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Voltage collapse prediction using an improved sensitivity approach
Electric Power Systems Research, 28 (1994) 181 190
181
Voltage collapse prediction using an improved sensitivity approach O. Crisan and M. Liu* Department of Electrical Engineering, University of Houston, Houston, TX 77204 (USA)
(Accepted October 5, 1993)
Abstract An improved sensitivity model is developed for predicting the proximity of the power system steady state to the voltage stability boundaries. A voltage stability index for evaluating the degree of voltage instability is constructed on the basis of the improved sensitivity model. A voltage collapse predictor for maximum power transfer prediction is modeled using the improved sensitivity model. The improved sensitivity approach can predict voltage collapse and suggest corrective action, thus providing valuable information in both the planning and operating environments. In the model analysis and formulation, the improved sensitivity approach includes the physical constraints on the power system, especially load characteristics, dispatch strategy and reactive power generation limits, which makes the model pertinent to the actual system condition. The model is flexible for different load increase patterns and can be calculated directly using the Newton-Raphson power flow results. Numerical tests indicate the improved sensitivity approach is sensitive and accurate for voltage collapse prediction. Key words: Voltage collapse prediction; Voltage stability index; Maximum power transfer; Sensitivity matrix model
1. I n t r o d u c t i o n Without a corresponding increase of transmission capacity, the growth of power system loading forces many power systems closer to their voltage stability boundaries. Studies of voltage stability and voltage collapse, therefore, become crucial to the power industry. While both static and dynamic factors are involved in voltage collapse analysis [1, 2], the static aspect of voltage stability is of great importance to system security and stability assessment and it can be tackled systematically. The static aspect of voltage collapse prediction is investigated in this paper. In recent years, many voltage stability and voltage collapse prediction methods have been presented in the literature. An early representative report comes from Barbier and Barret [3], who proposed a criterion based on the limit of power transfer to a loading area. They studied the p h e n o m e n a of voltage collapse connected to operation at maximum power transfer. They ana*Author to whom correspondence should be addressed.
0378-7796/94/$7.00 ~ 1994 Elsevier Sequoia. All rights reserved SSDI 0378-7796(93)00798-U
lyzed a two-terminal network theoretically, and then generalized their result to a large transmission system. After that, many voltage collapse indices and steady-state stability margin predictors were proposed. In the steady-state arena, Jarjis and Galiana [4] quantified the degree of steady-state stability using a computable scalar stability margin. They defined the margin in terms of the 'geometrical' proximity of the operating point to the boundary of the steady-state stability region, which represents the set of all feasible injections. Their procedure required the solution of eigenvalues and optimization problems. T a m u r a et al. [5] investigated and proved analytically the important relationship between voltage instability and a closely located power flow solution pair. For gauging the system voltage stability, Borremans et al. [6] introduced different usable criteria, especially the sensitivity criteria. They pointed out that a voltage stability criterion has to take into account both the impact of a disturbance on the system and the reserve margin of the system. Using the voltage collapse proximity indicators defined for a twobus system, Carpentier et al. [7] developed a set
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of constraints and a voltage instability indicator, and then generalized their results for a multibus system. Using power flow results, Kessel and Glavitsch [8] proposed an index which varies from 0 (no system load) to 1 (voltage collapse). For defining security or operating constraints that prevent voltage collapse and abnormal voltages in a power system, Schlueter et al. [9], using the sensitivity approach, developed a unified theoretical framework. Their controllability and stability models were defined at the equilibrium point using steady-state data. Tiranuchit and Thomas [10] proposed the minimum singular value of the N e w t o n - R a p h s o n power flow Jacobian matrix as a security index. They also presented an optimization algorithm which could yield the largest minimum singular value of the corresponding power flow Jacobian matrix. To provide operators and planners with more information in terms of real physical quantities, that is, in volts and p.u., Iba et al. [11] suggested the closeness of two power flow solutions as a voltage collapse index. They proposed a method for solving a pair of near power flow solutions based on the convergent characteristics of the N e w t o n - R a p h s o n method in rectangular coordinates. Gupta et al. [12] identiffed some of the problems experienced by power system engineers. Using the real-time data for a wide range of system parameters and operating conditions, they demonstrated that steady-state voltage stability sensitivity investigations can provide power system engineers with a viable procedure to estimate the available system reserve margin to maintain adequate voltage security. Semlyen et al. [13] proposed a secant method based on increasing load admittances to evaluate the extreme loading condition. Using the minimum singular value of the power flow Jacobian matrix as a static voltage stability index, LSf et al. [14] proposed a fast method to calculate the minimum singular value and the corresponding singular vectors. Chebbo et al. [15] developed a voltage collapse indicator based on the optimal impedance solution at the maximum power transfer state. Their theoretical frame was similar to that of Barbier and Barret [3]. Their voltage collapse indicator was formulated and tested when the load at a particular node or the system load increases gradually. A new voltage stability assessment software package, VSTAB [16], was recently developed by the Electric Power Research Institute. VSTAB conducts voltage stability assessment using the voltage stability index developed by N e w t o n - R a p h s o n Jacobian matrix eigenvalue analysis [17].
The above results are representative of different works on voltage collapse. They reveal the voltage collapse mechanism from different aspects and assess voltage security from different points of view. Each of them has its own merit, though the system models adopted could be oriented more to actual system physical conditions. Some of them did not include the reactive power generation limits, although reactive power depletion is a major cause of voltage collapse. Some did not take the load characteristics into account, though load characteristics can be beneficial or deleterious to voltage stability. Others did not consider the active power flow effects on the voltage stability, though the active and reactive powers of a load are correlated and, together with the active power generation, contribute to the change of the voltage profile. The purpose of voltage collapse analysis is to develop methods and tools for predicting the proximity of the system to the voltage stability boundaries, providing guidance for the maintenance of the system voltage stability, and, ultimately, averting the voltage collapse. Among different methods, sensitivity analysis has demonstrated an advantage in its capability to provide a voltage collapse index and suggest corrective action. A properly modeled sensitivity index or predictor can be very helpful in the operating and planning procedures. A representative report on the sensitivity method has been given by Schlueter et al. [9]. They linearized the power balance equations at an equilibrium state and developed systematically a theoretical framework for the sensitivity model. Taking the same direction, but accommodating more power system physical conditions, the present research builds a new sensitivity model for voltage collapse prediction. The physical conditions that constrain the power system have been scrutinized and represented in the sensitivity model formulation. In particular, the load characteristics and generation dispatch strategy are explicitly included in this model. A voltage stability index, which reveals the voltage collapse mechanism and indicates the degree of the voltage instability, is constructed using the data from the improved sensitivity model. A voltage collapse predictor, which predicts the maximum power transfer of the power system, is also developed on the basis of this sensitivity model. The new sensitivity model has been tested numerically and proves to be accurate and sensitive for voltage stability analysis and voltage collapse prediction.
