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Measurement-based voltage stability monitoring for load areas

Kai Sun; Fengkai Hu University of Tennessee, Knoxville, TN, United States

Abstract

This article introduces a measurement-based voltage stability monitoring method for a load area fed by N tie lines. Compared to a traditional Thevenin equivalent (TE)-based method, the new method adopts an N + 1 buses equivalent system so as to model and monitor individual tie lines. For each tie line, the method solves the power transfer limit against voltage instability analytically as a function of all parameters of that equivalent, which are identified online from real-time synchronized measurements on boundary buses of the load area. Thus, this new method can directly calculate the real-time power transfer limit on each tie line. The method is first compared with a TE-based method using a four-bus test system and then demonstrated by case studies on the Northeast Power Coordinating Council's 48-machine, 140-bus power system.

Keywords

Load center; N + 1 buses equivalent; Parameter estimation; PMU; Power transfer limit; Thevenin equivalent; Voltage stability margin

1 Introduction

Growth in electrical energy consumption and penetration of intermittent renewable resources would make power transmission systems operate close to their stability limits more often. Among all stability issues, voltage instability due to the inability of the transmission or generation system to deliver the power requested by loads is one of the major concerns in today's power system operations. Voltage stability refers to “the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition” [1]. In contrast to generator rotors in the angular stability of a power system, the driving forces for voltage instability are mainly loads and the means by which voltage or reactive power are controlled to support loads. Usually, voltage instability initiates from a local bus but may develop to a wide area or even a systemwide instability. In a heavily stressed power system, voltage instability becomes a major concern in system operations and an impacting factor on the power transfer capability of a transmission network. When voltage instability leads to loss of voltage in a significant part of the system, such a process is called voltage collapse, in which many bus voltages in the system decay to an unrecoverable level. As a consequence of voltage collapse, the system may experience a power blackout over a large area. A system restoration procedure would then be undertaken to bring the blackout area back to service.

At present, some electricity utilities use model-based voltage stability assessment (VSA) online software tools to assist operators in foreseeing potential voltage instability and estimating voltage stability margin information. Based on a state estimate on the operating condition, those software tools employ power system models to simulate assumed disturbances such as contingencies and load changes by either steady-state analysis or dynamic analysis. However, such a model-based approach has its limitations: the fidelity of its results highly depends on the accuracy of power system models; and it needs a convergent, accurate state estimate in order to conduct a stability assessment, which may be hard to obtain under stressed system conditions. Real-time, accurate voltage stability margin information is critical for system operators to be aware of any potential or already developing voltage instability. However, it is challenging for model-based VSAs to provide such real-time margin information, especially for a large system. Many generators, buses, and transmission lines have to be modeled; also, lots of uncertainties need to be addressed such as load variations and intermittent behaviors of renewable generation.

Many countries are deploying synchronized phasor measurement units (PMUs) on transmission systems to provide wide area measurements for real-time stability monitoring. That leads to more interests in developing measurement-based VSA methods to directly assess voltage stability from measurements on monitored buses [2–23]. A family of measurement-based methods is based on Thevenin's Theorem: local measurements at the monitored load bus or area are used to estimate a Thevenin equivalent (TE) approximating the rest of the system, that is, a voltage source connected through a Thevenin impedance as illustrated by Fig. 1. The power transfer to the monitored load reaches its maximum when that Thevenin impedance ${\bar{Z}}_{Thevenin}$ ${\bar{Z}}_{Thevenin}$ has the same magnitude as the load impedance ${\bar{Z}}_{Load}$ ${\bar{Z}}_{Load}$ [4,5]. Note: a variable having a bar represents a complex-value variable or a phasor. Based on a TE, voltage stability indices can be obtained [6,7]. Multiple such equivalents may together be applied to a load area [8]. For practical applications, paper [9] demonstrates a TE-based method on a realistic EHV network, and some other works consider load tap changers and overexcitation limiters in their models for better detection of voltage instability [10–12]. Influences from the changes on the external system and measurement errors on TE estimation are concerned in [13]. The equivalent circuit considering HVDC integrated wind energy is studied in [14].

Fig. 1 Thevenin equivalent for a load bus.

Basically, a TE-based method works satisfactorily on a radially fed load bus or a transmission corridor, and its computational simplicity makes it suitable for real-time application. In recent years, some research efforts have been directed to how to extend the application of the TE to a broader transmission system, for example, the coupled single-port circuit or the Channel Components Transform [15–19]. Some other efforts tested TE-based methods on load areas. References [20,21] apply a TE-based method to a load center area. The method requires synchronized measurements on all boundary buses of the area in order to merge them into one fictitious load bus, for which a TE can be estimated and applied. That method was demonstrated on a real power transmission system in the real-time environment using PMU data [22]. Paper [23] improves the TE estimation for a load area to better tolerate the fluctuations in voltage phase angles and power factors measured at boundary buses. Paper [24] proposes a measurement- and model-based hybrid approach using the TE to assess voltage stability under N − 1 contingencies.

However, as illustrated in [25], a TE-based method may not provide an accurate voltage stability margin for each of the multiple tie lines together feeding a load area if it merges those tie lines, as done in [20,21]. Even when estimating the total transfer limit of multiple tie lines, the TE is accurate only if the boundary buses through which those tie lines feed the load area are strongly connected and the external system is coherent so as to be regarded as a single voltage source. Those are two necessary conditions for merging tie lines and boundary buses and developing a meaningful TE. If connections between boundary buses are weak, when the load of the area gradually increases, tie lines may reach their power transfer limits at different time instants. In other words, voltage instability may initiate near one of the boundary buses and then propagate to the rest of the area. Thus, only monitoring the total power transfer limit of all tie lines using a single TE may delay the detection of voltage instability.

