Abstract— With unprecedented capabilities for monitoring by synchronized sub-second measurements and processing them with widely distributed decision intelligence and providing quasi-continuous control capabilities throughout the grid wherever and whenever needed, the smart grid offers the potential to increase all voltage, dynamic and transient stability limits to a level above the relevant thermal limits. With stability thus assured, the smart grid can ultimately become as flexible as the internet. This paper presents a strategy for assuring voltage stability in the smart grid.

Index Terms— flexible smart grid, voltage stability, power system stability, power system control, fast local control, distributed autonomous intelligence, coordinated local and wide area situational awareness.

I. INTRODUCTION VER increasing interconnectivity and increasing power

transfers over longer distances have accentuated the importance of dealing with stability in the power grid. As stability is characterized by the ability of the grid to withstand disturbances, stability related problems have been classified by the nature of the disturbance of interest. Accordingly stability problems have been classified into voltage, dynamic (or small-perturbation) and transient (or large perturbation) stability problems.

Historically, assuring stability of the grid required complex

and time consuming computations as well as very fast control responses. These two conflicting requirements were reconciled by performing extensive offline analyses based on most recent available on-line conditions and potentially severe contingencies. The outcome of the studies was a list of operating limits subject to pre-specified initial conditions, disturbance events and remedial measures. The remedial measures have been hard-wired to meet the fast response times required.

However, in the context of the smart grid, it is possible to

obtain measurements from throughout the grid fast enough to analyze the grid conditions and identify the necessary control actions and implement them in sub-second time frames. Such quasi-continuous control capability offers an unprecedented capability to deal with stability problems in the power grid. This paper proposes a methodology for assuring voltage stability in the smart grid.

Ranjit Kumar is with InfSyn, LLC., ranjitkuma@aol.com

II. HISTORICAL BACKGROUND In general, the severity of stability problems of all the three

types (voltage, dynamic, and transient) increases with the equivalent electrical distance between the sending end and the receiving end of a given power transfer. In the context of transmission lines, the electrical distance is highly correlated with the geographical distance between the two ends. Hence, transferring larger amounts of power over longer distances has remained a challenging problem over the last century. In one of the earliest proposals for unfettered long distance power transfer, Baum [1] proposed to use 220kV transmission with synchronous condensers installed every 100 miles (160km) to maintain voltage at the scheduled value. His recommendation was based on empirical knowledge and consideration of voltage regulation (i.e., voltage drop and not voltage stability).

The relationship between loadability of a transmission line

(maximum power transfer capability) and the physical distance was captured in the St. Clair curve [2] where loadability is expressed in per unit of surge impedance loading (SIL), permitting a single curve to be applied to a range of voltage classes. The maximum permissible loading implied by the St. Clair curve was based upon practical considerations and experience of its author with transmission lines already designed and in service at that time. Dunlop [3] provided a theoretical basis for this curve and extended it for more modern systems (765kV and up). In general, upto about 50 miles, the thermal limits are the limiting ones, since the voltage drop is negligible. Beyond 50 miles, voltage drop becomes the more stringent constraint. Using certain reasonable assumptions based on empirical experience with respect to the margin from the maximum power transfer limit required to maintain transient stability, it was shown in [3] that beyond 200 miles dynamic and transient stability considerations become more constraining than voltage drop.

Kimbark revisited the problem of long distance power

transfer in 1983 [4] using shunt compensation for long distance power transfer. Based on certain reasonable assumptions based on experience regarding subsynchronous resonance, transient stability, desired maximum power transfer capability as well as credible configurations and contingencies, he established a linear relationship between the phase angle and distance between the sending and receiving end of a transmission line. Based on this relationship, he recommended that the voltage of the transmission system be maintained at scheduled value every 233 km (145 mi) or 16.80 of phase angle.

Assuring Voltage Stability in the Smart Grid Ranjit Kumar, Senior Member

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978-1-61284-220-2/11/$26.00 ©2011 IEEE

In this paper (section III), we arrive at a similar conclusion

from a purely theoretical (as opposed to empirical) viewpoint concerning voltage stability. We provide the basic conditions for voltage stability of a two bus system and develop the a criterion named Baum-Kimbark Criterion. In section 4, we extend the concepts for application to a large-scale power grid. Section 5 we examine the issues associated with the implementation of the improvements suggested by the criterion. Section 6 and 7 contain conclusions and references respectively.

