Tracking Maximum Loadability Conditions in Power Systems

2007 iREP Symposium - Bulk Power System Dynamics and Control – VII Revitalizing Operational Reliability August 19-24, 2007, Charleston, SC, USA

Tracking Maximum Loadability Conditions in Power Systems
C. D. Vournas N. G. Sakellaridis
School of Electrical & Computer Eng. National Technical University of Athens 9, Iroon Polytechniou 157 73 Zographou, Greece

Abstract
This paper deals with the problem of identifying emergency conditions during system operation in the case of voltage instability. A novel detection scheme is proposed based on measurement of the controlled voltage of bulk power delivery transformers equipped with Load Tap Changers. The detection is based on the comparison of successive post-tap-change voltages. The detection scheme is purely local, even though it senses a system-wide emergency condition. It is shown that the condition detected is a precursor to maximum loadability and voltage instability.

schemes for its reliability. Since the early 1990s the nature and geometrical properties of the loadability surface bounding the operating region of a power system in the load power (or “parameter”) space were identified and investigated. In particular it is well established that the loadability surface is also (in the case of loads restoring to constant power) a stability limit of the power system. Early references [3], [4] concentrated on the saddle-node “bifurcation surface”, on which the loadability is due to continuous dynamics and not to the enforcement of a limiter in the form of an inequality constraint. This analysis has since been expanded [5], [6] to cover systems represented by both equality and inequality algebraic constraints. Regardless of the type of limit encountered (saddle node, or inequality constraint) the general properties of the loadability surface give useful information on the nature of the instability encountered, as well as on the most effective way to steer away from it. Loadability conditions are relatively easy to identify during power system simulations routinely performed in the framework of VSA and, in particular, when quasi-steady state simulation is used [5]. It is conceivable that the same conditions could be identified in real time, during system operation. Clearly this information would be of great practical value in providing emergency controls that could avoid an approaching collapse. However, the problem of identifying loadability conditions in real time (and preferably locally without the need of centralized computations) is still open. A well-known loadability condition, namely that of load and Th´ evenin impedance matching at a load bus, has been proposed in the existing literature [9], [10], [11] for on-line stability monitoring. However, the Th´ evenin impedance is not easily measurable. Moreover, as will be shown in this paper, the impedance matching condition is necessarily met after the onset of instability and is thus not able to provide an advance warning. In this general framework, the scope of this paper is to propose a new emergency detection test, which is based on a simple comparison of the controlled (secondary) voltage of bulk power delivery transformers equipped with Load Tap Changers (LTCs) between successive tap changes. Contrary to the impedance matching approach, the condition detected

Introduction
Voltage Security Assessment (VSA) methods have matured considerably during the last decade and several on-line applications are currently in operation worldwide [1]. Typically an on-line VSA application at the Energy Management System level will evaluate voltage stability following a list of probable contingencies, based upon the current state estimation of the system provided by SCADA measurements. On-line VSA monitors secure operation of the system and may even propose corrective countermeasures in certain conditions, when the expected security margins are violated. It is a generally acceptable fact, however, that as power systems evolve under the continuous pressure of growing demand and insufficient investments, there is a growing need to protect against unforeseen conditions that exceed the loadability of the system, i.e. its ability to fully meet load demand at normal voltage levels. In particular we refer to voltage instability conditions, where the emphasis is placed on the capability of the system to supply loads after severe disturbances. Thus, while on-line VSA is a valuable tool for monitoring and managing security, present day power systems often require a more challenging operating strategy, in order to extend operational margins to the limit. According to this strategy, a certain risk of unstable operation might be willingly undertaken, provided that a “safety net” exists, such as a Special Protection Scheme (SPS) that will actively prevent a widespread collapse or blackout should a voltage instability actually occur. This challenge has been described as “walking close to the edge” [2]. Power systems operation is thus entering a stage where it will increasingly rely on instability protection

1-4244-1519-5/07/$25.00 c 2007 IEEE.

P2
1

Post-contingency loadability surface L D direction of stress
V

Short−term load characteristic

C

M C B A post-contingency trajectory

O

Load demand

P1
P

Fig. 1.

Critical point C

Fig. 2.

Critical point C on a PV curve

by this test is a precursor to loss of stability and is able to provide an emergency alarm in time to save the system from an imminent collapse. For reasons of comparison with the impedance matching approach, in this paper we consider power system models, which comply with the Th´ evenin theorem. The complications introduced when including nonlinear generator models (constant P injections) and especially their overexcitation limitation (finite reactive support capability) will not be considered in the analysis performed in this paper, but they do not hinder the applicability of the emergency detection scheme proposed to actual power systems, since the detection is based on the inability of the LTC to control distribution voltage, a condition easily identified locally and with no need to restore to system modelling assumptions. Considering dimensionality, the treatment is general and refers to any number of LTCs. However, the simulated example is restricted to a two-load system, so that the corresponding state and parameter spaces are easily visualized to facilitate understanding of the underlying concepts.

the load demand characteristic (Fig. 2). In its unsuccessful attempt to meet the long-term load demand, the short-term load characteristic increases the apparent load admittance. After point C the operating point enters the lower part of the PV curve and thus increased admittance leads to less power absorption, i.e. the system becomes unstable. In this sense the crossing of the critical point is considered a direct indication of instability. The critical point has interesting and useful properties that can be used to design emergency controls to enable the restoration of a stable equilibrium point [5], [7]. One such key property is that after crossing this point, the necessary control action to restore stability (e.g. load shedding) is increasing. Also it is generally accepted that this constitutes a point of no return, even though there are cases where a stable operating point can be achieved even after the critical point is crossed, especially when the subsystem affected by the instability is of minor importance [8]. In this paper we focus on the detection of an imminent instability along a post-contingency unstable trajectory using instead of the critical point another point of interest shown also in Fig. 1 as point M. As seen, at point M load power P2 of the most affected bus has reached a maximum and power consumption at bus 2 starts decreasing along the trajectory followed by the system. As we will see, this is not the limit of stability, but a point after which the trajectory starts to draw away from the equilibrium conditions for the corresponding bus.

