Interarea Mode Analysis for Large Power Systems Using Synchrophasor Data

Abstract

Interarea oscillations are predominantly governed by the slower electromechanical modes which, in turn, are determined by the coherent machine rotor angles and speeds. The issue is that, although these rotor angles and speeds provide the best visibility of such modes, currently they are not available from phasor measurement units (PMU). As such, the aim of this chapter is to demonstrate that interarea oscillations are observable in the network variables, such as voltages and line currents, which are measured by PMU. By analyzing the electromechanical modes in the network variables, we can trace how electromechanical oscillations are spread through the power network following a disturbance. Applying eigenvalue and sensitivity analysis, we provide an analytical framework to understand the nature of these network oscillations through a relationship termed network modeshapes. Using this relationship, a novel concept, “dominant interarea oscillation paths,” is developed to identify the passageways where the interarea modes of concern travel the most. We demonstrate the concept of the dominant path with an equivalent two-area system. We propose an algorithm for identification of the dominant paths and illustrate with a reduced model of a large-scale network. Finally, we end this chapter with an important application of the concept: feedback input signal selection for damping controller design.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

10.1 Introduction

Electromechanical modes are predominantly determined by the machine rotor angles and speeds [1–5], and as a result, they provide the best visibility of such modes. Electromechanical modes are also observable in the network variables, such as voltages and line current phasors, which are measured by Phasor Measurement Units (PMUs) [6] and observed through simulations [7]. In this study, by computing the electromechanical modes in the network variables, we can trace how electromechanical oscillations are spread through the power network following a disturbance. In particular, interarea modes can travel throughout a large network. These network variables have received less attention because they are algebraic variables which can change abruptly. Recently, an analytical framework to study interarea oscillations present in the bus voltage and frequency variables has been proposed [8]. This technique uses the linearized model of a power system with bus voltages and frequencies as output variables.

In the first part of this chapter, we aim to generalize the results in [8] to analyze power system oscillations which may be present in any of the network variables. To motivate, we first analyze the oscillations observed during a major system disturbance that took place on 2/26/2008 in the Florida Reliability Coordination Council (FRCC) service area [9, 10]. The modal components from distributed PMU measurements have a common feature: they do not peak at the same time instants. Therefore, there exists a time difference (or delay) between the oscillations at different points of the network. This time difference is effectively a phase shift between the network oscillations.

With the goal of explaining the origin of the phase shifts in the network variables, we use the multi-machine linearized power system electromechanical model [11] and extend the mode shapes to the network variables. We show that with no damping and constant impedance loads, all electromechanical oscillations are in phase. However, when damping or control equipment such as voltage regulator models are included, the eigenvector matrix will indicate phase shifts. The time delays related to these phase shifts show a strong resemblance to those observed in PMU data.

To obtain the mode shapes, we perform a detailed sensitivity analysis of the network variables and provide analytical expressions [12]. This analysis provides a theoretical understanding of oscillations as directly measured by PMUs on high-voltage transmission systems. Finally, network sensitivities and eigenvectors are used to obtain the modal components in the network variables, thus providing a rationale explaining the phase shifts observed in the modal components of the synchrophasor data.

In the second part of this chapter, we aim to present two different applications of the concept of dominant interarea oscillation paths: an algorithm for tracing the spread of interarea modes (i.e., identification of dominant paths) and an application for feedback control input signals selection. We demonstrate the algorithm with a reduced Nordic grid model while implementing a fundamental study on damping control design using different input signals, PMU’s and non-PMU’s, with a conceptualized two-area system. The results are useful for determining not only proper feedback input signals but also PMU siting.

The remainder of this chapter is organized as follows: In Sect. 10.2, we analyze the network oscillations originating from the 2/26/2008 FRCC disturbance. In Sect. 10.3, we determine the origin of the phase shifts in the electromechanical mode shape. In Sect. 10.4, we perform sensitivity analysis of the network variables and provide closed-form expressions of the network sensitivities. Afterward, in Sect. 10.5 we map the network sensitivities onto the electromechanical mode shapes and discuss the nature of the modal components in the network variables, illustrating this with a four-generator, two-area system [12, 13]. Applications for interarea mode tracing and feedback control input signal selection are demonstrated in Sect. 10.6. Conclusions are given in Sect. 10.7.

10.2 Nature of Network Oscillations Observed from Phasor Measurement Data

We study the power system interarea oscillations excited by the FRCC system disturbance by analyzing archived phasor measurement data. The PMUs considered in this analysis are: Manitoba, near the city of Winnipeg, Canada; Maine near Bangor; Florida near Jacksonville; West Tennessee (W. Tenn.) near Memphis; and East Tennessee (E. Tenn.) near Knoxville, as shown in Fig. 10.1a. In Fig. 10.1b we show the bus frequency measured during the disturbance by the PMUs, showing the wide-area impact of the disturbance, the electromechanical swing that propagated from Florida, to E. Tenn. and W. Tenn., and subsequently to Manitoba, and finally to Maine.

Fig. 10.1

PMU locations and bus frequency traces obtained during the 2008 Florida disturbance. a PMU locations. b Bus frequency traces

We aim to analyze the oscillatory components and characteristics contained in these measurements. We use the Eigensystem Realization Algorithm (ERA) [14] to identify the individual modal components in the voltage magnitude, voltage angle, and active power flow measurements available from each PMU [9]. The identified low-frequency interarea modes include frequencies of 0.22 and 0.49 Hz. In Fig. 10.2 the voltage phasor measurements of the PMU at W. Tenn. are shown along with their ERA approximation for the 0.22 and 0.49 Hz components of the signals.

Fig. 10.2

Identified oscillatory components in the voltage phasor at W. Tenn. a Voltage magnitude. b Voltage angle

All identified components for the 0.22 Hz mode in the voltage angle are shown in Fig. 10.3a, and those for the 0.49 Hz mode are shown in Fig. 10.3b. The starting time $t=0$ s corresponds to 18:10:04.333 h. From the voltage angles of the 0.22 Hz mode, we note that Florida oscillates against Maine and Manitoba: that is, it is a North versus South mode.

Fig. 10.3

Voltage angle interarea oscillations and projections. a $\varDelta \theta _i$ 0.22 Hz components. b $\varDelta \theta _i$ 0.49 Hz components

The 0.49 Hz mode is more difficult to analyze from this limited data set. However, it can be noted that the voltage angle at W. Tenn., E. Tenn., and Florida have the largest oscillations while Manitoba and Maine have a less significant contribution. More important, E. Tenn. and W. Tenn. are in anti-phase suggesting that the pivot node of the oscillation is located somewhere between them.

The most important observation that could be made about the oscillations discussed above is the following: for all of the network variables, the individual modal components do not peak at the same instants. There is in fact a time shift (or delay) between the modal components. This time delay can be viewed as a phase shift between the modal components in the frequency domain. In the next section, we will investigate this phase shift by analyzing the mode shapes of a test network.

10.3 The Electromechanical Mode Shape

10.3.1 Multi-Machine Systems Electromechanical Model

In this section, we investigate the origin of electromechanical mode phase shifts by performing eigenanalysis on different linearized models of multi-machine power systems. We start with the electromechanical model [1]. For an $N$-machine power system, the linearized electromechanical model in state-space form is given by

$$\begin{aligned} \underbrace{\left[ \begin{array}{ll}{\varDelta \dot{\delta }}\\ {\varDelta \dot{\omega }} \end{array} \right] }_{\dot{x}} = \underbrace{ \left[ \begin{array}{ll} {0} &{} \varOmega {I}_{(N\times N)} \\ {M}^{-1} {K} &{} {M}^{-1} {D} \end{array} \right] }_{\bar{A}} \underbrace{\left[ \begin{array}{ll}{\varDelta \delta }\\ {\varDelta \omega } \end{array} \right] }_{x} \end{aligned}$$

(10.1)

where

$$\begin{aligned} \begin{array}{rll} {\varDelta \delta } &{}=&{} \left[ \varDelta \delta _1 \; \cdots \; \varDelta \delta _N \right] ^{T}\!, \; (N\times 1)\\ [1.5mm] {\varDelta \omega }&{} =&{} \left[ \varDelta \omega _1 \; \cdots \; \varDelta \omega _N \right] ^{T}\!, (N\times 1)\\ [1.5mm] {M}^{-1}&{} =&{} \text{ diag }\left( \frac{1}{2H_i} \right) , \, (N \times N) \\ [1.5mm] {D} &{}=&{} \text{ diag }\left( D_i \right) , \, (N\times N); \quad {K} = \left[ K_{ij} \right] , (N \times N) \end{array} \end{aligned}$$

