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Determining the Most Dangerous Direction of Power System Stressing Michael Vaiman, Senior Member, IEEE1, Marianna Vaiman, Senior Member, IEEE1 Papic & Ciniglio
Abstract— This paper presents an effective practical approach for identifying the most dangerous direction during power system stressing. To determine the most dangerous direction of system stressing, the value of voltage stability margin should be computed. The most dangerous direction of stressing is the direction along which voltage stability margin is the smallest. Thus, the use of the proposed approach allows us to maximize voltage stability margin. The approach is based on incorporating phase angle variation in steady-state stability criterion for power flow equations, and uses PV-curves and QV-curves together with phase angle variation as one combined voltage stability criterion. Thus, Pδ-curves and Qδ-curves (where δ is phase angle) are included as a part of voltage stability analysis in addition to PV-curves and QV-curves analyses. The proposed approach includes analysis of surfaces δ(P,Q) and V(P,Q). Index Terms— Phase angle, voltage stability analysis, power system margin, stressing direction, power transfer analysis
I. INTRODUCTION Real-time tools, such as real-time contingency analysis (RTCA) [1], have been an essential part of utilities’ operations. Usually, system response to credible contingencies is continuously monitored every 5 minutes in RTCA and actions are taken when RTCA shows unacceptable system performance. However, RTCA can only assess system performance under current operating conditions and is unable to assess post-contingency system performance at a higher system stress level beyond its current operating state. Analyzing system performance at higher stress levels is of great importance for operations planning because impending system vulnerabilities such as voltage collapse can be identified in advance, and operators can adjust the system and develop operating plans to respond adequately to potential system vulnerabilities. Therefore, utilities, ISOs and regional organizations are implementing real-time voltage stability analysis as an integral part of their real-time situational awareness, [2 – 4]. Voltage stability is one of the fundamental concepts of power system analyses, [5 – 10]. Physical (thermal and voltage constrains) and operational (voltage stability) margins are usually determined during contingency analysis, [11, 12]. Operational margin, i.e. closeness of the system to voltage instability, is defined based on the convergence of the Newton method to an equilibrium 1
Michael Vaiman and Marianna Vaiman are with V&R Energy Systems Research, Los Angeles, CA, USA (email:
[email protected],
[email protected]).
point. Conditions when voltage stability can be associated with convergence of the Newton method are explained in [13]. One of the well-known methods for voltage stability analysis is continuation power flow, [14, 15]. Though it has been widely employed, it has a number of disadvantages, including possible premature divergence and inability to identify weak elements that contribute to collapse. Another effective approach to voltage stability analysis is based on optimization techniques, which offer convergence robustness, [16]. However, such approaches are not used practically as optimization-based algorithms are computationally prohibitive, [17, 18]. Various system-wide and localized voltage stability indices have been also proposed to assess voltage instability. The system-wide indices are based on the power flow calculations, [19, 20], and have similar limitations to continuation power flow technique. Localized indices focus on individual buses and lines, and usually are not as accurate, [21, 22]. PV-curve and QV-curve analyses are the most commonly used techniques for voltage stability assessment, and are a part of current utility practice for analyzing voltage stability. However, traditional PV-curves analysis does not yield useful results under certain conditions when voltage remains almost constant followed by an uncontrollable decline. Based on the past system performance and engineering judgment, planners and operators address this problem by setting some arbitrary voltage stability margins within which the system operation is considered to be secure (e.g., 5%, 10%). To overcome the limitations of the traditional voltage stability analysis, a new voltage stability criterion was introduced in [23, 24]. The approach incorporates phase angle variation into voltage stability criterion during system stressing. It uses PV-curves and QV-curves together with phase angle variation as one combined voltage stability criterion. Thus, Pδ-curves and Qδ-curves (where δ is phase angle) are included as a part of voltage stability analysis in addition to PV-curves and QV-curves. This criterion allows to more accurately perform voltage stability analysis of a bulk power system. The remainder of this paper is divided into five sections. Section II is devoted to a new voltage stability criterion. A concept of the most dangerous direction of the system stressing is presented in Section III. Section IV describes a procedure to determine the most dangerous direction. Section V presents the results of computing the most dangerous direction. Section VI summarizes the authors’ conclusions. II. A NEW COMBINED VOLTAGE STABILITY CRITERION Steady-state voltage stability analysis is based on solution of power flow equations. Voltages and angles are state
2
variables, while real power, reactive power or total power, are stressing parameters. Power system stressing includes any variation of real and reactive power, such as real and/or reactive components of load; real and/or reactive generator power output, power transfer through and interface. When power system is stressed, the stressing parameter is increased until voltage instability. Some equations for real and/or reactive power, and therefore the system of algebraic equations, become incompatible at the level at which voltage instability occurs. When power flow equations become incompatible, voltages experience uncontrollable decline and angles (or angle differences) experience uncontrollable change at the same value of the stressing parameter , [23, 24]. Therefore, voltage instability may be predicted based on variation of voltages and phase angles (or phase angle differences). Voltage magnitude and phase angle are equal indicators of proximity to voltage collapse, [23, 24]. A curve with a steeper slope (PV or Pδ, and QV or Qδ) is more effective for the use in voltage stability analysis. Fig. 1 shows a PV-curve (green) and a Pδ-curve (blue). From PV-curve and Pδ-curve analyses it follows that the decline in voltage and increase in absolute value of phase angle occur at the same stressing level of 4560 MW. While voltage changes insignificantly, angle at the same bus experiences a significant change, and Pδ-curve has a steeper slope. Therefore, in this case, it is more effective to monitor angle for voltage stability analysis.