III. BAUM-KIMBARK VOLTAGE STABILITY CRITERION For convenience of reference we like to call any stability

criterion that explicitly relates voltages and phase angles a Baum-Kimbark Criterion in honor of their pioneering work on this topic.

For ease of discussion, we consider a two bus system

depicted in Fig. 1. Without loss of generality, we can assume a reactance of 1p.u. for the line and no resistance or shunt capacitance, a voltage of 1p.u at bus 1. We also use the nomenclature: P is the power transferred from bus 1 to bus 2, δ is the phase angle between the two buses, V is the voltage at bus 2 and Q1 and Q2 are the reactive power going into the transmission line at buses 1 and 2 respectively.

Fig. 1: Two Bus System

Then the relevant equations for real power and reactive

power are: P= V.Sin δ (1) Q1= 1 - V.Cos δ (2) Q2= V2 – V.Cos δ (3)

The relationships among the sensitivities of these quantities can be obtained by taking the differentials of the above expressions with respect to δ as follows:

ΔP = Sin δ. ΔV + V.Cos δ. Δδ (1a) Δ Q1 = - Cos δ. ΔV + V.Sin δ. Δδ (2a) Δ Q2 = 2.V. ΔV - Cos δ. ΔV + V.Sin δ. Δδ

(3a) Considering a constant power transfer P (i.e., ΔP=0), we

can deduce the following from equations (1a),(2a) and (3a): Δδ = - (1/V)(Sin δ/ Cos δ). ΔV (1b) Δ Q1 = - (1/Cos δ). ΔV (2b) Δ Q2 = - ((1 - 2.V. Cos δ)/Cos δ). ΔV

(3b)

Then, Δ Q1 / Δ Q2 = 1/(1 - 2.V. Cos δ) (4) Equation (4) implies the ratio of the additional MVAr

needed at bus 1 to compensate for the effects of a disturbance of 1 MVAr at bus 2 if there is no local compensation at bus 2 and to maintain a constant power transfer. Then, for voltage instability, the denominator in equation (4) should be zero, yielding:

V = 1/(2.Cos δ) (5) The value of V as given by this equation is depicted

graphically as curve A in Fig. 2 with the voltage of bus 1 indicated by a small circle at the point (0,1). Whenever the voltage at bus 2 falls below curve A, the system is unstable.

Fig. 2: Baum-Kimbark Criterion for Voltage Stability

Without loss of generality, we can say that if the voltage at bus 2 is taken as 1p.u., then the voltage at bus 1 would be given by the inverse of the right hand side value of equation (5). This expression is shown in Fig. 2 as curve B. If the voltage at bus 2 falls within the area enclosed between the two curves A and B, then the system is voltage stable. Otherwise it is unstable.

From Fig. 2, one can observe that a system can be unstable

even when all voltages are at the same scheduled value when the phase angle between the buses exceeds ±600. At first this appears counter intuitive, given the common assertion that the stability limit of a single line system considered here would be at 900. However, the common assertion is based on the assumption of adequate reactive resources available at both buses to maintain their voltages at constant values. Under those conditions, there would be no voltage instability and the stability limit at 900 is actually the dynamic (small-signal) stability limit.

Lines C and D correspond to voltage regulation within a

band of ±5% of scheduled values. This band is depicted in Fig. 2 by dash-dot lines C and D. As can be seen from the shaded area within this band, the corresponding stability limit would be at ±55.60. However, if one wants to maintain a safety margin of 5% in the power flow magnitude, then the limit would be at ±51.60.

Even in the most ideal conditions, there will be a dispersion

of voltages along a transmission path. The voltage at the ends of a transmission line (as a function of the phase angle across the line) is depicted in Fig. 2 by dotted curves E and F taking

the voltage at the midpoint as the reference. It is interesting to note that this ideal dispersion yields approximately the same stability limit as the regulation band.

It is very interesting to note that during the major balckout

in North America on August 14, 2003, the phase angle between Cleveland and Western Michigan increased gradually from approximately 120 to 500 over about an hour and then suddenly diverged to 1200 within 90 sec [5].

If there is only one line available for power transfer, then

one would use it to the above derived limit and trip the generator when the line is lost. However, if there are several parallel paths, then one should operate according to the N-1 security criterion, implying that the post-contingency system should conform to the above stability criterion. Table 1 presents the corresponding pre-contingency criteria. Figure 2 shows the limits for N=5 and N=3 respectively. In addition, the limits suggested in [1 and 4] are included in Table 1 and shown by circular icons in Fig. 2 for reference purposes. Table 1: Pre-contingency Voltage Stability Criterion to meet N-1 Security

Paths (N) Angle Limit Comment

5 38.8 Typically no special protection schemes are provided.