183
2. Definition of voltage stability in terms of sensitivity Definitions of voltage stability based on sensitivity matrices have been documented in recent literature [6, 7, 9]. They are proposed based on the heuristic cause/effect relationships that exist under normal conditions of a power system at load buses. Based on similar heuristic logic, if P and Q are bus power injections, the voltage stability is defined in terms of sensitivity as follows: when AVpv sw(t)--0, any nonzero nonnegative power injection disturbance, APL(t) and AQL(t), causes the state AV(t) to become nonnegative; where APL(t) and AQL(t) are power injection increments at load buses, and the A Vpv sw(t) are voltage increments at P V buses and the swing bus. Here, a P V bus is a bus with specified active power generation and sufficient reactive power support to keep its voltage magnitude at a specified level, and a swing bus is a bus with sufficient active and reactive power support to keep its voltage magnitude and phase angle at a specified level. This definition considers both APL(t) and AQL(t) as the disturbance variables affecting the system state. Although the most direct reason for voltage change is reactive power flow, including APL(t) in voltage collapse analysis is not trivial. This is because load disturbances APL(t) and AQL(t) may have a ratio corresponding to the load characteristics. That ratio may override the fact that voltages are mostly affected by the reactive power flow. Therefore, the A V(t) resulting from the active and the reactive parts of the injected power disturbance may be comparable in magnitude, and such that the AV caused by the APL(t) has to be considered. On the other hand, the active power generation dispatch strategy has a significant effect on the system power flow and hence the system voltage profile. Generally, there is a certain relationship between the active power generation dispatch strategy and the change of the active load and transmission losses. Therefore, the voltage collapse analysis has to represent the relation between the active load change and the coordinated generation dispatch correctly, as is done in the next section.
sitivity matrix model is analyzed and constructed in this section. The disturbance variables considered in the model are
d = [PpQ, QpQ] where PpQ and QpQ are vectors with dimension npQ, where npQ is the total number of PQ buses whose active and reactive powers are specified for power flow. The PQ bus group may contain load buses, network buses and PQ buses converted from P V buses because of their reactive generation limit violations. The control variables are
U :PPv where PPv is a vector with dimension npv which is the number of generating P V buses whose active powers are specified for power flow. The derivation of the voltage sensitivity relative to the disturbance and control variables is based on the physical constraints on the power system. At an equilibrium state, the power system is constrained by the network law equality constraints (e.g. bus active and reactive power balances) and automatic control imposed equality and inequality constraints (e.g. specified bus voltages, specified line flows and other operation limits). When the loads change, the system voltage profile and the automatic generation control respond to maintain the system power balance. Therefore, it is necessary to represent certain physical relationships in the voltage stability analysis. These physical relationships include load-voltage characteristics and the generation dispatch strategy. Load-voltage characteristics determine the relationship between the active power, the reactive power and the voltage magnitude. As functions of voltage, the active power and reactive power consumed by the load may change as the voltage changes, and thus the power factor of the load may change with the voltage. In the following sensitivity analysis, this relationship is represented as
where 51
F= 3. Voltage stability sensitivity matrix model Following the voltage stability definition given in the previous section, the voltage stability sen-
(la)
QpQ = F~PpQ
=
52 5npQ
with dimension npQ × npQ. The 5i = Qi/Pi are determined by the load characteristics. Later,
184
the load voltage characteristics, that is, PpQ = feE(V) and QpQ = fQL(V), are included again in the Jacobian matrix formulation. The generation dispatch strategy determines the relation between the load change and the generation distribution. Different generation dispatch strategies satisfy different operating requirements and lead to different equilibrium states. In the power flow, for any type of dispatch strategy, the relationship between the changes of active power generation and the changes of load can be represented by
APpv =
np~ APi
__
where i -- 1 , . . . , ng, and n~ is the total number of generators available for economic dispatch. The swing bus is included in these generators. Since 27~1 PGi is equal to the system load P~y~load, it follows that the active power generation of generator k is 1 e G k = P s y s load "-g l ~-
i~l Cii
n~ 1 \i=1 ~
bk 2Ck
i~_1C~i
and thus its dispatch coefficient is APGk flk = #k APsysload
i=1
1
1 Ch ~/k n g ~
rtpv i=l
or
(lb)
A P p v = F , APpQ
where the coefficient matrix is
F~=-
f12 PV
...
fi2
"""
[~npv
with dimension n p v x neQ. In the F, formula, the negative sign results from the bus injection assumption in the power flow calculation (i.e. generated power is positive, while load power is negative). The elements of F¢ are the dispatch coefficients fli =
APG~ n~° A P i i=1
In the above formula, parameter PGi is the active power generation at P V bus i, which is obtained at an equilibrium point according to the dispatch strategy such as inertial power flow, governor power flow or economic dispatch. Taking the economic dispatch (ECD) as an example, if the cost function of a generator is f(PG) = a + bPG + cPa 2
the system loads are dispatched to the generators according to the equal-increment rule. For a certain constant 2, all the generators satisfy df(PGi) _ dPa~
that is, bi + 2 c i P G i :
The coefficient Ph is a factor that accounts for the transmission loss increment together with the load increment. It is determined by the dispatch strategy; for example, it may be estimated from the previous power flows or approximated by experience. If, during the power flow, the equalincrement rule forces some generators to hit their generation limits, those generators are set to the violated limits with their dispatch coefficients set to zero. The other generators get the undispatched load dispatched according to the equal-increment rule. The dispatch procedure continues till no generator is violating its constraints. The resulting active power generation PGk settings for all the generators are used for power flow. The resulting dispatch coefficients flk are used for the sensitivity matrix modeling. For sensitivity analysis, the equality constraint equations are linearized at an equilibrium point. These equations include the active and reactive power balance equations and the specified line flow equations. In the linearization of these equality constraint equations, the state variables are the voltage phase angles at all the buses except the swing bus, the voltage magnitudes at the uncontrolled load network buses and the constrained generator buses, the turns ratios of the load tap changers or the susceptances of the capacitor banks that are attached to the controlled load network buses and are well within their operating limits, and the phase angles of the phase shifters that are attached to the specified branch and are well within their operating limits. In the linearization, the disturbance and control variables include the specified active powers and the specified reactive powers. To simplify the notation, the power increments
.185 are denoted by [AYp, AYQ] T, whose subvectors are
I-AP, ]
AYp = LAPpvJ
I-AQ.q
and
AYQ = LAPij J
The state variables are denoted by [X~, X v ] w whose subvectors are
X,~ =[~1
and
Xv=
F1 tij
The linearized system (total differential of the equality c o n s t r a i n t equations) is
AYQA
jl-Ax ] LAx.j
(lc)
w h e r e the coefficient m a t r i x J is p a r t i t i o n e d as
tivity matrices Sp and SQ are derived by solving the linear system in (lc). The derivation is given below. L i n e a r i t y leads to the state variable AXv being the superposition of its responses to different disturbances or control inputs. Therefore, to obtain Sp, the voltage sensitivity relative to the active power flow, it is assumed t h a t AYQ = O. Then, from eqn. (1), it follows t h a t
AYp = J1 AX~ + J2 AXv
(2a)
0 = J3 AX~ + J4 A X v
(2b)
APpv = Fz hPpv
(2c)
To solve for the voltage sensitivity Sp, m a t r i x calculations are performed. To m a i n t a i n the acc u r a c y and robustness of the sensitivity model, the matrix inversions are only performed on square matrices. Equations (2a) and (2b) yield AX(vP) = - J4-'J3 [ J~ - J2 J4 :'J3 ] - ~AYp
with
Substituting (2c) into (3a), it follows t h a t