This article introduces an N + 1 buses equivalent system proposed in [26] for measurement-based voltage stability monitoring that extends and generalizes the TE. This new equivalent has N buses interconnected to represent a load area with N boundary buses and one voltage source representing the external system. In fact, the TE is its special case with N = 1. By modeling boundary buses separately for a load area, a new method based on that equivalent is developed and demonstrated in this article to calculate power transfer limits at individual boundary buses.

In the rest of this article, Section 2 introduces the new method in detail, including the N + 1 buses equivalent system, its parameter identification, and analytical solutions of power transfer limits of its tie lines. In Section 3, an online scheme to implement the new method is presented. Section 4 first uses a four-bus power system to illustrate the advantages of the new method over a traditional TE-based method, and then validates the new method on the Northeast Power Coordinating Council (NPCC) 48-machine, 140-bus power system. Finally, conclusions are drawn in Section 5.

2 N + 1 buses equivalent system

For a load area fed by N tie lines, voltage instability may be a concern with its boundary buses when power flows of those tie lines approach their transfer limits. As shown in Fig. 2, an N + 1 buses equivalent system is used to model those boundary buses and tie lines while reducing the network details both inside and outside the load area. Assume the external system to be strongly coherent without any angular stability concern. Thus, it is represented by a single voltage source with phasor $\bar{E}$ $\bar{E}$ connected by N branches with impedances ${\bar{z}}_{E 1} . . . {\bar{z}}_{E N}$ ${\bar{z}}_{E 1} . . . {\bar{z}}_{E N}$ (representing N tie lines) to N boundary buses, respectively. Each boundary bus is monitored and connects an equivalent load with impedance ${\bar{z}}_{ii}$ ${\bar{z}}_{ii}$ , modeling the portion of the load seen from that bus. Connection between any two boundary buses i and j is modeled by impedance ${\bar{z}}_{ij}$ ${\bar{z}}_{ij}$ . The power transfer limit of each tie line is a function of N(N − 1)/2 + 2N + 1 complex parameters of that equivalent system, including voltage phasor $\bar{E}$ $\bar{E}$ , N tie line impedances ${\bar{z}}_{Ei} ’ s$ ${\bar{z}}_{Ei} ’ s$ , N(N − 1)/2 transfer impedances ${\bar{z}}_{ij} ’ s$ ${\bar{z}}_{ij} ’ s$ , and N load impedances ${\bar{z}}_{ii} ’ s$ ${\bar{z}}_{ii} ’ s$ .

Let ${\bar{S}}_{i} = P_{i} + {jQ}_{i}$ ${\bar{S}}_{i} = P_{i} + {jQ}_{i}$ denote the complex power fed to boundary bus i and let ${\bar{V}}_{i}$ ${\bar{V}}_{i}$ denote the bus voltage phasor. Using synchronized measurements on ${\bar{S}}_{i}$ ${\bar{S}}_{i}$ and ${\bar{V}}_{i}$ ${\bar{V}}_{i}$ , all parameters of the equivalent can be identified online (e.g., every 0.1–1 s) using the latest measurements of a sliding time window. The rest of the section presents the algorithms for estimating the parameters of the external system (i.e., $\bar{E}$ $\bar{E}$ and ${\bar{z}}_{Ei}$ ${\bar{z}}_{Ei}$ ) and the load area (i.e., ${\bar{z}}_{ii}$ ${\bar{z}}_{ii}$ and ${\bar{z}}_{ij}$ ${\bar{z}}_{ij}$ ) and then derives the analytical solution of each tie line power transfer limit.

2.1 Identification of external system parameters

Assume that the sliding time window contains K measurement points. The external system parameters are assumed to be constant during the time window and hence are estimated by solving the following optimization problem. Nodal power injection Eq. (1) holds at each measurement point k of the time window.

${\bar{S}}_{i} (k) = {(\frac{\bar{E} - {\bar{V}}_{i} (k)}{{\bar{z}}_{E i}})}^{*} \times {\bar{V}}_{i} (k), k = 1 ~ K$ ${\bar{S}}_{i} (k) = {(\frac{\bar{E} - {\bar{V}}_{i} (k)}{{\bar{z}}_{E i}})}^{*} \times {\bar{V}}_{i} (k), k = 1 ~ K$

si22_e (1)

where ${\bar{S}}_{i} (k) = P_{i} (k) + {jQ}_{i} (k)$ ${\bar{S}}_{i} (k) = P_{i} (k) + {jQ}_{i} (k)$ and ${\bar{V}}_{i} (k) = V_{i} (k) ∠ θ_{i} (k)$ ${\bar{V}}_{i} (k) = V_{i} (k) ∠ θ_{i} (k)$ are, respectively, the received complex power and voltage phasor at boundary bus i at time point k. The magnitude of $\bar{E}$ $\bar{E}$ , denoted by E, can be estimated from measurements at each boundary bus i, whose estimation error is

$e_{i}^{ex} (k) = E - |(P_{i} (k) - {jQ}_{i} (k)) (r_{Ei} + {jx}_{Ei}) + {(V_{i} (k))}^{2}| / V_{i} (k)$ $e_{i}^{ex} (k) = E - |(P_{i} (k) - {jQ}_{i} (k)) (r_{Ei} + {jx}_{Ei}) + {(V_{i} (k))}^{2}| / V_{i} (k)$

(2)

The optimization problem in Eq. (3) computes the optimal estimates of E, r_Ei, and x_Ei.