The concept of critical point
A general view of the loadability surface is given in Fig. 1 in a two-dimensional load power space. The solid-line trajectory shows the power actually consumed by loads, whereas the power demand is shown as a dotted arrow starting from the initial operating point O. When, due to a contingency and/or a load demand increase, the load demand is no longer within the feasible operating region, the power consumed by loads is unable to meet the demand and thus it follows a trajectory that will eventually hit the loadability surface at point C, from where it is bound to come back. The point where the post-contingency trajectory touches the loadability surface is called “critical point” [4], [5]. After crossing this point stability is lost, even if the demand were restored to the current consumption, because the thus formed implicit equilibrium point would be unstable. For a single load case, the critical point corresponds to the tip of a PV curve, when the latter has no intersection with

Multi-LTC discrete system formulation
Discrete LTC modeling and system assumptions We consider a general meshed power system with m loads. All loads are connected to the secondary (controlled voltage) bus of LTC transformers (Fig. 3a). We define X ⊂ ℜm the space of min max permissible LTC ratios ri ≤ ri ≤ ri with i = 1, ..., m. The LTC mechanisms are discrete-time, discrete-data systems with a deadband. Even though tap ratios take on discrete values only, the space X is considered to be continuous, which allows the computation of equilibrium manifolds and equilibrium sets,

V’’ jXli Ai ri :1 Po i , Q o i
(a)

VBi

V Ak jrk 2X lk rk V Bk rk2 Zk
(b)

Zi

In this paper we adopt the following assumptions: Assumption 1: All loads are constant admittances Assumption 2: Generators can be represented by constant voltage sources Under Assumption 1, with constant taps rk for all transformers k = i the corresponding loads can be represented as equivalent impedances referred to transformer primary (Fig. 3b). Assumption 2 essentially implies that there is only one infinite bus supplying all loads, otherwise the constant real power production of individual generators destroys the linearity of the network. Assumptions 1 and 2 are made, so that the conditions of Th´ evenin theorem apply. As a consequence, the network seen from the primary of transformer i can be replaced by its Th´ evenin equivalent, as shown in Fig. 3c. Equilibrium Equations With the above assumptions the equilibrium condition for LTCi for given taps rk , k = i for a given VBi , reduces to a bi-quadratic equation:
4 2 2 VAi −[ET i − 2 (Poi RT i + Qoi XT i )] VAi 2 2 2 +(Poi + Q2 oi ) (RT i + XT i ) = 0.

E Ti R Ti

jXT i

VAi r i :1 Poi , Q oi
(c)

V’’ jXli Ai

VBi Zi

Fig. 3.

General power system with m LTCs

as will be seen later in this paper. In general the analysis is easier when all LTCs have the same time delay and step size. However, different step sizes ∆si and time delays ∆ti can be also accommodated, provided that they are constant for each LTCi . LTCs with inverse time delay characteristics can be approximated by a continuous time systems [5], [12]. Following [13] we define a control function for LTCi as follows:  max VBi > VBi  1, for min max 0, for VBi ≤ VBi ≤ VBi f (VBi ) = (1)  min −1, for VBi < VBi

(4)

Note that ET i , RT i , and XT i are functions of rk with k = i and Poi , Qoi functions of VBi . The positive solutions for VAi (when they exist) correspond to unique ri values given by:
′′ ri = VAi /VAi

(5)

min where VBi is the secondary voltage of the i-th LTC and VBi , max VBi are its lower and upper deadband limits respectively.

Equation (4) defines an m − 1 manifold in state space X which consists of two branches, one for high transmission side voltage VAi (high tap) and one for low transmission voltage (low tap). The two branches bifurcate at the point where the discriminant of (4) becomes zero. We call the corresponding manifold LTCi equilibrium manifold, and its two branches high- and low-voltage LTCi equilibrium branches respectively. The equilibrium manifolds min max corresponding to VBi and VBi define the equilibrium band for LTCi in state space X . Clearly the actual LTC state assumes only discrete values within the equilibrium band thus defined. Sufficient Stability Condition The above formulation preserves the quadratic nature of equilibrium conditions originally adopted in [15] and extended in [14]. According to the above references a stable equilibrium can exist only at the intersection of high-voltage LTC equilibrium branches. All equilibria defined by intersections of LTC equilibrium branches, at least one of which is a low-voltage branch, are unstable. We will investigate further the stability conditions using sensitivity analysis. Let as consider the least common multiple T of all LTC time delays ∆ti : T = ki ∆ti i = 1, . . . , m (6)

The discrete tap dynamics can be described by the following set of difference equations:
k+1 k ri = ri + fi (VBi )∆si

(2)

where ∆si is the step size and the transition from k to k + 1 is occurring at multiples of the constant time delay ∆ti . For each LTC the equilibrium condition according to (1) is min max VBi ≤ VBi ≤ VBi . Note that regardless of the load dependence on voltage the active and reactive power Poi , Qoi (Fig. 3) absorbed by the transformer at equilibrium assumes values lying also in a band min max Poi < Poi < Poi [14]. For impedance loads in particular, Poi , Qoi are given by (see Fig. 3a): ˆ ′′ Poi + jQoi = V Ai VBi ∗ = Zi VBi |Zi |
2

(Zi + jXli )

(3)

where VBi belongs to the voltage deadband and is taken as the phasor reference.