power system may be operated within a region bounded by one of the following three-dimensional surfaces, [23, 24]: • Surface V(P,Q) The power system may be operated until the Voltage Collapse Line is reached. This is the line beyond which an uncontrollable decline of voltage magnitude occurs. • Surface δ(P,Q) The power system may be operated until the Line of Uncontrollable Angle Change is reached. This is the line beyond which an irreversible change of phase angle occurs. Fig. 2 illustrates a surface V(P,Q) that represents variation of voltage magnitude (V, p.u.) as a function of real power (P, MW) and reactive power (Q, MVAr). Relationship between the Voltage Collapse Line and equal level lines of surface V(P,Q) is weak. Therefore, voltage collapse can’t be predicted based on the change in voltage magnitude.
Fig. 2. Surface V(P,Q) and the Voltage Collapse Line.
Fig. 3 illustrates a three-dimensional surface δ(P,Q) that represents variation of phase angle (δ, deg.) as a function of real power (P, MW) and reactive power (Q, MVAr).
Fig. 1. PV-Curve and Pδ- Curve Plotted for a Bus in a 4911-bus System.
When stressing a power system with different power factors, a three-dimensional surface V(P,Q) or δ(P,Q) should be used, see Fig. 2 and Fig. 3, respectively. PV-curves, QV-curves, and Pδ-curves and Qδ-curves represent respective sections of surface V(P,Q) in three-dimensional (VPQ) space and surface δ(P,Q) in three-dimensional (δPQ) space. Thus, a
Fig. 3. Surface δ(P,Q) and the Line of Uncontrollable Angle Change.
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Fig. 3 shows that the Line of Uncontrollable Angle Change is parallel to the equal level lines of phase angle. Thus, it is possible to predict voltage collapse based on the change in phase angle. From Fig. 2 and Fig. 3 it follows that approaching voltage collapse is visible on the surface of phase angle, so the traditional PV-curves and QV-curves would not predict voltage collapse in this case. III. A CONCEPT OF THE MOST DANGEROUS DIRECTION OF THE SYSTEM STRESSING To determine the most dangerous direction of system stressing the value of voltage stability margin should be computed. The most dangerous direction of stressing is the direction along which voltage stability margin is the smallest. Stressing can be performed with different values of a power factor. If a major component of stressing is reactive power, then it is more effective to monitor voltage. If a major component of stressing is real power, then it is more effective to monitor phase angle or phase angle difference. Since in real time environment the power factor is unknown, it is effective to monitor both voltage and phase angle or phase angle difference, and use a three-dimensional surface V(P,Q) or δ(P,Q). Creating two tables that describe surfaces V(P,Q) and δ(P,Q) is a time consuming procedure. To simplify the use of this approach, approximation of surfaces V(P,Q) and δ(P,Q) by corresponding planes is proposed. Let variable z be either V or δ depending on which surface V(P,Q) or δ(P,Q) we are considering (see Fig. 4). Let’s make a trial step p along axis P and compute = Z(p, 0), i.e. V(P = p, Q = 0), or δ(P = p, Q = 0). Let’s make a trial step q along axis Q and compute = Z (0, q), i.e. V(P = 0, Q = q) or δ(P = 0, Q = q).
Having three points , , and for each of the surfaces V(P,Q) and δ(P,Q), we can plot approximate planes for surfaces V(P,Q) and δ(P,Q). Let d be the direction of the steepest slope. This direction is denoted as for the surface V(P,Q) and for the surface δ(P,Q), as shown in Fig. 5. We project directions and on the plane (P, Q), then connect two vectors and , and determine the resultant direction, . Vector is the most dangerous direction of power system stressing. Depending on whether variation of V or δ is larger, resultant vector is closer to vector or vector , respectively.
,
Fig. 5. Vectors
,
.
IV. A PROCEDURE TO DETERMINE THE MOST DANGEROUS DIRECTION OF THE SYSTEM STRESSING A. The Equation for Approximate Plane The equation for approximate plane may be written as: |
|
( (
)
(
)
) ,
(1)
where ( ( ,
) , ) ,
, is either voltage V or angle δ. Let
be = V(P = p, Q = q) or
Fig. 4. Planes that Approximate Surfaces V(P,Q) and δ(P,Q).