$min J^{ex} = \sum_{k = 1}^{K} \sum_{i = 1}^{N} \frac{ω_{e}}{N} {[e_{i}^{ex} (k)]}^{2} + \sum_{i = 1}^{N} ω_{z} {(\frac{r_{E i}}{r_{E p}} - 1)}^{2} + \sum_{i = 1}^{N} ω_{z} {(\frac{x_{E i}}{x_{E p}} - 1)}^{2} s . t . E > 0, r_{E i} \geq 0$ $min J^{ex} = \sum_{k = 1}^{K} \sum_{i = 1}^{N} \frac{ω_{e}}{N} {[e_{i}^{ex} (k)]}^{2} + \sum_{i = 1}^{N} ω_{z} {(\frac{r_{E i}}{r_{E p}} - 1)}^{2} + \sum_{i = 1}^{N} ω_{z} {(\frac{x_{E i}}{x_{E p}} - 1)}^{2} s . t . E > 0, r_{E i} \geq 0$

si27_e (3)

The first term summates the estimation errors of E for all buses and all time points. The second and third terms respectively summate normalized differences in r_Ei and x_Ei between their estimates for the current and previous time window (“p” indicates the estimate from the previous time window). ω_e and ω_z are weighting factors, respectively, for variances of E and ${\bar{z}}_{Ei}$ ${\bar{z}}_{Ei}$ over the time window. For instance, if the network topology of the external system does not change, ${\bar{z}}_{Ei}$ ${\bar{z}}_{Ei}$ will be constant. Thus, there should be ω_z > ω_e to allow more changes in E. The sequential quadratic programming (SQP) method [27] is used to solve the optimization problem in Eq. (3).

The above optimization problem for the external system is actually a nonconvex problem, so it needs to select good initial values for the external system parameters. As long as the initial values are in a neighborhood of the optimum that is considered as the true solution, the SQP method can make sure to converge to that solution. However, the problem does have multiple local optima. In practice, the initial values of external system parameters for optimization are determined as follows: at the beginning when the new measurement-based method is performed or whenever a major disturbance, for example, a line outage and a generator outage, is detected on the external system, the least square method is applied to the K data points of the current time window to estimate the parameters as new initial values; otherwise, the new method selects initial values from the optimization results of the previous time window. Because of the nonconvex nature of this optimization problem, if the initial values selected at the beginning are far from the true solution, the optimization may converge to a different solution with errors in parameter estimation and, consequently, cause inaccurate transfer limits at the end. Those errors may last until the next time the least square method is performed to recreate new initial values. Later in a case study on the NPCC system, the results from different initial values with errors intentionally added are compared. In practice, the least square method may be performed at a certain frequency, for example, every a few minutes, even if a major disturbance is not detected.

2.2 Identification of load area parameters

Load area parameters include load impedance ${\bar{z}}_{ii}$ ${\bar{z}}_{ii}$ and transfer impedance ${\bar{z}}_{ij}$ ${\bar{z}}_{ij}$ (i and j = 1, …, N), whose admittances are ${\bar{y}}_{ii}$ ${\bar{y}}_{ii}$ and ${\bar{y}}_{ij}$ ${\bar{y}}_{ij}$ . In each time window, assume constant ${\bar{y}}_{ij}$ ${\bar{y}}_{ij}$ if there is no topology change in the load area, and allow ${\bar{y}}_{ii}$ ${\bar{y}}_{ii}$ to change. From power flow equations,

${\bar{y}}_{ii} (k) = \frac{{\bar{S}}_{i}^{*} (k) - \sum_{j = 1, \dots, N, j \neq i} [V_{i}^{2} (k) - {\bar{V}}_{i}^{*} (k) {\bar{V}}_{j} (k)] {\bar{y}}_{ij}}{V_{i}^{2} (k)}$ ${\bar{y}}_{ii} (k) = \frac{{\bar{S}}_{i}^{*} (k) - \sum_{j = 1, \dots, N, j \neq i} [V_{i}^{2} (k) - {\bar{V}}_{i}^{*} (k) {\bar{V}}_{j} (k)] {\bar{y}}_{ij}}{V_{i}^{2} (k)}$

si36_e (4)

A window of K data points has NK values of ${\bar{y}}_{ii}$ ${\bar{y}}_{ii}$ and N(N − 1)/2 values of ${\bar{y}}_{ij}$ ${\bar{y}}_{ij}$ to be estimated. Thus, there are totally N² − N + 2NK real parameters to estimate. K data points can provide 2NK nodal power injection equations in the real realm. Because N² − N + 2NK > 2NK, there are insufficient equations to solve all parameters. In each window, if we may assume a constant power factor for each ${\bar{y}}_{ii}$ ${\bar{y}}_{ii}$ , its conductance g_ii, and susceptance b_ii of ${\bar{y}}_{ii}$ ${\bar{y}}_{ii}$ at two adjacent time points will satisfy

$\frac{g_{i i} (k - 1)}{b_{i i} (k - 1)} = \frac{g_{i i} (k)}{b_{i i} (k)}, (k = 2 ~ K)$ $\frac{g_{i i} (k - 1)}{b_{i i} (k - 1)} = \frac{g_{i i} (k)}{b_{i i} (k)}, (k = 2 ~ K)$

si41_e (5)

Thus, K − 1 more equations are added to each bus and the entire load area needs to solve 3NK − N equations. From Eq. (6), there is K = N.

$N^{2} - N + 2 NK = 3 NK - N$ $N^{2} - N + 2 NK = 3 NK - N$

(6)

It means that each time window needs to have at least N data points to be able to solve all parameters of the load area. For instance, for load areas with N = 2, 3, and 4 boundary buses, we need at least the same numbers of data points to solve 10, 24, and 44 unknowns, respectively.