[(?Yp]
[OYp
J =L X J =
rSYo]
C~fpL(V)] J
[(?Yo OfQL(V)]
J4=L
J
w h e r e Zl is a square matrix with dimension nye x nye, J2 is a r e c t a n g u l a r m a t r i x with dimension nyp x nyo , Ja is a r e c t a n g u l a r matrix with dimension nyo x nyp, and J4~is a square matrix with dimension ny 0 x nYo. In' the above formulas, nvp is the n u m b e r of equality constraints on bus active powers, and nvo is the n u m b e r of the r e m a i n i n g equality constraints, tha,t is, bus reactive power balances and/or specified line flows. The expressions for these partial derivatives are the same as those in the N e w t o n - R a p h s o n power flow J a c o b i a n matrix, which is well k n o w n and is not repeated here. At an equilibrium state obtained from an operating state or a power flow solution based on a sufficient set of given conditions, assuming linearity in the vicinity of a system equilibrium point, the voltage stability sensitivities are evalu a t e d by _
(3a)
FAPpQ]
w h e r e Sp is the active power related voltage sensitivity m a t r i x with dimension nvQ x nvp, and SQ is the reactive power r e l a t e d voltage sensitivity m a t r i x with dimension nyQ × nyQ. The sensi-
AX~ ) = -J4
1Ja[J1- J2Jn-lJa]-
F e APPo
w h e r e APpo are load level increments, and the voltage sensitivity relative to the active power flow is Se, w h e r e Se = -J4-1J3[J, - J2J4-1J~]' i
(3b)
}
Similarly, to obtain SQ, the vo]tage sensitivity relative to the reactive power flow, it is assumed that A Y p = O. Then, from eqn. (I), it follows that 0 = J~ AX~ + J2 A X v
(4a)
AYQ = J3 AX~ + J4 AXv •
(4b)
This yields AX(vQ) ----[ - J 3 J 1 - 1 J 2 -i- J4] -1 AYQ
If Pij is specified, it follows t h a t l
AXe) = [ - J 3 J l - l J e + J4]
1
XL~j AQpQ
(5a)
w h e r e AQpQ are load level increments, and the second coefficient m a t r i x has rows relative to the specified line flows set to zero. The voltage sensitivity relative to the reactive power flow is SQ, where Sq = [ - J 3 J l - l J 2 + J4] -lII01
(5b)
Combining the voltage sensitivities relative to both the active a n d reactive power flows, the voltage sensitivity equations solved from (1) are
186
hXv = AX~ ) + AXe) =Sp[I l APpQ -~So AVPo
(6a)
Substituting (la) into (6a), it follows t h a t AX V : SpQ APpQ
(6b)
where the voltage stability sensitivity matrix is
SpQ:Sp[I ]~- SQF~
(6c)
The sensitivity matrix SPv and its corresponding sensitivity equations provide accurate sensitivity data for both the voltages at the PQ buses and the parameter settings of the automatic regulators included in Xv. In a power flow, the PV bus reactive powers, QPV, are resources to keep the Vpv at a presumed level. Because of thermal limitations or other types of operating limit, when generators or reactive compensators violate their operating limits, the PV buses lose their ability to maintain their voltage levels. This procedure is reflected in the system reactive support depletion, which is the primary cause of voltage collapse. In the power flow calculation, these PV buses have to be converted to PQ buses and, consequently, the voltage stability sensitivity matrices have to be rearranged according to the conversion. In other words, the PQ group is enlarged by the buses converted from PV to PQ buses, while the PV group shrinks by the same number of buses. The regulating devices may also hit their operating limits. As a result, the dimensions and contents of the Jacobian matrix used in the sensitivity model change and cause corresponding changes in the sensitivity matrices. According to the voltage stability definition in the previous section, the system is voltage stable when the voltage stability sensitivity matrix SPv columns relative to buses in the PQ group are all nonnegative. When a matrix inversion in the SPv does not exist, or SpQ contains negative components relative to the load bus voltage, the system is voltage unstable. In this case, when loads increase or, equivalently, generations decrease, the system state undergoes the bifurcation t h a t may lead to voltage collapse. The voltage stability sensitivity matrix discussed above has included load characteristics and other power system physical constraints. In the steady-state stability domain it provides a robust foundation for voltage collapse analysis. The partial sensitivity matrix Sp reflects the effect of active load changes and the relative generation
changes on the system voltage level, while the partial sensitivity matrix SQ reflects the effect of the reactive load changes and reactive resource dispatch on the voltage level. When the voltage stability sensitivity matrix SpQindicates that the system has a voltage problem, Sp and SQ provide more detailed information concerning the voltage sensitivity and help in the decision making which may avert a voltage collapse, for example, resource dispatch, load shedding or brown-out.
4. Improved sensitivity approach-application of voltage stability matrix In the previous section, an improved voltage stability sensitivity matrix has been modeled and its formulation, based on the system condition, has been analyzed. This section discusses the utilization of the improved voltage stability sensitivity matrix in the voltage stability analysis and voltage collapse prediction.
4.1. Voltage stability index As discussed in the previous section, theoretically, the loss of voltage stability can be detected by the sign of the voltage sensitivity from the sensitivity matrix SpQ. In practice, when the voltage stability sensitivity matrix SpQ contains negative components in the columns relative to PQ buses, the system voltage levels are already extremely low. The power system equilibrium points are hardly located there. Instead, when voltage levels drop below the operating limits endurable by the power system equipment, the system state bifurcation usually occurs, resulting from the changes in load operating mode or power system equipment setting. Therefore, the sign shift from the voltage stable state to the voltage unstable state, described in the previous section, may not be helpful under most power system circumstances. The most dangerous case of voltage depression, which most probably causes voltage collapse, is marked by the sudden overshoot of the acceleration of the voltage decrease relative to the load increase or, equivalently, the generation decrease. In this case the system does not have enough margin to alert and avert the voltage collapse. Near the current equilibrium point, the voltage stability sensitivity matrix provides the sensitivity of voltage decrease relative to the load change. It can be used to estimate the acceleration of the voltage decrease caused by the load increase, and thus predict the proximity of
187
the steep voltage depression that may lead to a voltage collapse. For a practically sized system, it is very difficult and unnecessary to monitor the whole sensitivity matrix. A practical solution is to identify the weak buses in the system and monitor their voltage sensitivities. The weak bus identification is determined not only by the power systern analysis, but also by the system engineering circumstances. An example of a weak bus is one where there is a large electrical distance (the transmission impedance weighted by the intermediate bus loading) between it and the nearest generating P V or swing bus. The transfer of power to a weak bus causes further transmission loss and voltage depression together with phase shift. This will also occur when a bus is at the end of a long transmission line between voltage control areas. The voltage stability sensitivity matrix indicates this case numerically by the largest diagonal element corresponding to the weakest bus. The second type of weak bus is that at which there is a sustained loading increase. The power system voltage problem and voltage control related to reactive power flow tends to be localized. Therefore, the voltage at a bus will be affected most directly if the bus sustains increasing loading. The voltage stability sensitivity matrix reflects this case by a fast growing diagonal element corresponding to the bus. Finally, the buses of importance or high priority should also be considered as weak buses and monitored. A power flow simulation proved that the sensitivity matrix SpQ predicts a sharp voltage drop by a jump of order five (i.e. equal to or larger than five times) in the rate of change of the monitored SpQ components corresponding to the weak buses. The steps of voltage collapse prediction using the voltage stability index dSpQ(i, i)/ dx can be summarized as follows. Step 1. Use (6c) to obtain SpQ(i, i) at the current state, where i is the row number relative to the weak bus. Step 2. Use (6c) to obtain SpQ(i, i) at a state with a disturbance such as a load increment in a specified pattern. Step 3. Obtain dSpQ(i, i)/dx, where x is the normalized load increment; if it is of the order of five times higher than its base-case value, the system is confronting a voltage collapse.