Let variable for the base case values P = P0 and Q = Q0 (e.g., initial values before the stressing is performed), be , i.e. = V(P0, Q0) or = δ(P0, Q0).
= δ(P = p, Q = q).
Values of , , and are calculated to determine approximate planes for surfaces V(P,Q) and δ(P,Q), see Fig. 6. Trial steps P = p and Q = q are chosen such that the accuracy of approximation of these surfaces does not exceed | | , where ε is the accuracy of approximation.
(2)
4
directions and , chosen based on which variable, voltage V or angle δ, changed more. | and | | are used to Differences | determine the averaged direction (see Fig. 8), where: is voltage calculated at the end of the vector
,
is the base case value, is angle calculated at the end of the vector
,
is the base case value.
Fig. 6. Values of
,
,
and
.
B. The Direction of the Steepest Slope The direction of the steepest slope (see Fig. 7) may be located in the first and the third quadrants ab (Case 1), or in the second and the fourth quadrants a’b’ (Case 2). In both Cases, the direction of the steepest slope, d or d’ should be chosen in the first quadrant. Fig. 8. The Most Dangerous Direction of System Stressing.
Secure variation of voltages is permissible within a smaller range than secure angle variation. Therefore, we introduce voltage scale factor in Eq. 5: | |
| |
|
| |
|
| |
|
|
(5)
The most dangerous direction of stressing is closer to vector or depending on which value of differences | | or | | is larger. V. COMPUTATION RESULTS
Fig. 7. The Direction of the Steepest Slope.
For Case 1 (the first and the third quadrants): (
|
|
√
|
|
√
)
(3)
If we consider the left part of Eq. 1 as a function, Eq. 3 becomes the component of a gradient vector, which allows us to determine the direction of the steepest slope.
(
)
| |
)
| |
| |.
Coordinates of the vector are = (0.997355, 0.072691) Voltage scale factor is
For Case 2 (the second and the fourth quadrants): (
The proposed approach was applied to determine the most dangerous direction of system stressing for a power flow case that contains 9013 buses. Surfaces V(P,Q) and δ(P,Q) for this example are given in Fig. 2 and Fig. 3, respectively. Coordinates of the vector are = (0.047666, 0.998863)
| |
where
is low voltage limit in p.u.;
p.u.
(4)
As a result, we determine vectors and that are used for computations described below in Section IV.C. C. Computation of the Most Dangerous Direction The resultant vector is the most dangerous direction of power system stressing. It is an averaged direction between
The most dangerous direction of system stressing is
Vector is closer to vector since contribution of the surface δ(P,Q) is more significant than contribution of the surface V(P,Q), as shown in Fig. 9.
5 [8]
[9]
[10]
[11] [12]
[13]
[14]
[15]
Fig. 9. The Most Dangerous Direction in 9013-Bus System.
VI. CONCLUSIONS The results of using an effective approach to automatically determine the most dangerous direction of the power system stressing is described in this paper. The approach is based on a novel voltage stability criterion that incorporates voltage and phase angle for predicting steady-state instability, and uses surfaces δ(P,Q) and V(P,Q). These surfaces are approximated by corresponding planes. Direction of the steepest slope for each of these surfaces is determined using an approximate plane. Resultant vector of the direction of the steepest slope for each of these surfaces is the most dangerous direction of stressing. Use of the proposed approach allows us to maximize voltage stability margins. This approach can be also used to analyze power transfer limit through two interfaces and determine the most dangerous direction while simultaneously stressing two interfaces. The approach can be further generalized for n interfaces, where n > 2. VII. REFERENCES [1]
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VIII. BIOGRAPHIES Michael Vaiman (SM’17) received his MSEE degree from Kaunas Polytechnic University, Lithuania, Ph.D. degree from Moscow University of Transportation Engineering, Russia, and D.Sc. degree from St. Petersburg Polytechnic University, Russia. He has 50 years of power industry experience. Dr. Vaiman is a President and Principal Engineer at V&R Energy. His main areas of interest are power system stability and control, power flow and optimal power flow analysis, computer modeling of power system networks, selection of remedial actions for stability preservation, dynamic stability analysis, and predicting cascades. He leads the development of the POM Suite, consulting, and research and development activities at V&R Energy. He has authored over 100 publications devoted to the issues of power system stability and control. Marianna Vaiman (SM’17) received her BSEE and MSEE degrees from Moscow University of Transportation Engineering, Russia. She has over 25 years of power industry experience. In 1992 she joined V&R Energy, where she is currently Principal Engineer and Executive Vice President. She leads the work in the following areas at V&R Energy: software development, consulting activities, research and development activities. She has more than 30 publications devoted to the issues of power system stability and control.