In order to investigate local stability properties we introduce the LTC sensitivity matrix, which is the Jacobian matrix of secondary voltages with respect to tap ratios: S = Dr VB = ∂VBi ∂rj (7)

the critical point C, condition (11) has to be violated for at least one LTC, and the violation of (11) is a precursor to maximum loadability conditions, as it is also a precursor to instability for both continuous and discrete system formulation. Following these observations, we will consider the violation of condition (11) as an indication of a voltage emergency condition. Maximum Power and Sign of Sensitivities In this subsection we examine the sign of the elements of the sensitivity matrix S and thus of the long-term state Jacobian A. Let us consider the system of Fig. 3c with only tap ri variable, i.e. with taps rj , j = i constant. In this system a power transfer is made through a constant impedance ZT = RT i + jXT i to a 2 variable one ZL = ri (Zi + jXti ), which has a constant power factor. It is well known (see for instance Chapter 2 of [5]) that for this system power is maximized when |ZL | = |ZT | (12)

and we assume that no change of the control functions f takes place within the common period T . As a consequence, each LTCi will make ki steps of ∆si during this period, so that the affected change in the vector of secondary voltages VB is: ∆VB = S diag [ki ∆si ] f (VB ) = T S diag ∆si f (VB ) (8) ∆ti

where the brackets indicate an m × m diagonal matrix with the corresponding elements, ki has been replaced using (6), and vector f (VB ) has elements fi (VBi ). We now define the long-term state matrix as: A = T Sdiag ∆si ∆ti (9)

and thus the incremental change of the controlled voltages vector is: ∆VB = Af (VB ) (10) The above formulation allows the determination of sufficient stability conditions in terms of the elements of the state matrix A. Clearly, stability of the linearized system is guaranteed if all voltage errors absolutely decrease at each time step. In particular the voltages above deadband should decrease and those below deadband should increase for stability. Thus a sufficient stability condition is that A is diagonally dominant with all its diagonal elements negative [13], that is:
m

which is known as the impedance matching condition. The corresponding tap ratio value is given by: r ¯i = |RT i + jXT i | |Zi + jXti | (13)

Note that in this particular case the load impedance is pro2 2 portional to ri and the consumed power proportional to VBi because of (3). Thus, at the impedance matching point, where (12) holds, secondary voltage VBi is also maximized. Clearly, since at point ri = r ¯i voltage VBi (ri ) is maximized, its derivative with respect to ri , i.e. the diagonal element of the sensitivity matrix (and thus of the state matrix A) becomes zero: ∂VBi =0 (14) aii = ∂ri ri =¯ ri Note that for the same maximum power conditions (Poi (¯ ri ), Qoi (¯ ri )) the discriminant of (4) becomes zero and there is only one solution for VAi . At this point the high- and low-voltage (tap) solutions (4) merge. For tap ratio (impedance) values above that corresponding to the maximum power (ri > r ¯i ) the sensitivity ∂VBi /∂ri is negative (aii < 0), meaning that consumed power increases with increased admittance, whereas for tap ratios lower than r ¯i the voltage sensitivity becomes positive (aii > 0). Clearly for ri < r ¯i the LTCi operation is unstable, even with all other LTCs inactive. The sufficient stability condition (11) can only be satisfied for ri > r ¯i , i.e. on the high voltage solutions of (4), where aii < 0. The sign of the off-diagonal sensitivities ∂VBi /∂rj can be indirectly assessed as follows: If we assume that all taps are on a high-voltage solution (ri > r ¯i for all i), then an increase in any tap rj will reduce

aii +
i=j

|aij | < 0 for i = 1, . . . , m

(11)

In this way each voltage outside the deadband is corrected by the corresponding tap (due to the dominant negative diagonal element) no matter how the other LTCs are moving. The condition (11) is not necessary for stability and is thus conservative. Note that (11) is also a sufficient stability condition for a continuous systems having the same state matrix according to Gershgorin’s theorem [16]. This theorem states that all eigenvalues lie into disks with centers given by the values of diagonal elements and radii equal to the sum of the absolute values of the non-diagonal elements of the corresponding row (or column). Thus if (11) holds, no eigenvalue can be zero or positive. Furthermore, the stability limit for the continuous system is met when the determinant of A becomes zero (saddle-node bifurcation necessary condition). As shown in [5] this condition is satisfied at the critical point C, i.e. when the bifurcation surface in load power space is encountered during an unstable trajectory (Fig. 1). Thus, before an unstable trajectory reaches

the secondary voltage and thus the load power consumed at bus j . Furthermore, if we assume also that all loads are not capacitive, the reduction in the load consumed at bus j will result in an increase of all transmission voltages VAi . If all other taps ri , i = j remain constant, this will have the result to increase the secondary voltages VBi , i = j , as well. Thus the off-diagonal sensitivity matrix (and state matrix) elements are usually positive (aij > 0, for i = j ). When this condition holds, the absolute value in the sufficient stability condition (11) can be dropped, so that it simply becomes: m aij < 0
i=1

for i = 1, . . . , m

(15) (a)