(10.2)

where $i$, $j=1,\ldots ,N$. We will refer to this model (10.1) as the electromechanical model with damping, and to matrix ${\bar{A}}$ as the state matrix with damping. If the damping terms are neglected ($D_i \approx 0$), the model becomes

$$\begin{aligned} \begin{array}{c} \underbrace{\left[ \begin{array}{c} {\varDelta \dot{\delta }}\\ {\varDelta \dot{\omega }} \end{array} \right] }_{{\dot{x}}} = \underbrace{\left[ \begin{array}{cc} {0} &{} \varOmega {I}_{(N\times N)} \\ {A}_\omega &{} {0} \end{array}\right] }_{{A}} \underbrace{\left[ \begin{array}{c} {\varDelta \delta }\\ {\varDelta \omega } \end{array}\right] }_{{x}} \\ {A}_\omega = {M}^{-1} {K} \end{array} \end{aligned}$$

(10.3)

We will refer to ${A}$ as the state matrix without damping. Many properties of ${\bar{A}}$ and ${A}$ have been studied [11, 15]. In this investigation, some characteristics of the eigenvectors of ${\bar{A}}$ and ${A}$ are analyzed. These characteristics have strong effects on the phase shift of network variables as discussed later.

10.3.2 Eigenvectors of the Electromechanical Models

Consider the two-area, four-machine power system in Fig. 10.4 [12, 13] modeled using (10.3). In the resulting state matrix ${A}$, all elements corresponding to the machine damping ${M}^{-1} {D}$ are zero. Eigenanalysis is performed on ${A}$ resulting in the eigenvector matrix

$$\begin{aligned} {W}({A})=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0.5&{}-0.5930&{}0.7871&{}-0.0547\\ 0.5&{}-0.5997&{}-0.6167&{}-0.0512\\ 0.5&{}-0.4539&{}0.0155&{}0.9972\\ 0.5&{}0.2876&{}-0.0015&{}-0.0041 \end{array} \right] \end{aligned}$$

(10.4)

Fig. 10.4

Two-area four-machine power system and voltage magnitude interarea mode shape (no damping)

Note that only the components corresponding to the machine angles are shown. All the components of ${W}({A})$ are real. Column 2, the interarea mode mode shape, shows that $G_1$, $G_2$, and $G_3$ are oscillating against $G_4$: that is, Area 1 oscillates against Area 2.

Column 3, the mode shape for Local Mode 1, indicates that $G_1$ and $G_2$ are mostly oscillating against each other. Finally, Local Mode 2 (Column 4) is mostly confined within Area 1 with $G_3$ oscillating against $G_1$ and $G_2$. The most important characteristic to note is that the oscillations are either completely in-phase or anti-phase.

Using (10.1), we analyze the effect that machine damping has on the eigenvectors. The elements corresponding to the machine damping ${M}^{-1} {D}$ in ${\bar{A}}$ are given by

$$\begin{aligned} {M}^{-1} {D} = \text{ diag }\left( \left[ \begin{array}{cccc} -0.001&-0.3&-0.015&-0.2\end{array}\right] \right) \end{aligned}$$

(10.5)

and all other elements remain unchanged with respect to ${A}$.

Eigenanalysis is performed on ${\bar{A}}$, yielding the eigenvector matrix

$$\begin{aligned} {W}({\bar{A}})=\left[ \begin{array}{cccc} 0.5\angle 0^{\circ }&{}0.5882 \angle 172.6196^{\circ } &{} 0.7878 \angle 0^{\circ } &{}0.0548 \angle -178.5539^{\circ }\\ 0.5\angle 0^{\circ }&{}0.6066 \angle -180^{\circ } &{} 0.6084\angle 165.8682^{\circ } &{}0.0507 \angle 172.6111^{\circ }\\ 0.5\angle 0^{\circ }&{}0.4548 \angle 176.1869^{\circ } &{} 0.0951\angle 68.6262^{\circ } &{}0.9972 \angle 0^{\circ }\\ 0.5\angle 0^{\circ }&{}0.2814 \angle -6.3759^{\circ } &{} 0.0072\angle -116.0791^{\circ } &{}0.0041 \angle 176.6467^{\circ }\\ \end{array} \right] \end{aligned}$$

(10.6)

Note that the eigenvector matrix is now complex. For convenience, it is shown in polar form. The main oscillatory characteristics discussed for the mode shapes from ${A}$ are maintained for the mode shapes of ${\bar{A}}$. However, the components of each mode shape now present a phase shift due to the inclusion of machine damping. This phase shift is readily observed in the phasor diagrams in Fig. 10.5a, b. A more important observation is that in the time domain these phase shifts translate to time delays, making the oscillations in each mode peak at different instants as shown in Fig. 10.5c, d. It is interesting to note that the time-shifts shown in Fig. 10.5c, d strongly resembles those of the measurement data presented in the previous section.

Fig. 10.5

Machine angle mode shapes with damping—phasor diagram (a, b), and time response (c, d). a Interarea mode. b Local mode 1. c Interarea mode. d Local mode 1

Why do the network measurements exhibit phase shifts similar to those in the machine angle mode shapes of ${\bar{A}}$? In the following section, we answer this question by building upon the understanding gained from the mode shapes from ${\bar{A}}$, and generalize the results obtained in [8].

10.4 Sensitivity Analysis of Network Variables

Network sensitivities are used to provide linear relationships that predict changes in the network variables given a small perturbation in the power system. Here, we provide closed-form expressions for the network sensitivities [12]. This analysis gives a rationale about the nature of the network oscillations directly measured by PMU data.

10.4.1 Network Sensitivities

By computing the sensitivities of the entire network, it is possible to predict the incremental behavior of the network variables when a small perturbation occurs in the power system. This is of particular interest because PMUs are capable of measuring synchronized network variable changes across wide-areas of the power system that emerge from small perturbations. Thus, an understanding of network sensitivities can provide insight for PMU data analysis of small signal oscillations occurring in large-scale power networks.

To illustrate, consider the $n$-bus, $N$-machine power system shown in Fig. 10.6. Buses $i$ to $(i+4)$ are transmission and load buses remotely connected to generators $j=1,\ldots ,N$. Regardless of their distance from the generators, these buses will be affected by changes in the internal machine angle of any generator. For example, a change in the internal angle of Machine $j$, $\varDelta \delta _j$, will be reflected in the voltage magnitude at Bus $i$, $V_i$, in a proportion dictated by the network sensitivity $\left. (\partial V_i / \partial \delta _j ) \right| _{(0)}$.¹ The total change in the voltage magnitude at Bus $i$, $\varDelta V_i$, will be the sum of all the changes in the machine internal angles scaled by their corresponding network sensitivity.

Similarly, if the current flow among Buses $i$ to $(i+4)$ is as indicated in Fig. 10.6, the change in the current flow will also be affected by the change in machine internal angles. The complex line current flow changes will be proportionally distributed satisfying Kirchhoff’s current law, i.e., $\varDelta {\tilde{I}}_{i(i+1)} = \varDelta {\tilde{I}}_{(i+1)(i+2)}+\varDelta {\tilde{I}}_{(i+1)(i+3)}+\varDelta {\tilde{I}}_{(i+1)(i+4)}$. Thus, it becomes possible to trace how the current oscillations are being divided among multiple lines/paths and propagated across the entire network.