4.2. Maximum power transfer predictor The power balance equations are nonlinear and the P - V curves from power flow simulations have a parabolic shape. This observation leads to
an equivalent quadratic model P(V) = r2 V 2 + rl V -~- r0 where P is the local or system load level, and r0, rl, and r2 are the unknown coefficients of the quadratic function. If the quadratic model is correctly built, the system maximum transfer can be readily obtained as the maximum value of the quadratic function. To obtain the coefficients r0, rl, and r2 in the quadratic model, different curve fitting methods can be used. By including the load characteristics, dispatch strategy and reactive generation limits, the new sensitivity model provides a solid basis for the sensitivity analysis method. A new sensitivity approach is thus developed to solve the quadratic function and obtain the maximum transfer. The coefficients of the equivalent quadratic function, expressed in sensitivities, are 1 d2P r 2 - ~ dV 2'
dP rl= dV-2r2V
(7) ro = P -
r2V 2 - r l V
The predicted maximum transfer is rl 2
Pmax = ro -- - 4r2
(8)
Obviously, the manner of obtaining the sensitivity data, that is, dP/dV and d2P/dV 2, at an equilibrium state is a key factor in making a correct prediction of the maximum transfer. Here the sensitivities in the above formulas are obtained using the improved voltage stability sensitivity model developed in this research, which can be evaluated at a simulated system state or at an operational equilibrium state. For a single-load variance case, the sensitivity equations give directly AVL = SpQ(L, L) APL
(9)
where the sensitivity SpQ(L,L) is the diagonal element of the voltage sensitivity matrix SpQ corr e s p o n d i n g to the load bus. Then it follows that the first-order load sensitivity is d P L ,., A P a
1
d~L ~ AVL
SpQ(L, L)
(10a)
To obtain the second-order sensitivity, the power flows have to be solved for the same singleload case at two nearby loading levels, that is, p(~-1) and P(~). The results are used to calculate dP(~-I)/dVL and dP(~)/dVL, and then the secondorder sensitivity is obtained as
188
d2pL dP<~)/dV(~I)-dP(~L L
1)/dV(~
dYE 2
"
1)
(10b)
The maximum transfer can then be readily obtained using eqns. (7) and (8). For a system with a group of loads increasing by a similar a m o u n t x relative to the base case, the sensitivity equation gives A V = SpvPbase Ax
(11)
where Pbase is a column v e c t o r containing the base loadings of the load group. Selecting VL of a bus with a relatively low voltage as the independent variable of the equivalent q u a d r a t i c function and x as the dependent variable, then the first-order sensitivity is e v a l u a t e d by dx Ax = ~ AVL
transfer using eqn. (8) (or (8) with P replaced by x). The maximum transfer prediction method discussed above has been applied to several test systems. The predicted maximum transfers are very a c c u r a t e at different states, t h a t is, states a w a y from or a d j a c e n t to the simulated maximum transfer state. This means t h a t the maximum transfer and the corresponding critical voltage can be predicted by the data obtained at a couple of equilibrium states t h a t are not necessarily near the voltage collapse state or riding on the knee o f a P Vcurve.
5. T e s t r e s u l t s
(12a)
where the sensitivity coefficient is dx Ax 1 (12b) d VL "" AVE [SpQ(L, ")]gbase with [8pQ(L, ')] as the row of Spv relative to VL and Phase. In the same w a y as for the single-load case, the power flows have to be solved with the same load group at two n e a r b y loading levels, x~ -1) and x~ ). The results are used to calculate dx(i-1)/dVL and dx(i)/dVL, and then the secondorder loading sensitivity is e v a l u a t e d by d2x dx(i)/dV(~) -- d x ( i - 1 ) / d V ( ~ -1) dVL2~ V~ ) _ _ V ~ i _ 1) (12c) The predicted apex value of x can be readily obtained using eqns. (7) and (8) with the dependent variable P c h a n g e d to x. The maximum loading at each load bus of the load group is Xm~x times its base-case value. In the case of a group of loads increasing correlatively, the maximum transfer can be obtained with the same m e t h o d o l o g y as illustrated above for the groupload increase case. As a s u m m a r y of the above discussion, the steps for the application of the quadratic model in a maximum transfer prediction are listed below. Step 1. For a single-load increase case (or a group-load increase case), solve the sensitivity data needed for eqn. (10) (or (12)) at the c u r r e n t state or a projected state. Step 2. For a single-load increase case (or a group-load increase case), solve for the coefficients in the equivalent function using eqns. (10) and (7) (or (12) and (7) with P replaced by x). Step 3. For a single-load increase case (or a group-load increase case), o b t a i n the m a x i m u m
The test results from the I E E E 30-bus test system are presented in this section to illustrate the performance of the improved sensitivity approach. The l o a d - v o l t a g e characteristics and s h u n t c a p a c i t o r b a n k s were added to the test system to extend the simulation cases. The simulation data provided the P - V curves for bus voltage versus load level. As an example, the system load increase P - V curves for a relatively w e a k bus, bus 30, which is furthest a w a y from the generation, and a relatively strong bus, bus 7, which is located in the center of the generation area, are given in Fig. 1. The curves of bus voltage phase angle versus system load level are given in Fig. 2. In Fig. 1, at the knee or the steep voltage drop of the P V curve, the voltage m a g n i t u d e of bus 7, 177, is still quite high, a b o u t 0.9 p.u., and the phase angle of bus 7 in Fig. 2 is far below 90 °, the
l.I l.O
0.9 A v >
0.8 0.7 V30 ,I
0.5 0.5 0.5
V7
I
I
I
I
!