Geometrical interpretation
Equilibrium manifolds and equilibrium sets An equilibrium for the m-LTC system is achieved when all LTC secondary voltages lie inside their corresponding deadbands simultaneously. This defines the following equilibrium set in state space:
min max RS = {r : VBi ≤ VBi ≤ VBi for i = 1, . . . , m}

(b)
Fig. 4. Equilibrium manifolds and equilibrium sets

(c)

(16)

As stated in the previous Section for each value of VBi = Vo , equation (4) defines an equilibrium manifold in the state space of LTC tap ratios r. In 2-dimensional systems, points inside the area bounded by the equilibrium manifold correspond to higher voltages (VBi > Vo ) and points outside to lower voltages (VBi < Vo ). In other words, for increased Vo the area inside the equilibrium manifold shrinks, since load is also increased with increased secondary voltage. Note that this may not necessarily be true for capacitive loads. In particular, the equilibrium manifolds corresponding to upper min max and lower deadband limits VBi and VBi form an equilibrium band for LTCi . The controlled voltage VBi is above the max deadband for points inside the equilibrium manifold for VBi and below the deadband for points outside the equilibrium min manifold for VBi . The equilibrium sets lie on the intersection of all LTC equilibrium bands. For a 2-dimensional space this is shown in Fig. 4. The discussion made in the previous Section for high- and lowvoltage branches of equilibrium manifolds, is readily extended to LTC equilibrium bands. Thus, the stable equilibrium set S lies on the intersection of all the high-voltage LTC equilibrium bands, whereas the unstable equilibrium set U is defined by the intersection of the high-voltage equilibrium band of LTC1 and the low-voltage equilibrium band of LTC2 (Fig. 4a). Note that up to 3 disjoined unstable equilibrium sets may exist in a two load system (2m − 1 in the general case). Loss of stability of the discrete LTC system under slow parameter variation is quite similar to saddle-node bifurcation of continuous systems. Thus the two equilibrium sets

S and U may merge following parameter variations and in particular under increased system loading. Fig. 4b shows a case of a unique equilibrium set SU , which is the discrete system counterpart of a saddle-node. Similar to the latter, the equilibrium set SU is unstable, as there exists a direction of initial conditions in the neighborhood of the equilibrium, for which the trajectory moves away from it. The direction of tap movement is shown in Fig. 4b with arrows. It should be noted that in discrete systems operation is possible even on unstable equilibrium sets, provided that no disturbance causes a departure from the equilibrium set. This is not the case in practical continuous systems, as infinite precision is necessary to achieve operation on an unstable equilibrium point. Finally Fig. 4c shows a more severe instability condition, in which even the unstable equilibrium is lost. In this particular case the equilibrium manifolds corresponding to the lower deadband limits are tangent to each other and the equilibrium set disappears. Note that this is a condition for a saddle-node bifurcation of the continuous system having the lower deadband limits as the unique equilibrium condition for secondary voltages. The sensitivity properties discussed in the previous section can be readily recognized in the state space by observing the slopes of the high-voltage branches of equilibrium manifolds. Indeed, it is easily observed that for high r values (e.g. in the upper right corner of Fig. 4), when r1 is decreasing, the LTC1

deadband is approached from the right meaning that voltage VB 1 is increased (negative diagonal element of S or A). By the same movement, the trajectory is getting away from LTC2 deadband, thus voltage VB 2 is decreased (positive off-diagonal element of S or A). Similar conditions hold for r2 . Clearly the diagonal sensitivity is reversed when considering the lowtap branch of the equilibrium manifold, e.g. in the left part of Fig. 4a. Assuming equal tap steps and time delays, condition (11) is first violated on an equilibrium manifold at the point where the slope of the manifold becomes 1 (angle formed by tangent is 45o ), i.e. when a change ∆r1 = ∆r2 results in ∆VBi = 0. These are shown as points M1 and M2 in min Fig. 4a for equilibrium manifolds corresponding to VB 1 and min VB 2 respectively. On the other hand, the impedance matching condition is satisfied at point r ¯i , i.e. at the point where highand low-voltage solutions are met (zero or infinite slope for manifolds of LTC1 or LTC2 respectively). These points are also marked in Fig. 4a. Region of Attraction and Sufficient Stability Condition The conservative nature of (11) can be also demonstrated in the prediction of the region of attraction of the stable equilibrium set in state space. For instance, in the two-load system case considered in [14] the region satisfying conditions (11) is smaller than the exact region of of attraction of the stable equilibrium set. This is shown in Fig. 5 where the shaded area corresponds to the actual region of attraction, whereas the darker shaded area corresponds to points satisfying the sufficient stability conditions (11). The latter is bounded by the branches I and II of curves a11 + |a12 | = 0 and a22 + |a21 | = 0 respectively and inside this region both a11 + |a12 | < 0 and a22 + |a21 | < 0 hold. The lighter shaded area is the part of the exact region of attraction that is being missed by the sufficient stability criterion (11). Following the discussion of previous sections, it can be seen that curves a11 + |a12 | = 0 and a22 + |a21 | = 0 intersect with min min manifolds VB and VB respectively, at points where the 1 2 slope of these manifolds is equal to 1.