4 36.0 Typically no special protection schemes are provided.

3 31.5 Can be increased with optional special protection schemes.

2 23.1 Can be increased with optional special protection schemes.

1 51.6 Must be supplemented by special protection schemes.

Baum [1] 20.0 Based on voltage regulation consideration only

Kimbark [4]

16.8 Based on transient stability, and transmission with 2 series segments with 2 parallel paths in each segment.

In pre-contingency operating conditions, for voltage

stability, the voltage should be within the regulation band of Fig. 2 and within the angle limits of Table 1. In addition, during the period of post-contingency corrections, the voltage can deviate beyond these limits but not outside of the stability region defined by curves A and B in Fig. 2. However, during transient electro-mechanical oscillations (over a few seconds) voltages and phase angles can wander out of the stability region depicted in Fig. 2.

IV. BAUM-KIMBARK CRITERION FOR LARGE-SCALE GRIDS

Theoretically, even a power grid with bus phase angles ranging anywhere between -1800 to 1800 (or even beyond these limits subject to interpretation) can be stable. However, for the purposes of ascertaining voltage stability at a bus, according to the Baum-Kimbark criterion, buses within 600 from the bus of interest (for convenience, referred to as reference bus) and with adequate reserve capacity of dynamically controllable reactive power can contribute meaningfully to maintaining stability at the reference bus.

For convenience, we call the buses with adequate reserves

of dynamically controllable reactive power “pilot” buses. For ascertaining voltage stability at the reference bus we need to identify as many pilot buses as necessary to provide the required reserve within 600 distance. This angular distance is to be measured along the shortest path where the length of a branch is given by the absolute value of the phase angle across the branch. A visual mechanism of the related path identification method is given below.

At any point of time, the power grid is a directed graph in

the sense that there are no physical circular power flows in any loop of the graph of the network. (The commonly used term “loop flow” is not a physical flow, but only a value calculated for convenience in the discussion of certain transmission issues). So one can visualize the power system graph drawn with buses placed along the x-axis according to their phase angles with upstream buses to the left and downstream buses to the right (or vice versa). All bus voltages are then scaled such that the reference bus has a value of 1. Now the buses are placed along the y-axis according to their voltages. Now all pilot buses are marked (say in color red). Now take a mask with a hole in the shape of the stability region of Fig. 2 and place it on the map of the grid such that the reference bus would be at (0,1) in the stability region (the highlighted circle in Fig. 2). If there are no pilot (red ) buses visible through the stability region, then the reference bus is voltage unstable unless it is a pilot bus itself. When there are nearby pilot buses (i.e. within the criterion angle limit), one has to analyze if the available reserve capacity is adequate.

It is essential to provide adequate dynamically controllable

reactive power reserves at each pilot bus. The reserve required at a node should be determined based on the contingencies to be dealt with. When there are many contingencies considered sequentially, the reserve requirement at a node should be sufficient to meet the worst contingency (i.e., with the largest requirement at the node). The requirement at the node is given by the sum of the reserve required for each branch at the node. The reserve for each branch at each end node is approximately given by:

Qreserve = P1.(cos θ1 – cos θ2)/sin θ1

where P1 and θ1 are the pre-contingency power flow and phase angle across the branch and θ2 is the post-contingency

phase angle across the branch. It is also essential to adjust the reactive power injections

continuously to maintain voltages at the at the pilot buses at the scheduled values. With synchrophasor measurements at all pilot buses,and distributed local decision intelligence it is possible to deliver the necessary fast control signals (~20Hz).

The scheduled voltages at all pilot buses must be equal with

exception of allowing for transformer ratios. From the perspective of voltage stability high voltages are just as bad as the low voltages. This can be seen from the fact that the Baum-Kimbark criterion developed here considers only relative magnitudes of voltages and not the absolute values. This is in accordance with the best practice of “keeping voltage profiles high and flat” recommended by Taylor [6]. A single grid-wide voltage schedule can be implemented in a manner similar to the single frequency schedule.

V. IMPLEMENTATION ISSUES In essence, to assure voltage stability in the smart grid, one

has to have: (a) Pilot buses with adequate reactive reserves at the

stipulated distances, typically every 150 to 200 (equivalent physical distances of approximately 13.87km/degree or 8.67miles/degree) during normal operations can be sufficient for most circumstances. Based on the remarkably stable operation of the current grid, one has to conclude that with few exceptions such reserve capability already exists.