1.0
1.5
2.0
2.5
3.0
3.5
x (P / Po) Fig. 1. P - V b u s 7.
c u r v e s ( s y s t e m l o a d i n c r e a s e c a s e ) for b u s 30 a n d
189
.,0
-20
•~ -30
-40
I*
-50 0.5
,
,
1.0
1.5
delta'/
,
2.0 X (P / Po)
,
,
2.5
3.0
3.5
Fig. 2. B u s 30 a n d b u s 7 v o l t a g e p h a s e a n g l e s (delta) v e r s u s s y s t e m loading.
angular limit. It is clear that neither the voltage magnitude nor the phase angle indicates the voltage collapse directly. With Fig. 1 as a reference, let us first look at the test result of the voltage stability index. The monitored weak bus is bus 30, whose corresponding voltage stability index, dSpQ(30,30)/dx, that is, the rate of change of SpQ(30, 30), versus system loading level x (P/Po) is shown in Fig. 3. The index value near x = 3, which is at the knee of the P - V curve in Fig. 1, is extremely high (about 30) and is not shown in the Figure. In Fig. 3, at a load level greater than 2.3, the voltage stability index of bus 30, dSpQ(30, 30)/dx, rises to about five times its base-case value. This
index value indicates that the system is confronting the knee of the P - V curve, that is, the sudden drop in voltage level. This prediction is proved by the simulated case shown in Fig. 1. In Fig. 3, the voltage stability index values of bus 30 for all the states from Fig. 1 clearly indicate that the voltage stability index increases when the system states move nearer to the knee of the P - V curve. At the knee it jumps to an extremely large value, which shows that the voltage stability index is very sensitive to voltage collapse. Next, with bus 30 as the monitored weak bus, the test results of the maximum transfer predictor are demonstrated. The adjustment of the voltage regulator has a significant effect on the system maximum transfer and voltage stability. As the voltage problems arise, the voltage regulators and compensators are fully adjusted and thus their limit settings are used in the following discussion. With Fig. 1 as a reference, the predictions are performed at states both away from and near the knee of the P - V curve, with both high and low voltage levels. The predicted maximum transfers and the corresponding critical voltages are very consistent. The predicted P - V curve for a state at a high voltage level near the base case is given in Fig. 4. Also in Fig. 4, for comparison with the predicted P - V curve, a simulated case is shown with the regulators and compensators set to their limit settings, taken from the simulated knee point of the P - V curve in Fig. 1. In Fig. 4, it can be seen that the prediction is very accurate, that is, there is only a 2% error in the predicted Xmax and only 5% in the corresponding critical voltage. The maximum transfer
10
1.2 state where prediction is made 1.0
~" 6
A
t:~ 4"
V o 0.8 o~ ;>
0.6
0 0.5
1.5
2.5
3.5
x (P / Po) Fig. 3. V o l t a g e s t a b i l i t y i n d e x dSpQ(30, 30)/dx v e r s u s s y s t e m l o a d i n g level x (P/Po).
¢
0.4 0.5
Xpredict~l
i
!
1.5
2.5
3.5
X (P / PO) Fig. 4. C o m p a r i s o n of t h e s i m u l a t e d a n d t h e p r e d i c t e d P - V curves.
190
predictions demonstrated in the current literature have not achieved this kind of accuracy at states away from the knee of the P - V curve. The accuracy of the predictor from this research results from the correct modeling of the system physical conditions and the careful development of the predictor modeling technique. The improved sensitivity approach was tested with various coefficients of the load-voltage characteristics. It predicted the voltage instability and maximum transfer in agreement with the corresponding power flow simulation results. Test results from different cases, for example, single-load increase and group-load increase, also proved that the voltage stability index is sensitive and accurate for voltage collapse prediction, and the maximum transfer predictor provides consistently accurate results. These results indicate that the improved sensitivity approach is promising for voltage collapse analysis and prediction, and it is hoped that its application will lead to further refinements.
6. C o n c l u s i o n s An improved sensitivity approach has been developed. By explicitly including load characteristics and generation dispatch strategy, the sensitivity approach is made more pertinent to voltage collapse prediction and suggestions for corrective action. The sensitivity model can be calculated using the present power flow results, which can be easily obtained from the N e w t o n Raphson power flow program. The accurate solution provided by the method discussed above can be used with more approximate treatments in voltage collapse prediction, margin estimation and reactive resource planning.
Acknowledgements The authors would like to thank H. P. Nguyen, J. M. Adams and C. W. Fromen of Houston Lighting and Power Company for their collaboration during this research.
References I J. A. Momoh, A knowledge based support system for voltage collapse definition and prevention, E P R I Project R P 2473-36, Rep. No. EL-7168, Electr. Power Res. Inst., Palo Alto, CA, Feb. 1991. 2 M. K. Pal, Voltage stability conditions considering load characteristics, I E E E Trans. Power Syst., 7 (1992) 243-249. 3 C. Barbier and J. P. Barret, An analysis of phenomena of voltage collapse on a transmission system, Rev. Ggn. Electr., 89 (1980) 672-690. 4 J. Jarjis and F. D. Galiana, Quantitative analysis of steady state stability in power networks, I E E E Trans. Power Appar. Syst., PAS-100 (1981) 318-326. 5 Y. Tamura, H. Mori and S. Iwamoto, Relationship between voltage instability and multiple load flow solutions in electrical power systems, I E E E Trans. Power Appar. Syst., PAS-102 (1983) 1115 1125. 6 P. Borremans et al., Voltage s t a b i l i t y - - f u n d a m e n t a l concepts and comparison of practical criteria, CIGRE Rep. No. 38-11, Aug. 1984, pp. 1 8. 7 J. Carpentier, R. Girard and E. Scano, Voltage collapse proximity indicators computed from a n optimal power flow, Proc. 8th Power System Computation Conf. (PSCC), Helsinki, Finland, 1984, Butterworth, London, 1984, pp. 671-678. 8 P. Kessel and H. Glavitsch, Estimating the voltage stability of a power system, I E E E Trans. Power Delivery, PWRD.1 (1986) 346-354. 9 R. A. Schlueter, A. G. Costi, J. E. Sekerke and H. L. Forgey, Voltage stability and security assessment, E P R I Project R P 1999-8, Rep. No. EL-5967, Electr. Power Res. Inst., Palo Alto, CA, Aug. 1988. 10 A. Tiranuchit and R. J. Thomas, A posturing strategy against voltage instabilities in electric power systems, I E E E Trans. Power Syst., 3 (1989) 87 93. 11 K. Iba, H. Suzuki and M. Egawa, A method for finding a pair of multiple load flow solutions in bulk power systems, IEEE Trans. Power Syst., 5 (1990) 582 591. 12 R. K. Gupta, Z. A. Alaywan, R. B. Stuart and T. A. Reece, Steady state voltage instability operations perspective, IEEE Trans. Power Syst., 5 (1990) 1345-1354. 13 A Semlyen, B. Gao and W. Janischewskyi, Calculation of the extreme loading condition of a power system for the assessment of voltage stability, I E E E Trans. Power Syst., 6 (1991) 307-315. 14 P.-A. L6f, T. Smed, G. Andersson and D. J. Hill, Fast calculation of a voltage collapse index, I E E E Trans. Power Syst., 7 (1992) 54-64. 15 A. M. Chebbo, M. R. Irving and M. J. H. Sterling, Voltage collapse proximity indicator: behavior and implications, Proc. Inst. Electr. Eng., Part C, 139 (1992) 241-252. 16 P. Kundur, G. K. Morison and D. Maratukulam, EPRI's new voltage stability software available now, Power Syst. Dynamics, (7) (1992) 1-1. 17 B. Gao, G. K. Morison and P. Kundur, Voltage stability evaluation using modal analysis, I E E E Trans. Power Syst., 7 (1992) 1529-1542.