Fig. 5.

Exact region of attraction and aii + |aij | = 0 curves

that all taps will be decreasing simultaneously, i.e. function fi will be negative for all i. In the case where there are remote load buses not affected by the contingency in question, the corresponding voltages will not leave their deadband and thus fj = 0, so we can neglect altogether these LTCs without loss of generality. With the above assumptions, each secondary voltage (of the linearized system) is changing at each period T by an amount:
m

∆VBi = −
i=1

aij

(17)

As long as all ∆VBi are positive all voltages below deadband are being increased and the operating point is moving towards restoration. Notice that this condition is identical to the sufficient stability condition (15). If these conditions prevail during the whole transient, the system will eventually reach a stable equilibrium with all voltages restored. If, on the other hand, one of the row sums of A changes sign and becomes positive during the transient, the corresponding voltage VBi has reached a maximum. It will stop increasing and will start decreasing moving away from equilibrium conditions. Notice that even if this is not a sufficient condition for instability, it is however an indication that the restoration process for the specific bus voltage is not possible with the current direction of tap movement. Under these conditions and in view of the fact that the nonrestoration of the secondary voltage will result anyway in load reduction, it is even justified to shed some amount of non critical load of the affected bus i, in order to allow the restoration of the remaining demand. The change of sign of ∆VBi is thus an indication of an emergency voltage situation. The interesting point with this indicator is that even though it refers to the whole system (sum of a long-term Jacobian row is changing sign) and is a precursor to Jacobian singularity, it

Voltage Emergency Detection
The purpose of this paper is to identify emergency conditions during severe disturbances and in particular to provide a timely, dependable, and reliable indication of an imminent instability, similar to that of the critical point crossing discussed earlier in this paper. Thus we will now confine our analysis to cases following severe disturbances, where all transmission voltages (at least within an affected geographical area) have been substantially decreased. Under these conditions, we can reasonably assume

E jX 1

jX 3 V A1 jX 2

V A2
X1 0.175 X2 0.25

TABLE I Two-load system data (p.u) X3 0.45 E 1.05
min VBi max VBi

∆s i 0.01

∆ti (s) 10

0.985

1.015

r1 :1 V B1 G1
Fig. 6. Two load system

r2 :1 V B2 G2
2.2

Loadability surface

power consumtion at final demand at V =V min
B B

B

2

C M2 A IM2 M1

1.8

P

2 1.6

power consumtion at min initial demand (V =V )
B B

1.4

1.2

is simply based on a local measurement of a readily available variable, namely the secondary LTC transformer voltage. This voltage of course has to be compared with a previous measurement so that the total effect of all acting LTCs is included in the thus computed ∆VBi . It should also be taken into account that in a real system voltage drops result also from events other than tap changes, such as reactive limit enforcement on overexcited generators. Thus, in practice, the emergency detection has to wait for at least two periods of LTC operation, (i.e. a time delay in the range of 20 seconds), to have a more reliable estimate of a decreasing ∆VBi . Note that as the instability evolves with time, it is possible that the critical point C is overstepped during the delay time for emergency detection. However, this delay is relatively small and certainly smaller than that corresponding to the impedance matching condition. Thus, condition ∆VBi = 0 is considered a timely indication of emergency.

IM1
1 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3

P1

Fig. 7.

Load power space for severe instability

Case 1: Severe instability Scenario The simulated trajectory for a sudden load increase of ∆G1 = ∆G2 = 0.585 pu in the two-load system is shown in Fig. 7 in the load power space. Note that the load demand (for voltage restored to lower deadband bound) is beyond the loadability surface in the load power space (point B). As seen in Fig. 7 the trajectory gradually departs from the desired direction of load increase and eventually the power consumed by load 2 is maximized at point M2 , following which the loadability surface is encountered at the critical point C. The impedance matching condition for load 2 is met at point IM2 , then the maximum power consumption for load 1 is encountered at point M1 and finally the impedance matching for load is met at point IM1 . Note that points M1 and M2 do not belong to the loadability surface because they are maxima encountered along the specific trajectory. For instance, if at point M2 the power consumption P1 could be kept constant, the power P2 could still increase slightly before reaching loadability, as seen in Fig. 7. As expected from the previous theoretical discussion, the maximum of P2 (brought about by the maximum of VB 2 ) is encountered before and in reality shortly before the crossing of the critical point C, whereas the impedance matching conditions are met way after the critical point crossing. The same trajectory is shown in Fig. 8 in the r1 − r2 state space. For clarity of the figure only the lower deadband curves min (VB = VBi ) are shown. As seen, due to the severity of the contingency, the equilibrium manifolds for LTC1 and LTC2