From Eq. (4), g_ii and b_ii are both functions of ${\bar{y}}_{ij} ’ s$ ${\bar{y}}_{ij} ’ s$ (i ≠ j). An error index on Eq. (5) is defined as

$e_{i}^{in} (k) = g_{ii} (k - 1) b_{ii} (k) - g_{ii} (k) b_{ii} (k - 1)$ $e_{i}^{in} (k) = g_{ii} (k - 1) b_{ii} (k) - g_{ii} (k) b_{ii} (k - 1)$

(7)

The second optimization problem is formulated as Eq. (8) for estimating load area parameters by minimizing three weighted summation terms. g_ij and b_ij are the estimated conductance and susceptance of ${\bar{y}}_{ij}$ ${\bar{y}}_{ij}$ ; the second and third terms, respectively, summate their normalized differences from the previous time window (“p” indicates the estimate from the previous time window); ω_pf and ω_y are the weighting factors, respectively, for variances of the power factor and ${\bar{y}}_{ij}$ ${\bar{y}}_{ij}$ over the time window.

$min J^{in} = \sum_{k = 2}^{K} \sum_{i = 1}^{N} \frac{ω_{pf}}{N} {[e_{i}^{in} (k)]}^{2} + \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{N} ω_{y} {(\frac{g_{ij}}{g_{ijp}} - 1)}^{2} + \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{N} ω_{y} {(\frac{b_{ij}}{b_{ijp}} - 1)}^{2} s . t . g_{ij} > 0$ $min J^{in} = \sum_{k = 2}^{K} \sum_{i = 1}^{N} \frac{ω_{pf}}{N} {[e_{i}^{in} (k)]}^{2} + \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{N} ω_{y} {(\frac{g_{ij}}{g_{ijp}} - 1)}^{2} + \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{N} ω_{y} {(\frac{b_{ij}}{b_{ijp}} - 1)}^{2} s . t . g_{ij} > 0$

si47_e (8)

The SQP method is also used to solve all ${\bar{y}}_{ij}^{’} s$ ${\bar{y}}_{ij}^{’} s$ for each time window. Then, calculate load admittance ${\bar{y}}_{ii}$ ${\bar{y}}_{ii}$ directly by Eq. (4).

For this second optimization on the estimation of load area parameters, the initial values of g_ij and b_ij are also required, which can be obtained directly from the reduced bus admittance matrix about the load area by eliminating all buses except boundary buses.

2.3 Solving the power transfer limit of each tie line

From real-time estimates of equivalent parameters, the active power transfer limit of each tie line can be solved analytically as a function of equivalent parameters.

The admittance matrix of the load area of the equivalent system is given by

$Y = [\begin{array}{c} {\bar{Y}}_{11} & \dots & {\bar{Y}}_{1 i} & \dots & {\bar{Y}}_{1 N} \\ ⋮ & ⋱ & ⋮ \\ {\bar{Y}}_{i 1} & {\bar{Y}}_{ii} & {\bar{Y}}_{iN} \\ ⋮ & ⋱ & ⋮ \\ {\bar{Y}}_{N 1} & \dots & {\bar{Y}}_{Ni} & \dots & {\bar{Y}}_{NN} \end{array}]$ $Y = [\begin{array}{c} {\bar{Y}}_{11} & \dots & {\bar{Y}}_{1 i} & \dots & {\bar{Y}}_{1 N} \\ ⋮ & ⋱ & ⋮ \\ {\bar{Y}}_{i 1} & {\bar{Y}}_{ii} & {\bar{Y}}_{iN} \\ ⋮ & ⋱ & ⋮ \\ {\bar{Y}}_{N 1} & \dots & {\bar{Y}}_{Ni} & \dots & {\bar{Y}}_{NN} \end{array}]$

si50_e (9)

where ${\bar{Y}}_{ij} = {\bar{Y}}_{ji} = - {\bar{y}}_{ij}$ ${\bar{Y}}_{ij} = {\bar{Y}}_{ji} = - {\bar{y}}_{ij}$ and ${\bar{Y}}_{ii} = \sum_{j = 1}^{N} {\bar{y}}_{ij}$ ${\bar{Y}}_{ii} = \sum_{j = 1}^{N} {\bar{y}}_{ij}$ .

Let Y_E be a column vector about all admittances ${\bar{y}}_{Ei} = 1 / {\bar{z}}_{Ei}$ ${\bar{y}}_{Ei} = 1 / {\bar{z}}_{Ei}$ (i = 1, …, N).

$Y_{E} = {[{\bar{y}}_{E 1} {\bar{y}}_{E 2} \dots {\bar{y}}_{EN}]}^{T}$ $Y_{E} = {[{\bar{y}}_{E 1} {\bar{y}}_{E 2} \dots {\bar{y}}_{EN}]}^{T}$

(10)

Use a superscript “D” to indicate a diagonal matrix whose elements are from a vector, that is,

$Y_{E}^{D} = [\begin{array}{c} {\bar{y}}_{E 1} & 0 \\ ⋱ \\ 0 & {\bar{y}}_{EN} \end{array}]$ $Y_{E}^{D} = [\begin{array}{c} {\bar{y}}_{E 1} & 0 \\ ⋱ \\ 0 & {\bar{y}}_{EN} \end{array}]$

si55_e (11)

A vector about the injected currents satisfies Eq. (12), where $V = {[\begin{array}{c} {\bar{V}}_{1} & {\bar{V}}_{2} & \dots & {\bar{V}}_{N} \end{array}]}^{T}$ $V = {[\begin{array}{c} {\bar{V}}_{1} & {\bar{V}}_{2} & \dots & {\bar{V}}_{N} \end{array}]}^{T}$ .