181
Voltage collapse prediction using an improved sensitivity approach O. Crisan and M. Liu* Department of Electrical Engineering, University of Houston, Houston, TX 77204 (USA)
(Accepted October 5, 1993)
Abstract An improved sensitivity model is developed for predicting the proximity of the power system steady state to the voltage stability boundaries. A voltage stability index for evaluating the degree of voltage instability is constructed on the basis of the improved sensitivity model. A voltage collapse predictor for maximum power transfer prediction is modeled using the improved sensitivity model. The improved sensitivity approach can predict voltage collapse and suggest corrective action, thus providing valuable information in both the planning and operating environments. In the model analysis and formulation, the improved sensitivity approach includes the physical constraints on the power system, especially load characteristics, dispatch strategy and reactive power generation limits, which makes the model pertinent to the actual system condition. The model is flexible for different load increase patterns and can be calculated directly using the Newton-Raphson power flow results. Numerical tests indicate the improved sensitivity approach is sensitive and accurate for voltage collapse prediction. Key words: Voltage collapse prediction; Voltage stability index; Maximum power transfer; Sensitivity matrix model
1. I n t r o d u c t i o n Without a corresponding increase of transmission capacity, the growth of power system loading forces many power systems closer to their voltage stability boundaries. Studies of voltage stability and voltage collapse, therefore, become crucial to the power industry. While both static and dynamic factors are involved in voltage collapse analysis [1, 2], the static aspect of voltage stability is of great importance to system security and stability assessment and it can be tackled systematically. The static aspect of voltage collapse prediction is investigated in this paper. In recent years, many voltage stability and voltage collapse prediction methods have been presented in the literature. An early representative report comes from Barbier and Barret [3], who proposed a criterion based on the limit of power transfer to a loading area. They studied the p h e n o m e n a of voltage collapse connected to operation at maximum power transfer. They ana*Author to whom correspondence should be addressed.
0378-7796/94/$7.00 ~ 1994 Elsevier Sequoia. All rights reserved SSDI 0378-7796(93)00798-U
lyzed a two-terminal network theoretically, and then generalized their result to a large transmission system. After that, many voltage collapse indices and steady-state stability margin predictors were proposed. In the steady-state arena, Jarjis and Galiana [4] quantified the degree of steady-state stability using a computable scalar stability margin. They defined the margin in terms of the 'geometrical' proximity of the operating point to the boundary of the steady-state stability region, which represents the set of all feasible injections. Their procedure required the solution of eigenvalues and optimization problems. T a m u r a et al. [5] investigated and proved analytically the important relationship between voltage instability and a closely located power flow solution pair. For gauging the system voltage stability, Borremans et al. [6] introduced different usable criteria, especially the sensitivity criteria. They pointed out that a voltage stability criterion has to take into account both the impact of a disturbance on the system and the reserve margin of the system. Using the voltage collapse proximity indicators defined for a twobus system, Carpentier et al. [7] developed a set
182
of constraints and a voltage instability indicator, and then generalized their results for a multibus system. Using power flow results, Kessel and Glavitsch [8] proposed an index which varies from 0 (no system load) to 1 (voltage collapse). For defining security or operating constraints that prevent voltage collapse and abnormal voltages in a power system, Schlueter et al. [9], using the sensitivity approach, developed a unified theoretical framework. Their controllability and stability models were defined at the equilibrium point using steady-state data. Tiranuchit and Thomas [10] proposed the minimum singular value of the N e w t o n - R a p h s o n power flow Jacobian matrix as a security index. They also presented an optimization algorithm which could yield the largest minimum singular value of the corresponding power flow Jacobian matrix. To provide operators and planners with more information in terms of real physical quantities, that is, in volts and p.u., Iba et al. [11] suggested the closeness of two power flow solutions as a voltage collapse index. They proposed a method for solving a pair of near power flow solutions based on the convergent characteristics of the N e w t o n - R a p h s o n method in rectangular coordinates. Gupta et al. [12] identiffed some of the problems experienced by power system engineers. Using the real-time data for a wide range of system parameters and operating conditions, they demonstrated that steady-state voltage stability sensitivity investigations can provide power system engineers with a viable procedure to estimate the available system reserve margin to maintain adequate voltage security. Semlyen et al. [13] proposed a secant method based on increasing load admittances to evaluate the extreme loading condition. Using the minimum singular value of the power flow Jacobian matrix as a static voltage stability index, LSf et al. [14] proposed a fast method to calculate the minimum singular value and the corresponding singular vectors. Chebbo et al. [15] developed a voltage collapse indicator based on the optimal impedance solution at the maximum power transfer state. Their theoretical frame was similar to that of Barbier and Barret [3]. Their voltage collapse indicator was formulated and tested when the load at a particular node or the system load increases gradually. A new voltage stability assessment software package, VSTAB [16], was recently developed by the Electric Power Research Institute. VSTAB conducts voltage stability assessment using the voltage stability index developed by N e w t o n - R a p h s o n Jacobian matrix eigenvalue analysis [17].
The above results are representative of different works on voltage collapse. They reveal the voltage collapse mechanism from different aspects and assess voltage security from different points of view. Each of them has its own merit, though the system models adopted could be oriented more to actual system physical conditions. Some of them did not include the reactive power generation limits, although reactive power depletion is a major cause of voltage collapse. Some did not take the load characteristics into account, though load characteristics can be beneficial or deleterious to voltage stability. Others did not consider the active power flow effects on the voltage stability, though the active and reactive powers of a load are correlated and, together with the active power generation, contribute to the change of the voltage profile. The purpose of voltage collapse analysis is to develop methods and tools for predicting the proximity of the system to the voltage stability boundaries, providing guidance for the maintenance of the system voltage stability, and, ultimately, averting the voltage collapse. Among different methods, sensitivity analysis has demonstrated an advantage in its capability to provide a voltage collapse index and suggest corrective action. A properly modeled sensitivity index or predictor can be very helpful in the operating and planning procedures. A representative report on the sensitivity method has been given by Schlueter et al. [9]. They linearized the power balance equations at an equilibrium state and developed systematically a theoretical framework for the sensitivity model. Taking the same direction, but accommodating more power system physical conditions, the present research builds a new sensitivity model for voltage collapse prediction. The physical conditions that constrain the power system have been scrutinized and represented in the sensitivity model formulation. In particular, the load characteristics and generation dispatch strategy are explicitly included in this model. A voltage stability index, which reveals the voltage collapse mechanism and indicates the degree of the voltage instability, is constructed using the data from the improved sensitivity model. A voltage collapse predictor, which predicts the maximum power transfer of the power system, is also developed on the basis of this sensitivity model. The new sensitivity model has been tested numerically and proves to be accurate and sensitive for voltage stability analysis and voltage collapse prediction.