Fig. 10.6

Changes in the network variables with respect to changes in $\delta _j$’s

To obtain the change of the voltage magnitude ($\varDelta V_i$) and angle ($\varDelta \theta _i$) at Bus $i$ of the network with respect to the change of machine internal angles, we obtain the Taylor series expansion of $V_i$ and $\theta _i$ about an equilibrium. Ignoring the higher order terms, the change of voltage magnitude at Bus $i$ due to the change in the machine internal angles is given by

$$\begin{aligned} \varDelta V_i = \left. \left( \frac{\partial V_i}{\partial \delta _1} \right) \right| _{0} \varDelta \delta _1 + \left. \left( \frac{\partial V_i}{\partial \delta _2} \right) \right| _{0} \varDelta \delta _2 + \cdots + \left. \left( \frac{\partial V_i}{\partial \delta _N} \right) \right| _{0} \varDelta \delta _N \end{aligned}$$

(10.7)

where $\left. \left( \partial V_i / \partial \delta _j \right) \right| _{0}\,$ is the sensitivity of the $i$-th bus voltage magnitude to the $j$-th machine angle at the equilibrium point. In matrix form, the voltage magnitude changes at Buses $i=1,\ldots ,n$ due to the change in the machine internal angles are given by

$$\begin{aligned} \begin{array}{ccc} \underbrace{\left[ \begin{array}{c} \varDelta V_1 \\ \vdots \\ \varDelta V_n \end{array} \right] }_{{\varDelta V}} = &{} \underbrace{ \left[ \begin{array}{ccc} \left. \left( \frac{\partial V_1}{\partial \delta _1} \right) \right| _{(0)} &{} \ldots &{} \left. \left( \frac{\partial V_1}{\partial \delta _N} \right) \right| _{(0)} \\ \vdots &{} \ddots &{} \vdots \\ \left. \left( \frac{\partial V_n}{\partial \delta _1} \right) \right| _{(0)} &{} \ldots &{} \left. \left( \frac{\partial V_n}{\partial \delta _N} \right) \right| _{(0)} \end{array} \right] }_{{C}_{V\delta }} &{} \underbrace{ \left[ \begin{array}{c} \varDelta \delta _1 \\ \vdots \\ \varDelta \delta _N \end{array} \right] }_{{\varDelta \delta }} \end{array} \end{aligned}$$

(10.8)

where $\small {\varDelta V}$ is the vector of bus voltage magnitude changes of size $n\times 1$, and ${C}_{V\delta }$ is the bus voltage magnitude sensitivity matrix of size $n\times N$.

Similarly, sensitivities may also be obtained for any other network variable. For the bus voltage angle changes, we have

$$\begin{aligned} {\varDelta \theta } ={C}_\mathrm{{\theta \delta }} {\varDelta \delta } \end{aligned}$$

(10.9)

where ${C}_\mathrm{{\theta \delta }}$ is the bus voltage angle sensitivity matrix of size $n\times N.$

In a power system with a total number of $\ell$ transmission lines connecting the sending end buses, $f$, to the receiving end buses, $t$, the current magnitude and angle changes with respect to the machine angle changes are given by

$$\begin{aligned} {\varDelta I}_{ft}={C}_{I_{ft} \delta } {\varDelta \delta } \end{aligned}$$

(10.10)

$$\begin{aligned} {\varDelta \phi }_{ft} = {C}_{\phi _{ft} \delta } {\varDelta \delta } \end{aligned}$$

(10.11)

where ${\varDelta I}_{ft}$ and ${\varDelta \phi }_{ft}$ are the current magnitude and angle changes, respectively. The matrix ${C}_{I_{ft} \delta }$ is the current magnitude sensitivity matrix while ${C}_{\phi _{ft} \delta }$ is the current angle sensitivity matrix, both of size $\ell \times N$.

In general, the sensitivity matrices ${C}_{V\delta }$ and ${C}_{\theta \delta }$ are obtained by numerical perturbation using simulation software such as PST [16]. Analytical expressions may provide further insight about their properties and the parameters affecting them. In the next section, we provide closed-form expressions of different network sensitivities.

10.4.2 Analytical Derivation of the Network Sensitivities

The first step to develop closed-form expressions of the different sensitivities is to obtain a general expression of the network variables as a function of the machine internal nodes. To this aim, it is possible to write nodal voltage equations that extend to the machine internal nodes. To include $E_{j}^{\prime }$, the voltage behind transient reactances $x_{dj}^\prime$, we add $N$ buses to the $n$-bus power system network, thus extending the admittance matrix to the machine internal nodes.

The internal machine buses are denoted by $n+1, \ldots , n+j, \ldots , n+N$, which are the buses behind the transient reactances, $x_{dj}^\prime$. The resulting admittance matrix differs from the admittance matrix used in load flow analysis in that the additional internal machine buses are included to account for the machine internal voltages, ${\tilde{E}}_{j}^\prime$. In addition, loads are modeled as constant admittances and included in the diagonal elements of the admittances of the extended ${\tilde{Y}}$ matrix. As a result, injected currents in all nodes other than the generator internal nodes are zero, i.e., ${\tilde{I}}_{i} = 0$,   $i=1,\ldots ,n$.

Denoting all the generator current injections as ${\tilde{I}}_{N}$, generator voltages as ${\tilde{E}}_{N}^\prime$, and bus voltages as ${\tilde{V}}_{n}$, the node voltage equations are

$$\begin{aligned} \left[ \begin{array}{c} {0}\\ {\tilde{I}}_{N} \end{array} \right] = \left[ \begin{array}{cc} {\tilde{Y}}_{nn} &{} {\tilde{Y}}_{nN} \\ {\tilde{Y}}_{nN}^T &{} {\tilde{Y}}_{NN} \end{array} \right] \left[ \begin{array}{c} {\tilde{V}}_n\\ {\tilde{E}}_{N}^\prime \end{array} \right] \end{aligned}$$

(10.12)

From (10.12) an expression for the bus voltage phasors as a function of the machine internal voltages is obtained as

$$\begin{aligned} {\tilde{V}}_n = -{\tilde{Y}}_{nn}^{-1}{\tilde{Y}}_{nN}{\tilde{E}}_{N}^\prime \end{aligned}$$

(10.13)

where ${\tilde{Y}}_{nn}^{-1}{\tilde{Y}}_{nN}$ is referred to as the bus voltage reconstruction matrix and has a size of $(n\times N)$. The bus voltage reconstruction coefficient matrix is given by

$$\begin{aligned} {\tilde{\kappa }}=-{\tilde{Y}}_{nn}^{-1}{\tilde{Y}}_{nN} = {\kappa }\angle {\gamma } \end{aligned}$$

(10.14)

where ${\kappa }$ and ${\gamma }$ are the magnitude and angle of ${\tilde{\kappa }}$. We can now obtain a generalized expression relating the voltage phasor at Bus $i$ with the machine internal voltages using (10.14)

$$\begin{aligned} \tilde{V}_{i} = {\tilde{\kappa }}_{i1}{\tilde{E}}_{1}^{\prime } + {\tilde{\kappa }}_{i2}{\tilde{E}}_{2}^{\prime } + \cdots + {\tilde{\kappa }}_{iN}{\tilde{E}}_{N}^{\prime } = \sum _{j=1}^N {\tilde{\kappa }}_{ij}{\tilde{E}}_{j}^{\prime } \end{aligned}$$

(10.15)

Hence, the voltage at the $i$-th bus is a function of the machine internal angles $\delta _j$. In this expression, $\tilde{V}_i$ depends on the value of the machine internal voltage magnitudes at the equilibrium, ${E}_j^\prime$ , and the admittances in the voltage reconstruction coefficient matrix ${\tilde{\kappa }_{ij}}$, with $j=1,\ldots ,N$.