Case study (2-load system)
A three-bus test system with two loads is shown in Fig. 6, with data given in Table I. This system conforms with Assumptions 1 and 2 made in this paper. Thus, analytical expressions for the Th´ evenin equivalents seen by each transformer primary can be derived directly using circuit analysis. These, as well as the voltage sensitivities and the elements of the long-term state Jacobian, are all computed during simulation to illustrate the concepts and properties presented so far. Two unstable contingencies are considered and analysed in this Section. The first is an abrupt (step) load admittance increase at time t = 10s causing a severe voltage instability. The second is a marginally unstable case with a gradual load admittance increase bringing the system demand just beyond the loadability limits. Note that load demand in the load power space is computed as the power consumption for voltage equal to the LTC lower min . deadband limit VBi

0.5

1.2
app2

r

0.4

2
1

,Z

VB1=VB1

min

(r1, r2) VB2=VB2
min

o

o

app1

0.3

r2/G
2

2

r2 /G 1 1

TH1

M C
0.6

2

,Z

0.8

IM2

TH2

,Z

Z
0.2

T2

ZT1
0.1

IM2 IM1

V
0.4

B1

at M

M IM1

1

1

V

B2

at M

2

Z

0

0

50

100

150

200

250

300

350

400

450

500

0.2 0.2

time
0.4 0.6 0.8

r1

1

1.2

1.4

Fig. 10.

Load and Th´ evenin impedances for case 1

Fig. 8.

Severe instability case in state space

2.1

1

2

M

C

2

C V
0.95

IM

2

M

B1

1

P

VB2

2

1.9

M

2

C IM

IM

1
1.8

0.9

VB2

M2

2

M1
1.7

M1

0.85

initial load demand at V =V
B 1.6
0 50 100 150 200 250 300 350 400 450

min B 1.85 1.9 1.95 2

1.7

1.75

1.8

P

1

time

Fig. 9.

Voltages VB1 , VB2 evolution for case 1

Fig. 11.

Load power space for gradual load increase

do not intersect and the trajectory is passing through the gap left between the two manifold leading to voltage instability. Note that tap limits are not considered here, as the idea is to give the general properties of instability and its detection.

is crossed at time t = 170s. Impedance matching conditions for bus 2 are met at time t = 270s. The calculated Th´ evenin and load admittances during simulation are shown in Fig. 10.

Case 2: Marginal Voltage Instability The same points as in Fig. 7 are marked on the trajectory of Fig. 8. Note that at points M1 and M2 the trajectory (moving on a straight line with a slope of 1, since both tap ratios constantly decrease with the same rate) is tangent to the constant VB curves corresponding to the voltage values at these points (shown in dashed curves). Note also that after crossing M2 the trajectory moves further and further away from LTC2 deadband, thus clearly showing the approaching instability and the need for emergency measures. The evolution of the secondary voltages VB 1 , VB 2 with time is shown in Fig. 9. Condition (11) is violated at point M2 at time t = 130s. Following this, voltage VB 2 starts decreasing for the first time at the next tap operation (t = 140s). The critical point C, where the state Jacobian A becomes singular In this case the load admittances are increased gradually with a slope ∆G/∆t = 0.0001 pu/s until the values G1 = 2.056 pu and G2 = 2.0631 pu, which correspond to power consumption min (for VB ) slightly outside the feasible region, as shown in the load power space picture of Fig. 11. Because of the gradual demand increase, the LTCs have time to operate and restore voltage within their deadband. They also operate at different time instants, as the voltages VB 1 and VB 2 depart from their deadbands at different times. Therefore the trajectory of Fig. 11 is zigzaging instead of following a smooth curve. Of course, once the instability has prevailed the taps start operating continuously and the trajectory becomes smoother. However, close to the onset of instability, the lack of synchronization between the LTCs makes the picture hard to follow.

0.989

0.988

1.2
0.987

0.986

1

(ro, ro) VB1=VB1
min 1 2

V

B2

0.985

M C IM
2

r
M
2

2

2

0.8

0.984

0.983

0.6

min VB2=VB2

0.982

M
C
0.4

1

0.981 3360 3380 3400 3420 3440 3460 3480 3500 3520

IM1
time
0.2 0.2 0.4 0.6 0.8 1 1.2 1.4

r

Fig. 12.

Evolution of VB2 during critical time for Case 2 Fig. 13.

1

Marginal instability case in state space

Points C and M1 in Fig. 11 are marked based on the corresponding properties of the state Jacobian A analytically computed during simulation. Point C, being a loadability limit, lies always on the loadability surface. However, point M1 is not on a local maximum of the trajectory, because the latter does not follow the assumptions set in the previous section, namely that all LTCs move together. This delays the possibility of detecting the emergency through VB 2 measurement, but does not rule it out. This is shown in Fig. 12 where VB 2 is plotted for the time near the onset of instability . As seen in this figure, the second row sum of state matrix A becomes zero at t = 3400s. At this point LTC2 is able to restore VB 2 within its deadband again, because LTC1 is not acting. However, the movement of r2 forces VB 1 out of its deadband and thus LTC1 is activated initiating a cascade of alternating tap changes of the two LTCs. During this time period point C is crossed at time t = 3450s. LTC2 is able to restore voltage for two more times, but then the decline is evident. Note that LTC action is able to increase VB 2 even after this point (negative a22 ), but the diverse effect of LTC1 is prevailing (a21 + a22 > 0). The state-space view of the same scenario is shown in Fig. 13 where the points M1 , C, IM1 , etc. are also marked.

previous tap.
k −1 k ∆Vik = VBi − VBi

(18)

