. Next, the time τ in the ball was increased to drive the moment of inertia scaling factor to its critical value. This effectively moved the post-fault initial conditions onto the post-faul
图5.轨迹与控制不稳定平衡点（CUEP）的距离随时间的变化。CUEP的半径为r = 10.5。改变惯性矩比例因子，驱动轨迹在球内停留固定时间τ = 6.5秒。稳定性范围这是通过第四节中提出的预测器校正器延续过程来完成的。图6显示了随着球内时间的增加，惯性矩比例因子。请注意，正如该理论所建议的那样，随着球内时间的增加，比例因子会迅速收敛到其临界值。该过程产生一个临界惯性矩比例因子，其相关的故障后初始条件位于稳定边界上。
Figure 5. The distance of the trajectory from the controlling unstable equilibrium point (CUEP) as a function of time. The radius of the ball around the CUEP is r = 10.5. The moment of inertia scaling factor was varied to drive the trajectory to spend a fixed time τ = 6.5 sec inside the ball. stability boundary. This was accomplished via the predictorcorrector continuation process presented in Section IV. Figure 6 shows the moment of inertia scaling factor as the time
in the ball is increased. Note that the scaling factor converges to its critical value quite quickly as the time inside the ball increases, as suggested by the theory. This process yields a critical moment of inertia scaling factor whose associated postfault initial conditions lie on the stability boundary
既然找到了参数的临界值，就可以将一对参数作为自由变量，并计算它们的参数边界。对于这个测试用例，选择发电机1和发电机2的惯性矩作为两个自由参数。如前所述，采用类似延拓的方法，对参数边界进行了数值计算。图7描述了这个边界。对于边界以上的任何参数值，系统将从故障中恢复，并返回到原来的SEP。相反，对于边界以下的任何参数值，系统将不会从指定的扰动中恢复。虽然这个例子的参数边界几乎是线性的，但延拓方法能够追踪具有高曲率[2]的曲线。需要进一步的研究来确定在这个例子中的表观线性的原因
Now that a critical value of the parameter has been found, a pair of parameters can be treated as free variables, and their parameter boundary computed. For this test case, the moments of inertia of generators one and two were selected as the two free parameters. Using an analogous continuation method, as described earlier, the parameter boundary is numerically computed. Figure 7 depicts this boundary. For any parameter values above the boundary, the system will recover from the fault, returning to the original SEP. Conversely, for any parameter values below the boundary, the system will not recover from the specified disturbance. Although the parameter boundary turns out to be almost linear for this example, the continuation method is capable of tracing curves that have high curvature [2]. Further investigations are required to determine the reason for the apparent linearity in this example
罗马数字 7结论本文考虑了一个特定扰动驱动系统进入稳定平衡点吸引区域边界的临界参数值的计算。这些临界参数值构成了系统将从干扰中恢复的值和它将不会恢复的数值之间的参空间的边界。提出了一种利用特殊控制不稳定平衡点（CUEP）的存在来数值计算这些临界参数值的算法。CUEP的特性使得通过一个优化问题来定位临界参数值成为可能。先验
VII. CONCLUSION
This work considered the computation of critical parameter
values for which a specified disturbance drives the system onto
the boundary of the region of attraction of the stable equilibrium point. These critical parameter values form the boundary
in parameter space between values for which the system will
recover from a disturbance and values for which it will not.
An algorithm was developed with the goal of numerically
computing these critical parameter values by exploiting the
presence of a special controlling unstable equilibrium point
(CUEP). The properties of the CUEP make it possible to locate
critical parameter values via an optimization problem. Prior
图7.发电机1和发电机2惯性矩的参数边界。边界以上的所有参数值都会导致故障恢复，而边界以下的所有参数值都会导致不稳定。工作[3]为CUEP的存在性和所期望的性质提供了理论证明。该算法已成功地应用于一个由三台同步机器组成的测试用例中。它能够找到一个惯性矩比例因子的临界值。采用延拓过程跟踪两个发电机惯性矩的临界参数边界。由于同步机器通常是弱阻尼和高振荡的，这个例子证明了算法的可靠性。未来的工作将考虑混合行为的影响，包括开关和限制，对临界参数值的计算。
Figure 7. The parameter boundary for the moments of inertia of generators one and two. All parameter values above the boundary lead to fault recovery, whereas all parameter values below the boundary lead to instability. work [3] provides theoretical justification for the existence and desired properties of the CUEP. The algorithm was successfully applied to a test case consisting of three synchronous machines. It was able to find the critical value for a moment of inertia scaling factor. A continuation process was used to trace a critical parameter boundary for the moments of inertia of two of the generators. Since synchronous machines are typically weakly damped and highly oscillatory, this example demonstrated the reliability of the algorithms. Future work will consider the impact of hybrid behaviour, including switching and limits, on the computation
of critical parameter values.