It is also possible to develop a generalized expression for the complex line current flow through any line of the power network. Consider the $\pi$-equivalent of a transmission line: the current from Bus $f$ to Bus $t$ is given by

$$\begin{aligned} {\tilde{I}}_{ft} = \left( {\tilde{y}}_{ft} + {\tilde{y}}_{f0} \right) {\tilde{V}}_f - {\tilde{y}}_{ft}\tilde{V}_t \end{aligned}$$

(10.16)

Letting ${\tilde{Y}}_{ft0}= {\tilde{y}}_{ft} + {\tilde{y}}_{f0}$ and ${\tilde{Y}}_{ft} = {\tilde{y}}_{ft}$, and writing the voltage for Buses $f$ and $j$ in terms of the internal machine voltages using (10.15), we obtain

$$\begin{aligned} {\tilde{I}}_{ft} = \sum _{j=1}^{N} \left( {\tilde{Y}}_{ft0} {\tilde{\kappa }}_{fj} - {\tilde{Y}}_{ft} {\tilde{\kappa }}_{tj}\right) {\tilde{E}}_j^\prime = \sum _{j=1}^{N} {\tilde{\varPsi }}_{FTj}{\tilde{E}}_j^\prime \end{aligned}$$

(10.17)

where ${\tilde{\varPsi }}_{FTj} = \varPsi _{FTj}\angle \psi _{FTj}$, and

$$\begin{aligned} \varPsi _{FTj} = \left| {\tilde{Y}}_{ft0} {\tilde{\kappa }}_{fj} - {\tilde{Y}}_{ft} {\tilde{\kappa }}_{tj} \right| , \; \psi _{FTj} = \angle \left( {\tilde{Y}}_{ft0} {\tilde{\kappa }}_{fj} - {\tilde{Y}}_{ft} {\tilde{\kappa }}_{tj}\right) \end{aligned}$$

(10.18)

The complex line current flow in Line $f$-$t$ is a function of the machine angles $\delta _j$. The phasor ${\tilde{I}}_{ft}$ depends on the value of the machine internal voltage magnitudes at the equilibrium, ${E}_j^\prime$, the admittances in the voltage reconstruction coefficient matrix ${\tilde{\kappa }}_{ij}$, and the admittances and shunts between Buses $f$ and $t$. Expressions (10.15) and (10.17) are used to derive closed-form expressions of the network sensitivities. A complete derivation can be found in [12].

10.4.2.1 Voltage Sensitivities

Given an $N$-machine power system, the bus voltage magnitude sensitivities for Bus $i$ with respect to the $j$-th machine angle are given by

$$\begin{aligned} \frac{\partial V_i}{\partial \delta _j} = \frac{1}{| \tilde{V}_i |} \left( \alpha \right) , \, j=p ; \quad \frac{\partial V_i}{\partial \delta _j} = \frac{1}{| \tilde{V}_i |} \left( -\alpha \right) , \, j\ne p \end{aligned}$$

(10.19)

where

$$\begin{aligned} | \tilde{V}_i | = \sqrt{\left( \sum \nolimits _{j=1}^N {\tilde{\kappa }}_{ij}{\tilde{E}}_{j}^{\prime } \right) ^2} \end{aligned}$$

(10.20)

$$\begin{aligned} \alpha =&-\sum \nolimits _{p=1}^{N-1} \sum \nolimits _{q=p+1}^{N} \kappa _{ip}\kappa _{iq}E_{p}^{\prime }E_{q}^{\prime }\sin \left( \delta _p + \gamma _{ip} - \delta _q - \gamma _{iq}\right) \nonumber \\&+\sum \nolimits _{\begin{array}{c} p=1\\ p\ne j \end{array}}^{N-1} \sum \nolimits _{\begin{array}{c} q=p+1\\ q\ne j \end{array}}^{N} \kappa _{ip}\kappa _{iq}E_{p}^{\prime }E_{q}^{\prime }\sin \left( \delta _p + \gamma _{ip} - \delta _q - \gamma _{iq}\right) \end{aligned}$$

(10.21)

For the bus voltage angle

$$\begin{aligned} \frac{\partial \theta _i}{\partial \delta _j} = \frac{1}{| \tilde{V}_i |^2} \left[ \kappa _{ij}^2E_{j}^{\prime 2} + \sum _{\begin{array}{c} q=1\\ q\ne j \end{array}}^{N} \kappa _{ij}\kappa _{iq}E_{j}^{\prime }E_{q}^{\prime }\cos \left( \delta _j + \gamma _{ij} - \delta _q - \gamma _{iq}\right) \right] \end{aligned}$$

(10.22)

is the closed-form formula to obtain the sensitivity of the $i$-th bus voltage angle to the $j$-th machine angle.

10.4.2.2 Current Sensitivities

The line current magnitude sensitivities in any line from Bus $f$ to Bus $t$ with respect to the $j$-th machine angle are given by

$$\begin{aligned} \frac{\partial I_{ft}}{\partial \delta _j} = \frac{1}{| {\tilde{I}}_{ft} |} \left( \beta \right) , \, j=p \, ; \quad \frac{\partial I_{ft}}{\partial \delta _j} = \frac{1}{| {\tilde{I}}_{ft} |} \left( -\beta \right) , j\ne p \end{aligned}$$

(10.23)

where

$$\begin{aligned} \begin{array}{ccl} | {\tilde{I}}_{ft} |&= \sqrt{\left( \sum _{j=1}^N {\tilde{\varPsi }}_{FTj}{\tilde{E}}_{j}^{\prime } \right) ^2} \end{array} \end{aligned}$$

(10.24)

is the value of the line current magnitude at equilibrium, and $\beta$ is given by (10.25)

$$\begin{aligned} \beta =&-\sum \nolimits _{p=1}^{N-1} \sum \nolimits _{q=p+1}^{N} \varPsi _{FTp}\varPsi _{FTq}E_{p}^{\prime }E_{q}^{\prime }\sin \left( \delta _p + \psi _{FTp} - \delta _q - \psi _{FTq}\right) \nonumber \\&+\sum \nolimits _{\begin{array}{c} p=1\\ p\ne j \end{array}}^{N-1} \sum \nolimits _{\begin{array}{c} q=p+1\\ q\ne j \end{array}}^{N} \varPsi _{FTp}\varPsi _{FTq}E_{p}^{\prime }E_{q}^{\prime }\sin \left( \delta _p + \psi _{FTp} - \delta _q - \psi _{FTq}\right) \end{aligned}$$

(10.25)

For the line current angle

$$\begin{aligned} \frac{\partial \phi _{ft}}{\partial \delta _j} = \frac{1}{| {\tilde{I}}_{ft} |^2} \left[ \varPsi _{FTj}^2E_{j}^{\prime 2} + \sum _{\begin{array}{c} q=1\\ q\ne j \end{array}}^{N} \varPsi _{FTj}\varPsi _{FTq}E_{j}^{\prime }E_{q}^{\prime }\cos \left( \delta _j + \psi _{FTj} - \delta _q - \psi _{FTq} \right) \right] \end{aligned}$$

(10.26)

is the closed-form formula to obtain the sensitivity of $\phi _{ft}$ with respect to the $j$-th machine angle.

Finally, the sensitivities of the real and imaginary part of the line current (10.16) are given by

$$\begin{aligned} \frac{\partial \mathfrak R \{ {\tilde{I}}_{ft} \}}{\partial \delta _j} = -\sum _{j=1}^N \varPsi _{FTj}E_{j}^{\prime }\sin \left( \delta _j + \psi _{FTj}\right) \end{aligned}$$

(10.27)

$$\begin{aligned} \frac{\partial \mathfrak I \{ {\tilde{I}}_{ft} \}}{\partial \delta _j} = \sum _{j=1}^N \varPsi _{FTj}E_{j}^{\prime }\cos \left( \delta _j + \psi _{FTj}\right) \end{aligned}$$

(10.28)

Similar expressions can be obtained for the current from Bus $t$ to Bus $f$ by substituting the subscripts $ft$ and $FT$ with $tf$ and $TF$, and using the appropriate coefficients and admittances.

10.4.3 Properties of the Network Sensitivities

We finalize this discussion by providing some intrinsic properties of the sensitivities discussed above. For the bus voltage sensitivities, the properties are²

$$\begin{aligned} \sum _{j=1}^{N} \left( \frac{\partial V_i}{\partial \delta _j} \right) =0,\quad \sum _{j=1}^{N} \left( \frac{\partial \theta _i}{\partial \delta _j} \right) =1 \end{aligned}$$

(10.29)

and for the line currents the sensitivity properties are

$$\begin{aligned} \sum _{j=1}^{N} \left( \frac{\partial I_{ft}}{\partial \delta _j} \right) =0,\quad \sum _{j=1}^{N} \left( \frac{\partial \phi _{ft}}{\partial \delta _j} \right) =1 \end{aligned}$$

(10.30)

Both voltage and current sensitivities have the same property: the sum of sensitivities of the magnitude of the phasor w.r.t. all the machine angles is zero, while the sum of the sensitivities of the angle of the phasor w.r.t. machine angles is equal to unity.