The emergency detection process is initialized with the first tap change after the voltage gets below deadband. This first 0 post-tap-change voltage is taken as the first reference VBi . If ∆Vi is negative in two subsequent tap changes k and k + 1 (i.e for two periods of LTC operation) an emergency detection signal is issued. The emergency signal can be used by a System Protection Scheme if one is available, or can be used for local emergency action such as load shedding. In this section we assume a closed loop load shedding system where a 5% of local bus load is shed each time an emergency detection signal is issued with a time delay of 3s. After each load shedding the emergency detection algorithm is reset. If voltage remains below deadband, the voltage immediately after the first post-shedding tap change is taken as the new first 0 reference VBi . Application to Cases 1 and 2 The above emergency detection and control algorithm is applied to the severe and marginal instability cases described in the previous Section. In Figs. 14, 15 the implementation of the algorithm is shown, for cases 1 and 2 respectively. Note that the emergency condition ∆VB < 0 is identified only on bus 2, and thus load is shed only at this bus. This is in accordance with the shape of the loadability surface of Fig. 7, where it is evident that restoration of the demand point B is easier to be achieved by decreasing load P2 instead of load P1 . Thus, the emergency detection gives also as a by-product the best location for corrective control. In both cases the method resulted in the restoration of both voltages, with load shed only from bus 2. The amount of

Emergency identification and load shedding
Emergency Detection and Control Algorithm The emergency identification algorithm introduced in this paper is very simple and is based on the sign of ∆VBi of (17). Implementation details, measurement problems and related issues are left for further research. When the controlled voltage VBi is below the deadband we assume that a measurement is taken immediately after each tap change and the voltage is compared to its value after the

0.98

2.2
0.97

2.15

power demand at VB2

min

Reset
0.96

2.1 2.05 2

VB2

0.95

P

power consumption

2

1.95 1.9 1.85 1.8

0.94

Successive V drops
B2

Load Shed

0.93

0.92 120 140

Load shed + Reset
160 180 200 220 240 260

1.75 1.7

50

100

150

200

250

300

time

time

Fig. 14.

Emergency Detection and load shedding (Case 1)

Fig. 16.
2.04 2.02

Power demand and consumption with load shedding (Case 1)

0.987

0.986

2
0.985

Reset

1.98 1.96

power consumption

0.984

VB2

0.983

P21.94
1.92 1.9

0.982

power demand at Vmin B2

0.981

1.88
0.98

Successive V drops
B2

Load Shed
1.86 1.84

0.979

3440

3450

3460

3470

3480

3490

3500

3510

3520

3530

time

2400

2600

2800

3000

3200

3400

3600

3800

4000

time

Fig. 15.

Emergency Detection and load shedding (Case 2)

Fig. 17.

Power demand and consumption with load shedding (Case 2)

the minimum load shedding required to return within the loadability surface and the actual amount of load shedding during the implementation of the algorithm are shown in Table II. The final load demand, the load demand at the loadability limit, and the load demand after load shedding are shown in this Table, as well. Note that demand and load shedding are given in terms of power consumed at voltage min equal to the lower deadband limit VB 2 . As seen in Figs. 14, 16 four load shedding operations at times t1 = 143s, t2 = 173s, t3 = 213s, and t4 = 273s were necessary to restore equilibrium in the severe instability Case 1, whereas only one load shedding operation at time t = 3513s was enough to save the marginally unstable case 2, as shown in Figs. 15, 17.

Figures 18, 19 show the restoration of VB 1 in its deadband for cases 1 and 2 respectively, with load shedding at bus 2 as above.

Conclusions and further research
This paper proposed a new detection scheme tracking emergency conditions in the case of an imminent loss of voltage stability. The scheme is based on identifying the start of a continuous decrease of the distribution side LTC voltage. The concept of the critical point, where an unstable trajectory hits the loadability surface was reviewed. The point tracked by the proposed emergency detection scheme, however, is not the critical point, but the one after which secondary voltage and load recovery on a bus reach a maximum along the trajectory that the system is following. It was shown that this point lies always before the critical point and can thus serve to provide an early warning. A formulation of multi-LTC discrete system was used, which assumes impedance loads restoring power consumption through LTC action. This complies with Th´ evenin theorem

TABLE II Minimum and implemented load shedding for cases 1 and 2 (pu) Case 1 2
f inal P2 min P2 ,s

P2,s 1.8169 1.9016

min ∆P2

∆P2 0.41375 0.1001

2.23065 2.0017

1.889 1.999

0.34075 0.0027

1.02

emergency detection. The proposed method was implemented in a two-load system, where two cases of severe and marginal instability respectively have been considered. In both cases a voltage emergency was detected only at the bus mostly affected by the instability. Thus the detection was timely and provided also information on the extend of the voltage emergency. The detection process took a longer time to identify the emergency for the marginal instability case, because the voltages were alternatively in and out of their deadbands in this case. This is a desirable feature, as speed of detection is needed only in severe instability cases, where a shorter time to act is available. In both cases, load shedding after the emergency detection was sufficient to restore stability. The amount of shedding by a closed-loop scheme was comparable to the theoretically minimum to restore loadability. The proposed method of emergency detection and control is decentralized, since only local measurement is needed, but refers to a system-wide emergency since the zero crossing of the sum of a row of the Jacobian matrix is traced. Moreover, it is quite easily applicable, because it requires measurements of a readily available quantity, namely the LTCs controlled voltage.
2600 2800 3000 3200 3400 3600 3800 4000

1.01

1

V

0.99

B1 0.98

0.97

0.96

120

140

160

180

200

220

240

260

280

time

Fig. 18.