(b) Fast sub-second control capability to maintain voltages at the scheduled value(s). This is capability is largely missing in the current grid. Apart from the fast voltage controls at generators, there are only a handful of other fast devices. According to the recommendation of [6], the reactive power from generators should be used only as reserve to deal with contingencies. The smart grid will have a much larger number of modern control devices such as the static VAr compensators, etc.

(c) Fast measurements to support the necessary fast controls. This capability is also lacking in the present grid. However, there will be a large number of synchrophasors available in the smart grid.

It can be seen from Table 2 that the suggested

improvements will greatly improve stability limits (Pmax at 900) without requiring excessive reactive reserves (Q/P). The results in Table 2 relate to a single transmission path with several equal segments where the end points of each segment are maintained at scheduled voltage of 1p.u. The reactance of the overall path is 1p.u. and the phase angle across the path is 600. It should be noted that a significant part of the required reactive power would be provided by the shunt capacitance of the transmission lines themselves.

VI. CONCLUSIONS This paper developed a voltage stability criterion named

“Baum-Kimbark Criterion” relating bus voltage and phase

angles. Following the derivation of the criterion using a simplified two bus system, a method was developed to apply the criterion in a large-scale power grid. It is demonstrated that the implementing the improvements suggested by the criterion would greatly increase stability limits with little additional reactive reserves. The smart grid will have the necessary widely distributed dynamically controllable reactive reserves combined with distributed local decision intelligence and the supporting fast measurements and controls. Table 2: Reactive Power and Maximum Power by Number of Segments

Segments (N)

Angle/ Segment Pmax P

Q / segment

Total Q

Total Q/P

1 60 1 0.866 1.000 1.000 1.155 2 30 2 1.000 0.536 1.072 1.072 3 20 3 1.026 0.362 1.086 1.058 4 15 4 1.035 0.273 1.090 1.053 5 12 5 1.040 0.219 1.093 1.051 10 6 10 1.045 0.110 1.096 1.048 60 1 60 1.047 0.018 1.097 1.047

The suggested “rapid control of voltage can greatly

improve the transient behavior” [4] and make the voltage and electromechanical stability limits exceed the corresponding thermal limits of the branches in the grid. This in turn will make “power transmission comparable to railway transportation, with a flexibility not possible in the ordinary system which does not have the constant voltage feature” [1]. This complete avoidance of the stability problems through autonomous automated controls will minimize the need for centralized control of the grid resulting in the flexibility that has been elusive since 1920’s [1].

VII. REFERENCES [1] Frank G. Baum, "Voltage Regulation and Insulation for Large-Power

Long-Distance Transmission Systems," A.I.E.E Trans., Vol. 40, pp. 1018-1077, June 22, 1921.

[2] H. P. St. Clair, "Practical Concepts in Capability and Performance of Transmission Lines," AIEE Transactions (Power Apparatus and Systems). Paper 53-338 presented at the AIEE Pacific General Meeting, Vancouver, B. C., Canada, September 1-4, 1953

[3] R. D. Dunlop, R. Gutman, and P. P. Marchenko, "An Analytical Development of Loadability Characteristics for EHV and UHV Transmission Lines," IEEE Trans. on P.A.& S., Vol. 98, No. 2, pp. 606-617, Mar./Apr. 1979.

[4] E. W. Kimbark, “A New Look at Shunt Compensation”, IEEE Trans. On PAS, vol.PAS-102, No.1, pp212-218, January 1983.

[5] http://www.phasor-rtdms.com/downloads/guides/RTDMSFAQ.pdf [6] C. W. Taylor, “Reactive Power: System Engineering, Reliability, Best

Practices, and Simplicity”, Comments on staff report, Principles for Efficient and Reliable Reactive Power Supply and Consumption, 4 February 2005, Docket No. AD05-1-000. Version: 13 April 2005, Federal Energy Regulatory Commission, United States of America.

VIII. BIOGRAPHY Ranjit Kumar received Ph.D. from the University of Missouri at Rolla

(now known as Missouri University of Science and Technology). He has over 30 years of experience in research and development of algorithms and software for the design, operation and real-time control of power systems, markets and smart grid. He has made several contributions related to power system stability, fuel resource scheduling, and dynamic security analysis.