$\{\begin{cases} I = VY \\ I = \bar{E} Y_{E} - Y_{E}^{D} V \end{cases}$ $\{\begin{cases} I = VY \\ I = \bar{E} Y_{E} - Y_{E}^{D} V \end{cases}$

si57_e (12)

Then, there is

$V = \bar{E} {(Y + Y_{E}^{D})}^{- 1} Y_{E} = \bar{E} \frac{adj (Y + Y_{E}^{D})}{det (Y + Y_{E}^{D})} Y_{E}$ $V = \bar{E} {(Y + Y_{E}^{D})}^{- 1} Y_{E} = \bar{E} \frac{adj (Y + Y_{E}^{D})}{det (Y + Y_{E}^{D})} Y_{E}$

si58_e (13)

For simplicity, let α_i denote the ith element of adj(Y + Y_E^D)Y_E and let γ = det(Y + Y_E^D). There is

$\bar{V_{i}} = \frac{\bar{E} α_{i}}{γ}$ $\bar{V_{i}} = \frac{\bar{E} α_{i}}{γ}$

si59_e (14)

The complex power transferred on each tie line is a function of elements of Y, $\bar{E}$ $\bar{E}$ and Y_E.

$S = {[\begin{array}{c} {\bar{S}}_{1} & {\bar{S}}_{2} & \dots & {\bar{S}}_{N} \end{array}]}^{T} = {(I^{*})}^{D} \times V = {(\bar{E} Y_{E}^{D} - V^{D} Y_{E}^{D})}^{*} V$ $S = {[\begin{array}{c} {\bar{S}}_{1} & {\bar{S}}_{2} & \dots & {\bar{S}}_{N} \end{array}]}^{T} = {(I^{*})}^{D} \times V = {(\bar{E} Y_{E}^{D} - V^{D} Y_{E}^{D})}^{*} V$

(15)

If changes on the external system are ignored, each complex power is a function of load admittances, that is,

${\bar{S}}_{i} ({\bar{y}}_{11} \dots {\bar{y}}_{NN}) = ({\bar{E}}^{*} {\bar{y}}_{Ei}^{*} - {\bar{y}}_{Ei}^{*} {\bar{V}}_{i}^{*}) {\bar{V}}_{i} = {|\bar{E}|}^{2} ({\bar{y}}_{Ei}^{*} - {\bar{y}}_{Ei}^{*} \frac{α_{i}^{*}}{γ^{*}}) \frac{α_{i}}{γ}$ ${\bar{S}}_{i} ({\bar{y}}_{11} \dots {\bar{y}}_{NN}) = ({\bar{E}}^{*} {\bar{y}}_{Ei}^{*} - {\bar{y}}_{Ei}^{*} {\bar{V}}_{i}^{*}) {\bar{V}}_{i} = {|\bar{E}|}^{2} ({\bar{y}}_{Ei}^{*} - {\bar{y}}_{Ei}^{*} \frac{α_{i}^{*}}{γ^{*}}) \frac{α_{i}}{γ}$

si62_e (16)

$P_{i} ({\bar{y}}_{11} \dots {\bar{y}}_{NN}) = {|\bar{E}|}^{2} Re [({\bar{y}}_{Ei}^{*} - {\bar{y}}_{Ei}^{*} \frac{α_{i}^{*}}{γ^{*}}) \frac{α_{i}}{γ}]$ $P_{i} ({\bar{y}}_{11} \dots {\bar{y}}_{NN}) = {|\bar{E}|}^{2} Re [({\bar{y}}_{Ei}^{*} - {\bar{y}}_{Ei}^{*} \frac{α_{i}^{*}}{γ^{*}}) \frac{α_{i}}{γ}]$

si63_e (17)

Based on the aforementioned constant power factor assumption over a time window for each load, P_i is a function of all load admittance magnitudes, that is, y_jj (j = 1, …, N). Its maximum P_{i, j}^Max with respect to the change of y_jj at bus j is reached when

$\frac{\partial P_{i} (y_{11} \dots y_{N N})}{\partial y_{j j}} = 0 i, j = 1 . . . N$ $\frac{\partial P_{i} (y_{11} \dots y_{N N})}{\partial y_{j j}} = 0 i, j = 1 . . . N$

si64_e (18)

If an analytical solution of y_jj is obtained from Eq. (18) as a function of equivalent parameters, an analytical expression of P_{i, j}^Max can be derived by plugging the solution of y_jj into Eq. (17). P_{i, i}^Max with i = j represents the maximum active power transfer to bus i with only the local load at bus i varying; P_{i, j}^Max with j ≠ i represents the maximum active power transfer to bus i with only the load at another bus j varying.

For a general N + 1 buses equivalent, the analytical solution of P_{i, j}^Max can be obtained by solving a quadratic equation. The proof based on Eq. (18) is given in the Appendix of [26].

This paper assumes that each y_jj may change individually, so there are N such transfer limits P_{i, 1}^Max ⋯ P_{i, N}^Max for each tie line. By estimating the real-time pattern of load changes, the limit that best matches that pattern will be more accurate and selected. The voltage stability margin on a tie line is defined as the difference between the limit and the real-time power transfer.

3 Online scheme for implementation

Compared with the traditional TE-based approach, this new measurement-based method uses a more complex N + 1 buses equivalent to model more details about the boundary of a load area. Synchronized measurements on all boundary buses are needed from either state estimation results if only steady-state voltage instability is concerned or PMUs for real-time detection of voltage instability. The parameters of the equivalent are identified using real-time measurements over a recent time window for several seconds. In a later case study on the NPCC system, results from 5 to 10 s time windows will be compared. If the scheme uses PMUs in order to speed up online parameter identification and filter out noise or dynamics irrelevant to voltage stability, the original high-resolution measurements (e.g., at 30 Hz) may be downgraded to a low-sampling rate f_s (e.g., 2 Hz) by averaging the raw data over a time internal of 1/f_s (i.e., 0.5 s for 2 Hz).