Fig. 10.7

Illustration of the sensitivity properties

These sensitivity properties can be explained by using the phasor diagram in Fig. 10.7. The voltage at the $i$-th bus of a two-machine system, $\tilde{V}_i=V_i\varepsilon ^{j\theta _i}$, is perturbed by introducing small changes to the machine angles, $\varDelta \delta _1$ and $\varDelta \delta _2$, resulting in the perturbed voltage $\tilde{V}_i^{*}=V_i^{*}\varepsilon ^{j\theta _i^{*}}$. When the perturbations introduced to the machine angles are identical, the resulting perturbation to the bus angle will be equal to the value used to perturb the machines. The perturbation of each machine is appropriately scaled by the corresponding sensitivity. Hence, the sum of all sensitivities is one. Note that if only one of the machines is perturbed, the bus angle will only be changed in the proportion dictated by the sensitivity and the perturbation value.

In addition, observe that the magnitude before the perturbation is the same as the magnitude after the perturbation, i.e., $V_i=V_i^{*}$. Note from the phasor diagram that $\varDelta \delta _1$ scaled by $\partial V_i/\partial \delta _1$ will oppose the change in the voltage magnitude from $\varDelta \delta _2$ scaled by $\partial V_i / \partial \delta _2$. Observe that the value of the bus voltage sensitivities for this case is identical. Because both sensitivities are equal with opposing signs, the change in the bus voltage magnitude, $\varDelta V = 0$.

10.5 Network Modeshapes: Sensitivity Mapping onto the Electromechanical Mode Shapes

In the previous section, we investigated the origin of network sensitivities, and how they can be used to compute the total change in the network variables resulting from small perturbations in the machine internal angles. These changes include all $(N-1)$ modal components of the power system. Measurements from PMUs are able to capture these changes in the network variables, and separation of their modal components can be performed with various techniques such as ERA or Prony analysis. In this section, we investigate how the eigenvector matrices discussed in Sect. 10.3, can be used to separate the components of each oscillatory mode contained in the total change of the network variables. This result was previously exploited in [8], where we showed that it is possible to compute the bus voltage magnitude and frequency mode shapes. Here we extend this concept to include any type of network variable, and to understand how by mapping the network sensitivities to a particular mode shape from ${\bar{A}}$, it is possible to replicate the phase shift observed in the modal components of PMU measurements.

By mapping the network sensitivities onto the right eigenvector, we obtain a network modeshape, which indicates the observability of a particular mode in a specific network variable. The mode shapes as observed in the bus voltage magnitudes and angles for all the network buses are given by

$$\begin{aligned} {S}_{{V}}&= {C}_{V \delta } {W}\end{aligned}$$

(10.31)

$$\begin{aligned} {S}_\mathrm{\theta }&= {C}_\mathrm{\theta \delta } {W} \end{aligned}$$

(10.32)

where ${W}$ is of size $N\times N$.

Similarly, for the line current magnitude and angle we have

$$\begin{aligned} {S}_{I_{ft}}&= {C}_{I_{ft} \delta } {W}\end{aligned}$$

(10.33)

$$\begin{aligned} {S}_{\phi _{ft}}&= {C}_{\phi _{ft} \delta } {W} \end{aligned}$$

(10.34)

where the subscript $ft$ indicates that these are the modal components from Bus $f$ to Bus $t$. Modal components for any other network variable can be obtained in similar fashion.

When the eigenvectors are computed from ${\bar{A}}$, the phase shifts due to damping will be mapped onto the bus voltages in a proportion dictated by the sensitivities, reproducing the phase shifts observed in the network variables measured by PMUs.

To illustrate, consider the two-area four-generator system discussed in Fig. 10.4, Sect. 10.3.2. The bus voltage magnitude sensitivities of Bus 7 w.r.t. all machine angles are given by

$$\begin{aligned} {C}_{V \delta _{(7,k)}} = [-0.0463\;\;-0.0270\;\;0.0146\;\;0.0587] \end{aligned}$$

(10.35)

Multiplying by the eigenvector column corresponding to the interarea mode, the interarea mode shape viewed from the bus voltage magnitude at Bus 7 is

$$\begin{aligned} \begin{array}{rl} S_{V_{(7,2)}} = &{} \frac{\displaystyle \partial V_7}{\displaystyle \partial \delta _1} W_{(1,2)} + \frac{\displaystyle \partial V_7}{\displaystyle \partial \delta _2} W_{(2,2)} + \frac{\displaystyle \partial V_7}{\displaystyle \partial \delta _3} W_{(3,2)} + \frac{\displaystyle \partial V_7}{\displaystyle \partial \delta _4} W_{(4,2)} \\ S_{V_{(7,2)}} = &{} -0.0463\; W_{(1,2)} -0.0270 \; W_{(2,2)} + 0.0146 \; W_{(3,2)} + 0.0587\; W_{(4,2)} \end{array} \end{aligned}$$

(10.36)

Note that each sensitivity scales its corresponding element of interarea mode shape.

The eigenvector matrix computed from (10.3) has the interarea mode shape given by Column 2 of (10.4)

$$\begin{aligned} W_{(j,2)}({A}) = [ -0.5930 \;\; -0.5997 \;\; -0.4539 \;\; 0.2876 ]^T \end{aligned}$$

(10.37)

Using this mode shape, it is possible to compute the interarea component of the bus voltage magnitude at Bus 7 (10.36)

$$\begin{aligned} S_{V_{(7,2)}} ({A})&= (-0.0463)(-0.5930) + (-0.0270)(-0.5997) \nonumber \\&\qquad + (0.0146)(-0.4539) + (0.0587)(0.2876) = 0.05391 \end{aligned}$$

(10.38)

Similarly, if the eigenvector matrix is computed from model (10.1), the interarea mode shape is given by Column 2 of (10.6), and the interarea component of the bus voltage magnitude becomes

$$\begin{aligned} S_{V_{(7,2)}} ({\bar{A}}) = 0.0534 \angle -5.25^{\circ } \end{aligned}$$

(10.39)

Comparing (10.39) to (10.5) helps in illustrating the origin of the phase shifts observed from phasor measurement data: that is, the presence of damping.

10.5.1 Illustration with the Two-Area Four-Machine System

We extend our discussion to consider the different network variables across the entire power system. The bus voltage magnitude mode shapes (the mapping of the sensitivities onto the electromechanical mode shapes) for all network buses are shown in the one-line diagrams in Figs. 10.4 and 10.8 for the interarea mode, for both the case without damping and with damping, respectively, and the phasor diagrams in Fig. 10.9. These figures clearly show that the damping in ${\bar{A}}$ will give rise to the phase shifts across all bus voltage magnitudes in the power system. Consider the phase shift introduced to the voltage magnitude at Bus 1: when no damping is included $S_{V_{(1,2)}}({A})=0.0544\angle 0^{\circ }$ and, with the effect of damping $S_{V_{(1,2)}}({\bar{A}})=0.0547\angle -7.14^\circ$. Thus the phase shift $-$7.14$^\circ$ is a result of damping. Similarly for Bus 4 $S_{V_{(4,2)}}({A})=0.0085\angle 180^\circ$, while $S_{V_{(4,2)}}({\bar{A}})=0.0085\angle 176.14^\circ$. The 3.86$^{\circ }$ of difference between these two last quantities are the result of including damping. A more interesting case is Bus 8, which lies at the right end of the tie-line. Note that $S_{V_{(8,2)}}({A})=0.0057\angle 0^\circ$, while $S_{V_{(4,2)}}({\bar{A}})= 0.0056\angle -6.05^\circ$, the $-6.05^\circ$ being a result of the inclusion of damping.

Fig. 10.8

Two-area four-machine power system and voltage magnitude interarea mode shape (damping)

Fig. 10.9

$V_i$ interarea mode shape with and without damping. a Without damping. b With damping

Fig. 10.10

$V_i$ Local 1 mode shape with and without damping. a Without damping. b With damping

It is worthwhile to note that this holds for any mode. For example, consider the bus voltage magnitude mode shape for Local Mode 1 shown in Fig. 10.10. In this case, there is a more prominent phase shift between Buses 1 and 12 than in Fig. 10.9 because Local Mode 1 mostly involves $G_1$ and $G_2$. This is also reflected in the voltage mode shapes. It is also important to highlight that the observations above also hold for any network variable. Consider the bus voltage angle mode shapes in Fig. 10.11. In the case without damping in Fig. 10.11a, it is observed that the bus voltage angles are either completely in phase or anti-phase. In contrast, Fig. 10.11b shows that the result of including damping is to have phase shifts in all the mode shapes. Another important feature of this mode shape is that when comparing Figs. 10.11b to 10.5b, the machine angles outline a boundary within which the bus voltage angles exist. In other words, a bus voltage angle close to a generator will have a smaller relative phase angle difference with the generator than other generators farther apart from it.