183
2. Definition of voltage stability in terms of sensitivity Definitions of voltage stability based on sensitivity matrices have been documented in recent literature [6, 7, 9]. They are proposed based on the heuristic cause/effect relationships that exist under normal conditions of a power system at load buses. Based on similar heuristic logic, if P and Q are bus power injections, the voltage stability is defined in terms of sensitivity as follows: when AVpv sw(t)--0, any nonzero nonnegative power injection disturbance, APL(t) and AQL(t), causes the state AV(t) to become nonnegative; where APL(t) and AQL(t) are power injection increments at load buses, and the A Vpv sw(t) are voltage increments at P V buses and the swing bus. Here, a P V bus is a bus with specified active power generation and sufficient reactive power support to keep its voltage magnitude at a specified level, and a swing bus is a bus with sufficient active and reactive power support to keep its voltage magnitude and phase angle at a specified level. This definition considers both APL(t) and AQL(t) as the disturbance variables affecting the system state. Although the most direct reason for voltage change is reactive power flow, including APL(t) in voltage collapse analysis is not trivial. This is because load disturbances APL(t) and AQL(t) may have a ratio corresponding to the load characteristics. That ratio may override the fact that voltages are mostly affected by the reactive power flow. Therefore, the A V(t) resulting from the active and the reactive parts of the injected power disturbance may be comparable in magnitude, and such that the AV caused by the APL(t) has to be considered. On the other hand, the active power generation dispatch strategy has a significant effect on the system power flow and hence the system voltage profile. Generally, there is a certain relationship between the active power generation dispatch strategy and the change of the active load and transmission losses. Therefore, the voltage collapse analysis has to represent the relation between the active load change and the coordinated generation dispatch correctly, as is done in the next section.
sitivity matrix model is analyzed and constructed in this section. The disturbance variables considered in the model are
d = [PpQ, QpQ] where PpQ and QpQ are vectors with dimension npQ, where npQ is the total number of PQ buses whose active and reactive powers are specified for power flow. The PQ bus group may contain load buses, network buses and PQ buses converted from P V buses because of their reactive generation limit violations. The control variables are
U :PPv where PPv is a vector with dimension npv which is the number of generating P V buses whose active powers are specified for power flow. The derivation of the voltage sensitivity relative to the disturbance and control variables is based on the physical constraints on the power system. At an equilibrium state, the power system is constrained by the network law equality constraints (e.g. bus active and reactive power balances) and automatic control imposed equality and inequality constraints (e.g. specified bus voltages, specified line flows and other operation limits). When the loads change, the system voltage profile and the automatic generation control respond to maintain the system power balance. Therefore, it is necessary to represent certain physical relationships in the voltage stability analysis. These physical relationships include load-voltage characteristics and the generation dispatch strategy. Load-voltage characteristics determine the relationship between the active power, the reactive power and the voltage magnitude. As functions of voltage, the active power and reactive power consumed by the load may change as the voltage changes, and thus the power factor of the load may change with the voltage. In the following sensitivity analysis, this relationship is represented as
where 51
F= 3. Voltage stability sensitivity matrix model Following the voltage stability definition given in the previous section, the voltage stability sen-
(la)
QpQ = F~PpQ
=
52 5npQ
with dimension npQ × npQ. The 5i = Qi/Pi are determined by the load characteristics. Later,
184
the load voltage characteristics, that is, PpQ = feE(V) and QpQ = fQL(V), are included again in the Jacobian matrix formulation. The generation dispatch strategy determines the relation between the load change and the generation distribution. Different generation dispatch strategies satisfy different operating requirements and lead to different equilibrium states. In the power flow, for any type of dispatch strategy, the relationship between the changes of active power generation and the changes of load can be represented by
APpv =
np~ APi
__
where i -- 1 , . . . , ng, and n~ is the total number of generators available for economic dispatch. The swing bus is included in these generators. Since 27~1 PGi is equal to the system load P~y~load, it follows that the active power generation of generator k is 1 e G k = P s y s load "-g l ~-
i~l Cii
n~ 1 \i=1 ~
bk 2Ck
i~_1C~i
and thus its dispatch coefficient is APGk flk = #k APsysload
i=1
1
1 Ch ~/k n g ~
rtpv i=l
or
(lb)
A P p v = F , APpQ
where the coefficient matrix is
F~=-
f12 PV
...
fi2
"""
[~npv
with dimension n p v x neQ. In the F, formula, the negative sign results from the bus injection assumption in the power flow calculation (i.e. generated power is positive, while load power is negative). The elements of F¢ are the dispatch coefficients fli =
APG~ n~° A P i i=1
In the above formula, parameter PGi is the active power generation at P V bus i, which is obtained at an equilibrium point according to the dispatch strategy such as inertial power flow, governor power flow or economic dispatch. Taking the economic dispatch (ECD) as an example, if the cost function of a generator is f(PG) = a + bPG + cPa 2
the system loads are dispatched to the generators according to the equal-increment rule. For a certain constant 2, all the generators satisfy df(PGi) _ dPa~
that is, bi + 2 c i P G i :
The coefficient Ph is a factor that accounts for the transmission loss increment together with the load increment. It is determined by the dispatch strategy; for example, it may be estimated from the previous power flows or approximated by experience. If, during the power flow, the equalincrement rule forces some generators to hit their generation limits, those generators are set to the violated limits with their dispatch coefficients set to zero. The other generators get the undispatched load dispatched according to the equal-increment rule. The dispatch procedure continues till no generator is violating its constraints. The resulting active power generation PGk settings for all the generators are used for power flow. The resulting dispatch coefficients flk are used for the sensitivity matrix modeling. For sensitivity analysis, the equality constraint equations are linearized at an equilibrium point. These equations include the active and reactive power balance equations and the specified line flow equations. In the linearization of these equality constraint equations, the state variables are the voltage phase angles at all the buses except the swing bus, the voltage magnitudes at the uncontrolled load network buses and the constrained generator buses, the turns ratios of the load tap changers or the susceptances of the capacitor banks that are attached to the controlled load network buses and are well within their operating limits, and the phase angles of the phase shifters that are attached to the specified branch and are well within their operating limits. In the linearization, the disturbance and control variables include the specified active powers and the specified reactive powers. To simplify the notation, the power increments