As shown by the flowchart in Fig. 3, the new method first identifies all branches connecting the load area with the rest of the system, which comprise a cut set partitioning the load area. Those branches are assumed to be coming from the same voltage source $\bar{E}$ $\bar{E}$ . Then, any branches coming to the same boundary buses are merged. Assume that M branches are yielded. The proposed method is able to calculate the transfer limit for each branch using an M + 1 buses equivalent. However, if M is large, it will result in huge computational burdens in estimating M(M − 1)/2 + 2M + 1 parameters and consequently calculating M limits. Different from the TE that merges all M branches to one fictitious tie line, this new approach may group some branches across the boundary and only merge each group to one fictitious tie line. The criteria of grouping are: the boundary buses in one group are tightly interconnected; the branches in one group reach limits almost at the same time; and it is not required to monitor the branches within a group individually. For the fictitious tie line representing a group of branches, only its total limit is calculated. Thus, after merging some groups of branches to fictitious tie lines, the final number of tie lines becomes N < M. A simpler N + 1 buses equivalent is used, which still keeps characteristics of the load area regarding voltage stability. The TE-based approach is a special case of this new method with N = 1.

The next two sections will test the new method on a four-bus power system and the NPCC 140-bus system using data generated from simulation. All computations involved in the algorithms of the new method are performed in MATLAB on an Intel Core i7 CPU desktop computer.

4 Demonstration on a four-bus power system

This section demonstrates the new method on a four-bus power system with one constant voltage source supporting three interconnected load buses representing a load area. The system is simulated in MATLAB with gradual load increases at three load buses. The simulation results on three load buses are treated as PMU data and fed to the new method. The system represents a special case of the system in Fig. 1 with N = 3. Let $\bar{E} = 1.0 ∠ 5^{\circ} pu$ $\bar{E} = 1.0 ∠ 5^{\circ} pu$ and three tie lines have the same impedance ${\bar{z}}_{E 1} = {\bar{z}}_{E 2} = {\bar{z}}_{E 3} = 0.01 + j 0.1 pu$ ${\bar{z}}_{E 1} = {\bar{z}}_{E 2} = {\bar{z}}_{E 3} = 0.01 + j 0.1 pu$ . At the beginning, three load impedances ${\bar{z}}_{11} = {\bar{z}}_{22} = {\bar{z}}_{33} = 1 + j 1 pu$ ${\bar{z}}_{11} = {\bar{z}}_{22} = {\bar{z}}_{33} = 1 + j 1 pu$ . Consider two sets of transfer impedances in Table 1, respectively, for weak and tight connections between three boundary buses.

Table 1

Values of transfer impedances
Set	${\bar{z}}_{12}$ ${\bar{z}}_{12}$ (pu)	${\bar{z}}_{13}$ ${\bar{z}}_{13}$ (pu)	${\bar{z}}_{23}$ ${\bar{z}}_{23}$ (pu)
A	0.01 + j0.1	0.015 + j0.15	0.005 + j0.05
B	0.0005 + j0.005	0.0008 + j0.0075	0.0003 + j0.0025

Table 1

Keep the impedance angle of ${\bar{z}}_{33}$ ${\bar{z}}_{33}$ unchanged but gradually reduce its modulus by 1% every 2 s until active powers on all tie lines meet limits. As shown in Fig. 4, three PV (power-voltage) curves (P_i vs V_i) are drawn for two groups of transfer impedances. For Set A, the transfer impedances between boundary buses are not ignorable compared with the tie line and load impedances, so three PV curves are distinct. However, when those impedances decrease to the values in Set B, three PV curves basically coincide, which is the case where a TE can be applied.

Fig. 4 PV curves for weak and tight connections between boundary buses. (A) Set A (weak); (B) Set B (tight).

Simulation results are recorded at a 1 Hz sampling rate. The new method is performed every 1 s using the data of the latest 10 s time window. All computations of the new method on each time window are finished within 0.05 s in MATLAB. The new method gives each tie line three limits for three extreme load increase assumptions. For two sets of impedances, Fig. 5 shows P₁ to P₃, the total tie line flow P_Σ = P₁ + P₂ + P₃, and their limits calculated by the new method. The limits from solutions of ∂ P₁/∂ y₃₃ to ∂ P₃/∂ y₃₃ match the actual load increase, so their sum is defined as the total tie line flow limit P_Σ^Max(New). For comparison, the TE-based method in [20,21] is also performed to give the total tie line flow limit as P_Σ^Max(TE) in Fig. 5.

Tests on those two groups of data show that when a TE-based method is applied to a load area, voltage instability is detected only when the total tie line flow meets its limit. However, due to the uneven increase of load, one tie line may be stressed more to reach its limit earlier than the others, which can successfully be detected by the new method. Another observation on Fig. 5 is that the curve of P_Σ^Max(New) is flatter than that of P_Σ^Max(TE), and P_Σ^Max(TE) is more optimistic and less accurate than P_Σ^Max(New) when the system has a distance to voltage instability.

For Set A with a weak interconnection between boundary buses, Fig. 5A shows that three individual tie line flows P₁, P₂, and P₃ meet their limits at t = 680 s, 676 s, and 666 s, respectively. The total tie line flow P_Σ meets P_Σ^Max(New) at t = 680 s. P_Σ^Max(TE) from the TE-based method is not as flat as P_Σ^Max(New), and it is met by P_Σ after t = 700 s. Fig. 6 gives details of Fig. 5A on each tie line flow and its three limits, calculated according to Eq. (18) for three load increase assumptions. The limit from the solution of ∂ P_i/∂ y₃₃ matches the actual load increase, and its curve is flat and met by P_i earlier than the other two. By using the new method, zero margin is first detected at t = 666 s on the third tie line and then on the other two tie lines as well as the total tie line flow. However, the TE-based method detects zero margin much later because, first, it only monitors the total tie line flow limit and second, the limit curve is not as flat as that given by the new method.