Fig. 10.11

$\theta _i$ Local 1 mode shapes with and without damping. a Without damping. b With damping

The relative time delay between the voltage magnitudes at Buses $p$ and $q$ can be calculated from the oscillatory frequency of the $k$-th mode and the phase shift at each bus by computing

$$\begin{aligned} \tau _{(p-q,k)} = \frac{\theta _{(p,k)} - \theta _{(q,k)}}{2\pi }\times \frac{1}{f_k} \end{aligned}$$

(10.40)

where $f_k$ is the frequency of the $k$-th mode of the system, and

$$\begin{aligned} \theta _{(p,k)}= \angle \left( S_{V_{(p,k)}} \right) , \; \; \theta _{(q,k)}= \angle \left( S_{V_{(q,k)}} \right) \end{aligned}$$

(10.41)

are the $k$-th mode phase shifts of the voltage magnitude oscillation in rad. at Buses $p$ and $q$, respectively, and $j=1,\ldots ,N$.

As an example, consider the oscillations from Local Mode 1 at Bus 12, $S_{V_{(12,3)}}$, and Bus 1, $S_{V_{(1,3)}}$, as shown in Fig. 10.10. The relative time delay is given by

$$\begin{aligned} \tau _{(12-1,3)} = \frac{ (5.6295) - (3.1291) }{2 \pi \times 0.21516} = 1.8495\,\mathrm s \end{aligned}$$

This time delay is shown in Fig. 10.12, where we compare oscillations for Local Mode 1 in the bus voltage magnitude at Bus 1 and Bus 12.

Fig. 10.12

Time delay between $S_{V_{(12,3)}}$ and $S_{V_{(1,3)}}$

Equation (10.40) can be used to compute the time delays for any network mode shape by selecting the sensitivity and mode of interest. Conversely, the relative phase shift between the buses can also be computed by knowledge of the time response of the network (or ERA results from processing PMU data) from

$$\begin{aligned} \theta _{(p,k)} - \theta _{(q,k)} = 2\pi f_k \tau _{(p-q,k)} \end{aligned}$$

(10.42)

where $f_k$ is the frequency of the $k$-th mode of the system. Note that this frequency is different for each system mode, i.e., the interarea mode frequency is different from the Local Mode 1 frequency. Using the expression above, and selecting a reference voltage magnitude, we obtain the mode shapes from the time response, or more important, from PMU data.

Also of interest is the mapping for the line complex current flow mode shape, shown in Fig. 10.13 for ${W}({\bar{A}})$. Note that the phase shifts are also mapped onto these network variables. The phase lags shown in the phasor diagram in Fig. 10.13a will translate to the different time delays that can be observed in Fig. 10.13b. More important, it is interesting to observe how the interarea mode distributes in different lines as shown in Fig. 10.13c. Note that it is possible to compute the current balance for the real and imaginary part, and that we can also determine how the interarea mode is distributed in each line. This new capability to “track” the oscillations has important implications as discussed next.

Fig. 10.13

Interarea mode${\tilde{I}}_{ft}$ mode shapes (model with damping, imaginary quantities are shown inside parenthesis). a Phasor diagram. b Time response. c One-line diagram

10.6 Applications for Interarea Mode Tracing: Identification of Dominant Interarea Oscillation Paths

One goal of a Wide Area Measurement System (WAMS) is to have tracking tools for oscillatory dynamics in an interconnected power grid, particularly those which are critical to operational reliability, i.e., interarea oscillations [17]. Insufficient damping of low-frequency interarea oscillations arises as weak interconnected power systems are stressed to meet up with increasing demand [18]. This inadequacy may lead to oscillatory instability, resulting in system collapse.

“Interaction paths,” defined in [17] as the group of transmission lines, buses, and controllers which the generators in a system use for exchanging energy during swings, are one important source of dynamic information necessary for WAMS. If the interaction paths of interarea swings can be identified, monitored, and tracked, proper preventive measures or control actions can be carried out to enhance the system’s transfer capacity while maintaining high security.

A characteristic of power oscillations is that, for every mode of oscillation, there exists a series of connecting corridors in which the highest content of the mode would propagate. For a particular case of interarea modes, the path is termed “dominant interarea oscillation path” [19], a concept based on the notion of interaction paths. These dominant interarea oscillation paths are deterministic [20]. Furthermore, signals from the dominant path are the most observable and have the highest content of interarea modes. Results from the study suggest that by using well-selected dominant path signals for wide-area control, adequate damping performance can be achieved.

10.6.1 Dominant Interarea Oscillation Paths

Dominant interarea oscillation paths are defined as the passageway containing the highest content of the interarea oscillations. Consider a simplified dominant interarea path, represented by the two-area system shown in Fig. 10.14, where $G_1$ and $G_2$ represent coherent groups of machines involved in the interarea swing while transformers and line impedances represent elements of the dominant path connecting the two areas.

Fig. 10.14

Simplified dominant interarea path represented with a two-area, two-machine system

The characteristics of dominant interarea paths³ can be demonstrated using the computed bus voltage magnitude ($S_V$) and angle modeshapes ($S_\theta$) as illustrated in Fig. 10.15a, b, respectively. Fig. 10.15c, d illustrate the corresponding magnitude ($|S_\theta |$) and phase ($\angle S_\theta$) of the voltage angle modeshapes, $S_\theta$. Two transfer levels, Case A and Case B, are compared in this figure. The labels “1,100” and “900” MW represent the amount of power transfer over the dominant path. The $x$-axis represents the bus number in the dominant path; the distance between buses is proportional to the line impedance magnitude. According to Fig. 10.15, important features of the dominant path are summarized below.

Fig. 10.15

Voltage magnitude and angle modeshapes of the dominant path in the two-area system. a Voltage magnitude modeshape, b Voltage angle modeshape, c Magnitude of voltage angle modeshape, d Phase of voltage angle modeshape

• The smallest $|S_{\theta }|$ element(s) (Fig. 10.15c) or the largest $S_V$ (Fig. 10.15a) indicates the center of the path. This center can be considered as the “interarea mode center of inertia” or the “interarea pivot” for each of the system’s interarea modes.

• The pivot divides the path into two groups where their respective phases are opposing each other (Fig. 10.15d).

• The difference between the $S_{\theta }$ elements of two edges of the path (Fig. 10.15b) are the largest among any other pair within the same path. In other words, the oscillations are the most positive at one end while being the most negative at the other end. Hence, they can be considered as the “tails” for each interarea mode.

• The $S_V$ elements of the edges (Fig. 10.15a) are the smallest or one of the smallest within the path.

• The interarea contents of the voltage magnitude modeshapes are more observable in a highly stressed system.

10.6.2 Dominant Interarea Oscillation Paths Identification Algorithm

As illustrated in Sect. 10.5.1, the current magnitude modeshapes ($S_{I_{ft}}$) indicate how much contents of the interarea modes are distributed among the transfer corridors. Thus, corridors having the highest content of current magnitude modeshapes signify the paths where the interarea oscillations will travel the most; hence the term “dominant interarea oscillation paths.” On the other hand, the magnitude of voltage magnitude and angle modeshapes ($S_V$ and $S_\theta$) indicates the modal observability of a signal. The larger in magnitude the modeshape is, the more observable the signal measured (from the dominant path) becomes. This will be helpful when selecting feedback signals having high interarea modal content.

An algorithm for identification of dominant interarea oscillation paths is proposed here. Note that this algorithm takes synchrophasor measurements into consideration by exploiting ambient data.⁴

Ambient measurements are synthesized by simulating the time response of the power system with random noise and small step inputs at all loads [23]. The algorithm for handling ambient measurements is described next.

Dominant Path Identification Algorithm using Ambient PMU data

• Step 1. Pre-process a parcel of ambient measurements, $\varDelta I_{ft}, \varDelta V$, and $\varDelta \theta$ by filtering all the measurements such that only the interarea modal content of interest is preserved in the signals.

• Step 2. Compute the power spectral densities (PSD) of $\varDelta I_{ft}$.

• Step 3. Select an appropriate window. Find the peak PSD for each signal within the selected window, and sort the contents in descending order.