Voltage evolution at bus 1 for Case 1

1.02

1.015

1.01

1.005

1

V

B1 0.995

0.99

0.985

0.98

0.975

0.97 2400

time

Fig. 19.

Voltage evolution at bus 1 for Case 2

Further research is needed for the method to be applied in larger systems, where generators and in particular their overexcitation limiters have to be modeled in detail. Measurement and filtering issues will also have to be closely investigated.

References
requirements, since we performed also a comparison with the loadability condition of load and Th´ evenin impedance matching at a load bus. Sufficient stability conditions were derived using sensitivity analysis and it was shown that the condition chosen for emergency detection, is equivalent to the violation of the sufficient stability condition at a bus, and coincides with the point where the sum of a row of the long-term Jacobian becomes zero. On the other hand, it was shown that the impedance matching condition is equivalent to the diagonal element of the sensitivity matrix becoming zero, since the sign of the diagonal elements is negative for values of tap ratios greater than those corresponding to maximum power delivery at a single bus. The information stemming from this analysis was used to design a voltage emergency detection scheme based on the measurement and comparison of controlled voltages of LTCs of two successive tap changing operations. If the controlled voltage decreases twice, the emergency detection signal is issued. To show the usefulness of this alarm for system protection, a load shedding was performed shortly after the
[1] CIGRE WG C4.6.01, “Review of On-line Dynamic Security Assessment Tools & Techniques”, CIGRE Technical Brochure, Apr. 2007. [2] T. Van Cutsem and C. D. Vournas, “Emergency Voltage Stability Controls: an Overview”, IEEE/PES General Meeting, Tampa, Jun. 2007. [3] S. Greene, I. Dobson, and F. L. Alvarado, “Sensitivity of the loading margin to voltage collapse with respect to arbitrary parameters”, IEEE Trans. Power Systems, vol. 12, pp. 262272, Feb. 1997. [4] T. Van Cutsem, “An approach to corrective control of voltage instability using simulation and sensitivity”, IEEE Trans. on Power Systems, vol. 10, no. 2, pp. 616-622, May 1995. [5] T. Van Cutsem and C. D. Vournas, Voltage Stability of Electric Power Systems. Norwell, MA: Kluwer, 1998. [6] M. E. Karystianos, N. G. Maratos and C. D. Vournas, “Maximizing Power System Loadability in the presence of Multiple Binding Complementarity Constraints”, IEEE Trans. on Circuits and Systems, to appear. [7] T. Van Cutsem, C. Moors, and D. Lefebvre, Design of load shedding schemes against voltage instability using combinatorial optimization”, IEEE/PES Winter Meeting, pp. 848–853, Jan. 2002. [8] C. D. Vournas and G. A. Manos, “Modelling of Stalling Motors During Voltage Stability Studies”, IEEE Trans. on Power Systems, Vol. 13, pp. 775–781, Aug. 1998. [9] M. Begovic, K. Vu, D. Novosel, M. Saha, “Use of local measurements to estimate voltage stability margins”, IEEE Trans. on Power Systems, vol. 14, pp. 1029–1035, Aug. 1999. [10] A. Holen, L. Warland, “Estimation of distance to voltage collapse: testing an algorithm based on local measurements”, Proc. 14th Power System Computation Conference, Sevilla, 2002, paper 38-3. [11] B. Milosevic, M. Begovic,“Voltage-stability protection and control

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using a wide-area network of phasor measurements”, IEEE Trans. on Power Systems, vol. 18, no. 1, pp. 121–127, Feb. 2003. P. W. Sauer, M. A. Pai, A Comparison of Discrete vs Continuous Dynamic Models of Tap Changing Under Load Transformers”, Proceedings: Bulk Power System Voltage Phenomena - III, pp. 643-650, Davos, Aug. 1994. J. Medanic, M. Ilic-Spong, and J. Christensen, “Discrete Models of slow voltage dynamics for under load tap-changing transformer coordination”, IEEE Trans. Power Syst., vol. PWRS-2, pp. 873–882, Nov. 1987. C. D. Vournas and N. G. Sakellaridis, “Region of Attraction in a Power System with Discrete LTCs”, IEEE Trans. Circuits Syst. I, vol. 53, no. 7, pp. 1610–1618, Jul. 2006. C. C. Liu, and K. T. Vu, “Analysis of tap-changer dynamics and construction of voltage stability regions”, IEEE Trans. Circuits Syst., vol. 36, pp. 575–590, Apr. 1989.

[16] A. Jennings, Matrix computation for engineers and scientists, John Wiley & Sons, 1977.

Costas D. Vournas is Professor at the School of Electrical and Computer Engineering of National Technical University of Athens, Greece. His research area is power system dynamics and control, including hybrid system modeling, voltage stability analysis and security assessment. Nikos G. Sakellaridis received the Diploma of Electrical and Computer Engineering from the National Technical University of Athens, Greece, in 2003. He is currently pursuing the Ph.D. degree in nonlinear power system dynamics in the Electrical Energy Systems Lab of the School of Electrical and Computer Engineering of NTUA.