Fig. 6 Line flows and limits for Set A. (A) P₁ and limits; (B) P₂ and limits; (C) P₃ and limits.

For Set B with a tight connection between boundary buses, Fig. 5B shows that all tie line flows reach their limits at t = 732 s. Also, P_Σ meets both P_Σ^Max(New) and P_Σ^Max(TE) at that same time.

5 Case studies on the NPCC test system

The proposed new method is tested on a NPCC 48-machine, 140-bus system model [28]. As highlighted in Fig. 7, the system has a Connecticut Load Center (CLC) area supported by power from three tie lines, that is, 73-35, 30-31, and 6-5. Line 73-35 is from the NYISO region and the other two are from north of the ISO-NE region. Powertech's TSAT is used to simulate four voltage instability scenarios about the CLC area:

• Scenario A: Generator trip followed by load increase leading to voltage instability.
• Scenario B: Generator trip followed by a tie line tip causing voltage instability.
• Scenario C: Two successive tie line trips causing voltage instability
• Scenario D: Shunt switching to postpone voltage instability.

The load model at each load bus adopts the default load model setting in TSAT, that is, 100% constant current for real power load and 100% constant impedance for reactive power load. Simulation results on the voltages at boundary buses 35, 31, and 5 and the complex powers of the three tie lines are recorded at 30 Hz, that is, the typical PMU sampling rate. The raw data are preprocessed by an averaging filter over 15 samples to be downgraded to 2 Hz. The processed data are then fed to the new method for estimating the external and load area parameters and calculating transfer limits. That data preprocessing improves the efficiency of two optimizations for parameter estimation while keeping the necessary dynamics on voltage stability in the data. The new method is performed every 0.5 s on data of the latest 5 s time window.

5.1 Scenario A: Generator trip followed by load increase leading to voltage instability

To create a voltage collapse in the CLC area, all its loads are uniformly increased by a total of 0.42 MW per second from its original load of 1906.5 MW with constant load power factors. At t = 360 s, the generator on bus 21 is tripped, which pushes the system to be close to the voltage stability limit. Shortly after a slight load increase, voltage collapse happens around t = 530 s, as shown in Fig. 8 on three boundary bus voltages. Fig. 9 indicates the PV curves monitored at three boundary buses. To better illustrate the PV curves, the figure is drawn using the data sampled at 25 s intervals until t = 500 s to filter out transient dynamics on the curves right after the generator trip and the voltage collapse at the end. Note that the generator trip causes a transition from the precontingency PV curves to the postcontingency PV curves with a more critical condition of voltage stability. Bus 35 is the most critical bus because the “nose point” of its postcontingency PV curve is passed.

Fig. 8 Voltage magnitudes at CLC boundary buses (Scenario A).

Fig. 9 PV curves monitored at the CLC boundary buses.

When the external system has strong coherency, the proposed new method can be applied to reduce it to one voltage phasor $\bar{E}$ $\bar{E}$ connected with boundary buses of the load area by N branches, respectively. Fig. 10 shows the voltage magnitudes and angles of all buses in Fig. 7B that are outside the CLC load center, indicating a strong coherency of the external system. Therefore, the proposed method is valid and may adopt a 3 + 1 buses equivalent to calculate the power transfer limits separately for three tie lines.

Fig. 10 External system. (A) Bus magnitudes; (B) bus angles.

Fig. 11 gives estimates of three load impedance magnitudes seen at the boundary buses. The figure shows that the load seen from bus 5 changes more significantly than the others. Hence, P_{35, 5}^Max, P_{31, 5}^Max and P_{5, 5}^Max calculated from ∂ P_i/∂ y₅ = 0 are more accurate and used to calculate the stability margin.

To compare with the new method, Fig. 12 gives the total transfer limit estimated using a TE-based method. The margin stays positive until the final collapse of the entire system.

Fig. 13 gives the results from the new method and each tie line has three transfer limits. Before the generator trip, all lines have sufficient margins to their limits. After the trip, more power is needed from the external system, so the active powers of the three tie lines all increase significantly to approach their limits. In Fig. 13A, P₃₅ of 73-35 reaches the limit P_{35, 5}^Max at t = 473.5 s. From Fig. 13B and C, the other two lines keep positive margins until the final voltage collapse. It confirms the observation from Fig. 9 that voltage collapse will initiate from bus 35. If the limit and margin information on an individual tie line is displayed for operators in real time, an early remedial action may be taken before voltage collapse. However, such information is not available from a TE-based method.

Once parameters of the N + 1 buses equivalent are estimated, transfer limits are directly calculated by their analytical expressions. The major time cost of the new method is with online parameter estimation. Fig. 14 gives the probability density about the times spent respectively on estimating the external system and the load area over one time window. According to the figure, parameter estimation over one time window can be accomplished within 0.02–0.1 s. The average total time cost for each cycle of the new method's online procedure (i.e., steps 2–5 in Fig. 3) is 0.0614 s, which includes 0.0221 s for external parameter estimation, 0.0271 s for load area parameter estimation, and 0.0122 s for transfer limits calculation. The times on measurements input and margin display are ignorable. The test results indicate that the new method can be applied in an online environment.

Fig. 14 Time performances on online parameter estimation. (A) Time for estimating external system parameters; (B) time for estimating load area parameters.