• Step 4. The dominant interarea oscillation path is determined from the signals having the largest PSD contents. Compare to the schematic diagram of the system of study and identify such paths.

• Step 5. Using one of the edges of the path in Step 4 as a reference, the cross power spectral densities (CPSD) of the preprocessed $\varDelta V$ and $\varDelta \theta$ of the dominant path are computed.

• Step 6. Select appropriate windows for each type of signal. Find the corresponding largest CPSD magnitude for each measurement within the selected window.

• Step 7. Verify the characteristics of the dominant path using its corresponding peak CPSD of voltage magnitude and angle. The results should resemble the dominant interarea oscillation path’s features shown in Fig. 10.15.

10.6.3 Algorithm Illustration with the KTH-NORDIC32 System

The system under study, namely KTH-NORDIC32 [24] (illustrated in Fig. 10.16), is a simplified model of the Swedish power system and its neighbors [25]. It has 20 generators and 52 transmission lines. Small signal stability analysis [25] reveals that the system has two lightly damped low-frequency interarea oscillations: 0.50 and 0.74 Hz. Note that to develop a fundamental understanding, we consider a case where controls such as exciters and turbine governors are disabled.⁵ Two scenarios, namely Case 1 and Case 2, are used to illustrate the algorithm throughout this study. Power flow from the northern to the southern regions for each respective scenario are 3,134, and 2,933 MW. Note that the step-by-step demonstration will be described only for Case 1.

Fig. 10.16

KTH-NORDIC32 system

Fig. 10.17

$\varDelta I_{ft}$ signals: before and after pre-processing. a Before pre-processing. b After pre-processing, (from 0 to 20 s)

Fig. 10.18

PSD of $\varDelta I_{ft}$, mode 1

• Step 1. $\varDelta I_{ft}$ measurements before and after pre-processing are illustrated in Fig. 10.17.

• Step 2. The computed PSD of $\varDelta I_{ft}$ are illustrated in Fig. 10.18. The dashed (red) lines indicate the cutoff frequencies used in the preprocessing.

• Step 3. The window of the selected data corresponds to the dashed lines in Fig. 10.18. The peak PSD for each $\varDelta I_{ft}$ are sorted in descending order; the ten largest values and their corresponding sending and receiving buses are summarized in Table 10.1.

• Step 4. The dominant interarea oscillation path for Mode 1 is identified to be 52-51-35-37-38-40-48-49-50⁶ as marked by the (yellow) stars in Fig. 10.16.

• Step 5. $\varDelta V_{52}$ and $\varDelta \theta _{52}$ are used as references for the CPSD computation of voltage magnitude and voltage angle measurements of the dominant path, respectively. The corresponding computed CPSD are illustrated in Fig. 10.19.

• Step 6. After selecting the appropriate windows, the largest (absolute) CPSD values for each measurement within the dominant path are used to reconstruct the path. Using the characteristics of the dominant path from Fig. 10.15c and d, the bus having the smallest magnitude of voltage angle deviations, $|\varDelta \theta |$, is the pivot of the path and thus used as the reference. The reconstructed dominant path of Mode 1 is as shown in Fig. 10.20a with blue dots for Case 1.

• Step 7. The characteristics of the obtained path is verified by comparing to that of Fig. 10.15. The main features of the dominant paths remain preserved, and, thus, the path is justified.

Table 10.1 Ten largest PSD of $\varDelta I_{ft}$

Fig. 10.19

Computed CSD of the dominant path. a Voltage magnitude. b Voltage angle

Fig. 10.20

Dominant interarea oscillation paths for both interarea modes, ambient measurements. a Mode 1. b Mode 2

Comparing the two case studies in Fig. 10.20a, Case 1 has an overall larger modal content in $\varDelta V$’s while $\varDelta \theta$’s are about the same in both cases. The differences between the two edges of both cases are comparable.

Repeating Step 1–7 for Mode 2, the dominant interarea oscillation path for Mode 2 is identified to be 50-49-44-47. Voltage magnitude and angle deviations of the path are illustrated in Fig. 10.20b. Similar to the result from Mode 1, Case 1 has larger modal content in $\varDelta V$’s.

As a final remark, note that Fig. 10.19b shows clear indications of the coherency groups: buses with positive CPSD peaks correspond to the group on the left and those with negative peaks to the group on the right of the dominant path. This is the characteristic shown in Fig. 10.15b for the two-area system. A similar analogy is valid for the CPSDs in Figs. 10.19a and  10.15a. In Fig. 10.19a, the CPSD peaks are all positive and with magnitudes according to the location of the buses within the dominant path where the signal is taken (see Fig. 10.15a). Observe that according to the voltage magnitude mode shape in Fig. 10.15a, those buses close to the “tails” of the dominant path have the lowest voltage oscillation (lowest CPSD magnitude) while those within the middle of the dominant path experience the highest voltage oscillations (highest CPSD magnitude). The fact that all CPSDs from the dominant path are positive corroborates that all voltage magnitude oscillations in a dominant path are either in phase or showing only a small phase shift between each other. Note that the shift in the CPSDs peak corresponds also to the phase shift illustrated in Sect. 10.5.1.

10.6.4 Damping Controller Design: Feedback Signals Selection

One major challenge in damping control design is the selection of feedback input signals. Conventionally, power system stabilizers (PSSs) use local measurements for input signals, such as active power in the outgoing transmission line, generator speed, and frequency at the terminal bus. With the availability of signals from PMUs, choices of inputs are not only limited to those local but now include wide-area signals. Several studies suggest that wide-area signals are preferable to local signals [27, 28]; therefore, the exploitation of PMU signals is desirable. However, the main issue is which signal, among all the available signals, would give satisfactory damping performance.

Results from previous studies on the concept of dominant interarea oscillation paths [19, 26] have suggested that effective damping control can be achieved by using signals from the dominant paths. The aim of this section is thus to carry out a fundamental study on feedback control using PMU signals from a dominant path. As such, the two-area system introduced in Sect. 10.6.1 is used to illustrate PSS control design for damping enhancement. Two types of signals, namely voltage angle and generator rotor speed, are used as inputs for a PSS controller. The first type represents the signals available from PMU while the latter represents one of the most commonly used signals in PSS damping design.⁷ Their corresponding performances are analyzed and compared. The results of this study offer promising and feasible choices of signals to be used in feedback control. Although only PSS is considered, the concepts are applicable for any other damping controllers.

In this study, the impact of different feedback input signals on the damping of the two-area test system will be evaluated. Each signal requires different controller (PSS) parameters, as well as different structures. The purpose of the design is to achieve a specific damping performance. As such, for each input signal, PSS parameters will be tuned such that the system achieves $\zeta = 15~\%$ damping ratio.

Voltage angle and generator speeds will be used as feedback input signals. Controller performance is evaluated considering the following factors: (1) effective gain (the cumulative gain of the PSSs which can be computed from $\alpha ^n K_d$), (2) overshoot ($M_p$), and (3) rise time ($t_r$).

The monitored signal is the bus voltage terminal at $G_1$, $V_1$, which will be used to evaluate the control performance.

10.6.4.1 Controller Structure

The objective of the design is to improve damping of the interarea mode by installing a PSS at $G_1$ modulating the AVR error signal. Following the design in [29], the structure of the PSS includes lead/lag compensators in the form

$$\begin{aligned} PSS = K_d\left[ \alpha \frac{s+z}{s+p} \right] ^n \left[ \frac{T_ws}{1+T_ws} \right] ^m \end{aligned}$$

(10.43)

where $n$ and $m$ are the number of compensator stages and $T_w$ is the washout filter time constant having the value of 10 s. The exponent $m = 1$ when the input signal is a generator rotor speed while $m=2$ when the input signal is a voltage angle difference.