Fig. 15 compares the estimated P_{35, 5}^Max using a 5 s sliding time window with that using a 10 s sliding window. The two results match very well, which indicates that the new method is not very sensitive to the length of the sliding time window.

To test the results of the new method using inaccurate initial values in the external system parameter estimation, Fig. 16 compares the tie line power limits for 73-35 with three different sets of initial values: 110%, 100%, and 90% of the estimates from a least square method on the first time window at the beginning. From the figure, a 10% error will cause a less than 5% error in the transfer limit estimation before the generator trip and about a 2% error in the limit after the generator trip. If the least square method is used to reestimate the initial values when the generator trip is detected, that 2% error can be eliminated and these three limits will merge to one limit associated with accurate initial values. Considering that in the real world, small errors in the initial values cannot be avoided completely, a small positive threshold rather than zero may be defined for the transfer margin as an alarm of voltage instability.

5.2 Scenario B: Generator trip followed by a tie line tip causing voltage instability

Before t = 400 s, this scenario is the same as Scenario A. At t = 400 s, tie line 73-35 is tripped to cause immediate voltage collapse. The voltage magnitudes of three boundary buses are shown in Fig. 17.

Fig. 17 Voltage magnitudes at CLC boundary buses (Scenario B).

Fig. 18 shows the tie line power flows and their limits for this scenario. After the generator trip, all tie line flows become closer to their limits, and tie line 73-35 carries 795 MW, which is higher than the total margin of 667 MW on tie lines 31-30 and 6-5. When tie line 73-35 is tripped at t = 400 s, its flow is transferred to the other two tie lines to cause them to meet limits. That explains why voltage collapse happens following that tie line trip. If the above tie line margin information is presented to the system operator before t = 400 s, the operator will be aware that the system following the generator trip cannot endure such a single tie line trip contingency and may take a control action.

5.3 Scenario C: Two successive tie line trips causing voltage instability

In this scenario, two successive tie line trips on 31-30 and 6-5 are simulated to test the adaptability of the new method to “N − 1” and “N − 2” conditions. During t = 0–100 s, all loads in the CLC area are uniformly increased by 0.43 MW/s from its original load of 1906.5 MW with constant power factors unchanged. At t = 100 s, the tie line 31-30 is tripped, and thus the voltages of three boundary buses drop significantly. During t = 100–200 s, loads keep increasing at a lower speed equal to 0.37 MW/s. At t = 200 s, the tie line 6-5 is tripped, causing voltage collapse as shown in Fig. 19.

Fig. 19 Voltage magnitudes at CLC boundary buses (Scenario C).

Fig. 20 gives the results on each tie line and the transfer limits. Before tie line 31-30 is tripped at t = 100 s, all tie lines have sufficient transfer margin. After that trip, P₃₁ = 0 is captured from measurements and the new method sets the corresponding y_E31 = 0 to adapt to the new “N − 1” condition. Then, the transfer limit on tie line 73-35 drops to 677 MW. Before the next tie line 6-5 is tripped at t = 200 s, tie lines 73-35 and 6-5 totally transfer 733 MW to the CLC area, which is higher than the limit 677 MW of tie line 73-35. Therefore, the second tie line trip causes zero margin on tie line 73-35, followed by a voltage collapse.

5.4 Scenario D: Shunt switching to postpone voltage instability

This scenario considers switching in a capacitor bank located in the CLC area to postpone voltage collapse. Everything of this scenario during t = 0–440 s is the same as Scenario A. At t = 440 s when the transfer margin on the most critical tie line 73-35 drops to 3%, a 50 MVAR capacitor bank at bus 33 is switched in. Due to its additional VAR support, the voltage of the CLC area increases, as shown in Fig. 21, about voltage magnitudes of three boundary buses. Fig. 22 shows the transfer limits on tie line 73-35 for this scenario. Both the tie line flow and limits increase after that switch. The slight tie line flow increase is caused by voltage-sensitive loads in the load area, which is smaller than the increase of the limit. Therefore, zero margin happens at t = 496.5 s, that is, 23 s later than the 473.5 s of Scenario A. This scenario demonstrates that adding VAR support in the load area will increase the voltage stability margin, which is correctly captured by the new method.

Fig. 21 Voltage magnitudes at CLC boundary buses (Scenario D).

6 Discussion and conclusions

This article has introduced a new measurement-based method proposed in [26] for real-time voltage stability monitoring of a load area fed by multiple tie lines. The new method is based on an N + 1 buses equivalent system whose parameters are estimated directly from synchronized measurements obtained at the boundary buses of the load area. For each tie line, the method calculates the transfer limit and margin against voltage instability analytically from that estimated equivalent system. The new method has been demonstrated in detail on a four-bus system and then tested by case studies on a 140-bus NPCC system model.

Compared to a traditional TE-based method for measurement-based voltage stability monitoring, the new method has two apparent advantages. First, the new method offers detailed limit and margin information on individual tie lines so as to identify the tie line and boundary bus with the smallest margin as the location where voltage instability more likely initiates. Second, as demonstrated on the four-bus system, before the voltage collapse point, the total tie line flow limit from the new method is more accurate than the limit from the TE-based method. The latter fluctuates more and is not as flat as the former because the TE-based method does not model the weak or strong connection between boundary buses. The above second advantage makes the new method more suitable for online monitoring and early warning of voltage instability, and the first advantage can help the system operator to identify the location where voltage instability more likely initiates and accordingly choose more effective control resources, for example, those having shorter electrical distances to the tie line with the smallest margin. Recent studies in [29] demonstrate that voltage stability margin information for an anticipated “n − 1” condition can also be provided by sensitivity analysis studies on the N + 1 buses equivalent.

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Table of Contents for 14: Measurement-based voltage stability monitoring for load areas

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