Note that generator speed, as well as angle difference, has high components of torsional modes [4]. Therefore, a torsional filter is added to the PSS structure when generator speed and bus voltage angle differences are used as feedback input signals. The impact of the torsional filter on PSS design is represented by a lower-order transfer function

$$\begin{aligned} G_{tor}(s) = \frac{1}{0.0027s^2 + 0.0762s + 1}. \end{aligned}$$

(10.44)

The controller parameters $\alpha$, poles ($p$), and zeros ($z$) can be computed from the following equations

$$\begin{aligned} \phi _m&= \frac{180^{\circ }-\theta _\mathrm{dep}}{n} \end{aligned}$$

(10.45)

$$\begin{aligned} \alpha&= \frac{1+sin(\phi _m)}{1-sin(\phi _m)} \end{aligned}$$

(10.46)

$$\begin{aligned} p&= \sqrt{\alpha } \omega _c, \quad z = p / \alpha \end{aligned}$$

(10.47)

where $\theta _ \mathrm{dep}, \phi _m,$ and $\omega _c$ represent the angle of departure of the interarea mode, the phase compensation required, and the frequency of the mode in rad/s, respectively.

Fig. 10.21

Root-locus plots of the system with $\varDelta \theta _{14}$ as feedback input signal. a No phase compensation. b With phase-lead compensation

10.6.5 Controller Design Illustration

In this illustration, the signal $\varDelta \theta _{14} = \theta _1 - \theta _4$, the voltage angle difference between Bus 1 and Bus 4 (see Fig. 10.14), is used as the feedback input signal. A root-locus plot of the open-loop system (no phase compensation) including two washout filters is shown in Fig. 10.21a. The angle of departure ($\theta _\mathrm{dep}$) of the interarea mode is $16.167^{\circ }$. Using this angle, the PSS parameters are computed using Eqs. (10.43–10.47). Applying the designed controller to the system, the root-locus plot is shown in Fig. 10.21b, which shows a stabilizing direction of the interarea mode; i.e., the damping of interarea mode is improved. The gain $K_d$ is obtained when moving along the branch of the root loci of the interarea mode until the desired damping ratio ($15~\%$) is reached. Finally, for the signal used, the obtained PSS has the form

$$\begin{aligned} PSS =0.00325\left[ 200.284 \frac{s+0.1659}{s+33.2357} \right] ^2 \left[ \frac{10s}{1+10s} \right] ^2. \end{aligned}$$

(10.48)

Fig. 10.22

Damping control performance using $\varDelta \theta _{14}$ as feedback input signal

The responses of the terminal voltage at Bus 1 with and without PSS are compared in Fig. 10.22.

10.6.5.1 Voltage Angle Differences as Input Signals

Using the same design process, the system performance using voltage angle differences as feedback input signals is summarized in Table 10.2. All angle differences in Table 10.2 use a 2-stage lead compensator. Note that $\theta _4$ is used as a reference, and the order of representation corresponds to the location of the buses in Fig. 10.14.

Table 10.2 System performance using voltage angle differences as feedback input signals.

According to the results in Table 10.2, it can be concluded that the overall performance corresponds to the voltage angle modeshape ($S_\theta$) relationship. That is, the larger the angle modeshape (difference), the lesser the gain is required, the smaller the overshoot is, and the faster the rise time becomes. In other words, the signal $\varDelta \theta _{42}$, having the largest $S_\theta$ difference, requires the smallest gain and has the smallest overshoot and rise time. This is because the signals have modal contents proportional to the voltage angle difference dictated by the voltage angle modeshape (see Fig. 10.15b).

Fig. 10.23

Damping control performance using $\varDelta \theta _{ij}$ as feedback input signals

Table 10.2’s corresponding responses of the terminal voltage at Bus 1 are illustrated in Fig. 10.23. Note that, in order to have the same sign, $\varDelta \theta _{14}$ is used instead of $\varDelta \theta _{41}$.

The possible signal combinations using angle differences between the two areas are shown in Table 10.3 where the average angle differences $\varDelta \theta _\mathrm{avg,1}$ represents $(\theta _1+\theta _4)-(\theta _2+\theta _5)$, and $\varDelta \theta _\mathrm{avg,2}$ represents $(\theta _{a}+\theta _4)-(\theta _{b}+\theta _5)$. Note that Bus $a$ is a bus in the middle between Bus 4 and Bus 3, whereas Bus $b$ is a bus in the middle between Bus 3 and Bus 5. The aim of using the average angle differences in the two areas is to increase the interarea mode content of the resulting feedback signal, in larger power networks it can be used to reduce the excitation of the local oscillations [30]. All two-area angle difference combinations use a 2-stage lead compensator.

Table 10.3 System performance using signal combinations of voltage angle differences as feedback input signals

Fig. 10.24

Damping control performance using signal combinations of $\varDelta \theta _{ij}$ as feedback input signals

According to the results in Table 10.3, by combining signals from both areas, the effective gains are significantly reduced compared to the results in Table 10.2. In addition, overshoots are slightly reduced, while rise times are similar. Overall, $\varDelta \theta _\mathrm{avg,1}$ requires the least amount of gain with comparable overshoot and rise time performance. However, in practice, $\theta _1$ and $\theta _2$ are generator buses and thus not usually available from PMUs (see [31]). Hence, for any practical implementation, the most feasible combination is $\varDelta \theta _\mathrm{avg,2}$, for which the desired damping performance can be achieved while requiring a smaller gain than those from individual signals.

The time responses of the terminal voltage at Bus 1 are illustrated in Fig. 10.24.

10.6.5.2 Rotor Speeds as Input Signals

For comparison purposes, we consider speed signals from generators, although only available locally and not commonly available from PMUs [31]. The system performance using generator speeds as feedback input signals is summarized in Table 10.4. All signals in Table 10.4 use a 2-stage lead compensator except for $\omega _{2}$ which requires a 3-stage lead compensator.⁸

According to the results in Table 10.4, overall, using speeds as feedback input signals not only results in larger overshoots and longer rise times but also requires considerably larger gain than using bus voltage angle difference signals, particularly using $\omega _2$.⁹

Table 10.4 System performance using generator speed as feedback input signals.

The time responses of the terminal voltage at Bus 1 are illustrated in Fig. 10.25. Note that because responses of $\omega _{1}$ and $\omega _1-\omega _2$ are exactly the same, only the response using $\omega _{1}$ is presented in the figure.

10.6.5.3 Input Signal Comparison

A comparison of Tables 10.2, 10.3, and 10.4 reveals that using bus voltage angle differences as feedback input signals for damping control is as effective (or even better) as using generator speeds, for interarea mode damping. From Tables 10.2, 10.3 and 10.4, it can be observed that using $\varDelta \theta _{ij}$ as input signals, similar damping performance can be obtained while having much lower overshoots and, more important, using much lower effective gain.¹⁰ Comparing $\varDelta \theta _{12}$ to $\omega _1-\omega _2$, the angle difference outperforms the speed signals in effective gain required, overshoot, and rise time.¹¹

Fig. 10.25

Damping control performance using $\omega _i$ as feedback input signal

Results from Sects. 10.6.5.1 and 10.6.5.2 indicate that angle difference is the most effective feedback input signal with superior overall performance compared to generator speed. Future work will investigate the use of relative generator speeds and bus frequencies for use as feedback input signals.

The overall performance of each signal of the voltage angle differences is in accordance with their corresponding network modeshapes (see Fig. 10.15). That is, the performance of signals having high network modeshape is superior to that of those with lower network modeshape.

Different loading effects are not yet considered in this study. This is relevant because, for different loading scenarios, the open-loop observability of the dominant path signals shifts depending on the loading level. Further work is necessary to determine if the closed-loop observability on different loading levels maintains the same properties as those revealed in this study.

The selection of the “right” input signals from PMUs is critical for effective damping control. However, in the case of signal loss (due to communication failures), the controller must be adjusted even if new signals are used to replace a lost signal so that the highest damping can be obtained. These adjustments must occur adaptively and must be initiated by an adequate switch-over logic that guarantees the continued operation of the damping controller. Depending on the types of signals as well as the signal combination, the controller structure must be adapted accordingly to achieve optimal damping. As such, “adaptive” controllers, which can automatically adjust their parameters for each input signal feeding in, are promising and desirable.

10.7 Conclusions

In this chapter, the fundamental results presented in the analysis of network modeshapes provide a novel understanding of power system oscillations as viewed from network variables. Furthermore, they have several applications for interarea mode tracing and monitoring, PMU placement, and damping of power system oscillations. An algorithm for identifying dominant paths using ambient PMU data was presented, and the use of signals from the dominant path for feedback control input signals was investigated for the case of PSS damping control. The demonstration of these applications corroborates the fact that interarea modes are visible in network variables measured by PMUs which are readily exploitable.