.185 are denoted by [AYp, AYQ] T, whose subvectors are
I-AP, ]
AYp = LAPpvJ
I-AQ.q
and
AYQ = LAPij J
The state variables are denoted by [X~, X v ] w whose subvectors are
X,~ =[~1
and
Xv=
F1 tij
The linearized system (total differential of the equality c o n s t r a i n t equations) is
AYQA
jl-Ax ] LAx.j
(lc)
w h e r e the coefficient m a t r i x J is p a r t i t i o n e d as
tivity matrices Sp and SQ are derived by solving the linear system in (lc). The derivation is given below. L i n e a r i t y leads to the state variable AXv being the superposition of its responses to different disturbances or control inputs. Therefore, to obtain Sp, the voltage sensitivity relative to the active power flow, it is assumed t h a t AYQ = O. Then, from eqn. (1), it follows t h a t
AYp = J1 AX~ + J2 AXv
(2a)
0 = J3 AX~ + J4 A X v
(2b)
APpv = Fz hPpv
(2c)
To solve for the voltage sensitivity Sp, m a t r i x calculations are performed. To m a i n t a i n the acc u r a c y and robustness of the sensitivity model, the matrix inversions are only performed on square matrices. Equations (2a) and (2b) yield AX(vP) = - J4-'J3 [ J~ - J2 J4 :'J3 ] - ~AYp
with
Substituting (2c) into (3a), it follows t h a t
[(?Yp]
[OYp
J =L X J =
rSYo]
C~fpL(V)] J
[(?Yo OfQL(V)]
J4=L
J
w h e r e Zl is a square matrix with dimension nye x nye, J2 is a r e c t a n g u l a r m a t r i x with dimension nyp x nyo , Ja is a r e c t a n g u l a r matrix with dimension nyo x nyp, and J4~is a square matrix with dimension ny 0 x nYo. In' the above formulas, nvp is the n u m b e r of equality constraints on bus active powers, and nvo is the n u m b e r of the r e m a i n i n g equality constraints, tha,t is, bus reactive power balances and/or specified line flows. The expressions for these partial derivatives are the same as those in the N e w t o n - R a p h s o n power flow J a c o b i a n matrix, which is well k n o w n and is not repeated here. At an equilibrium state obtained from an operating state or a power flow solution based on a sufficient set of given conditions, assuming linearity in the vicinity of a system equilibrium point, the voltage stability sensitivities are evalu a t e d by _
(3a)
FAPpQ]
w h e r e Sp is the active power related voltage sensitivity m a t r i x with dimension nvQ x nvp, and SQ is the reactive power r e l a t e d voltage sensitivity m a t r i x with dimension nyQ × nyQ. The sensi-
AX~ ) = -J4
1Ja[J1- J2Jn-lJa]-
F e APPo
w h e r e APpo are load level increments, and the voltage sensitivity relative to the active power flow is Se, w h e r e Se = -J4-1J3[J, - J2J4-1J~]' i
(3b)
}
Similarly, to obtain SQ, the vo]tage sensitivity relative to the reactive power flow, it is assumed that A Y p = O. Then, from eqn. (I), it follows that 0 = J~ AX~ + J2 A X v
(4a)
AYQ = J3 AX~ + J4 AXv •
(4b)
This yields AX(vQ) ----[ - J 3 J 1 - 1 J 2 -i- J4] -1 AYQ
If Pij is specified, it follows t h a t l
AXe) = [ - J 3 J l - l J e + J4]
1
XL~j AQpQ
(5a)
w h e r e AQpQ are load level increments, and the second coefficient m a t r i x has rows relative to the specified line flows set to zero. The voltage sensitivity relative to the reactive power flow is SQ, where Sq = [ - J 3 J l - l J 2 + J4] -lII01
(5b)
Combining the voltage sensitivities relative to both the active a n d reactive power flows, the voltage sensitivity equations solved from (1) are
186
hXv = AX~ ) + AXe) =Sp[I l APpQ -~So AVPo
(6a)
Substituting (la) into (6a), it follows t h a t AX V : SpQ APpQ
(6b)
where the voltage stability sensitivity matrix is
SpQ:Sp[I ]~- SQF~
(6c)
The sensitivity matrix SPv and its corresponding sensitivity equations provide accurate sensitivity data for both the voltages at the PQ buses and the parameter settings of the automatic regulators included in Xv. In a power flow, the PV bus reactive powers, QPV, are resources to keep the Vpv at a presumed level. Because of thermal limitations or other types of operating limit, when generators or reactive compensators violate their operating limits, the PV buses lose their ability to maintain their voltage levels. This procedure is reflected in the system reactive support depletion, which is the primary cause of voltage collapse. In the power flow calculation, these PV buses have to be converted to PQ buses and, consequently, the voltage stability sensitivity matrices have to be rearranged according to the conversion. In other words, the PQ group is enlarged by the buses converted from PV to PQ buses, while the PV group shrinks by the same number of buses. The regulating devices may also hit their operating limits. As a result, the dimensions and contents of the Jacobian matrix used in the sensitivity model change and cause corresponding changes in the sensitivity matrices. According to the voltage stability definition in the previous section, the system is voltage stable when the voltage stability sensitivity matrix SPv columns relative to buses in the PQ group are all nonnegative. When a matrix inversion in the SPv does not exist, or SpQ contains negative components relative to the load bus voltage, the system is voltage unstable. In this case, when loads increase or, equivalently, generations decrease, the system state undergoes the bifurcation t h a t may lead to voltage collapse. The voltage stability sensitivity matrix discussed above has included load characteristics and other power system physical constraints. In the steady-state stability domain it provides a robust foundation for voltage collapse analysis. The partial sensitivity matrix Sp reflects the effect of active load changes and the relative generation
changes on the system voltage level, while the partial sensitivity matrix SQ reflects the effect of the reactive load changes and reactive resource dispatch on the voltage level. When the voltage stability sensitivity matrix SpQindicates that the system has a voltage problem, Sp and SQ provide more detailed information concerning the voltage sensitivity and help in the decision making which may avert a voltage collapse, for example, resource dispatch, load shedding or brown-out.
4. Improved sensitivity approach-application of voltage stability matrix In the previous section, an improved voltage stability sensitivity matrix has been modeled and its formulation, based on the system condition, has been analyzed. This section discusses the utilization of the improved voltage stability sensitivity matrix in the voltage stability analysis and voltage collapse prediction.
4.1. Voltage stability index As discussed in the previous section, theoretically, the loss of voltage stability can be detected by the sign of the voltage sensitivity from the sensitivity matrix SpQ. In practice, when the voltage stability sensitivity matrix SpQ contains negative components in the columns relative to PQ buses, the system voltage levels are already extremely low. The power system equilibrium points are hardly located there. Instead, when voltage levels drop below the operating limits endurable by the power system equipment, the system state bifurcation usually occurs, resulting from the changes in load operating mode or power system equipment setting. Therefore, the sign shift from the voltage stable state to the voltage unstable state, described in the previous section, may not be helpful under most power system circumstances. The most dangerous case of voltage depression, which most probably causes voltage collapse, is marked by the sudden overshoot of the acceleration of the voltage decrease relative to the load increase or, equivalently, the generation decrease. In this case the system does not have enough margin to alert and avert the voltage collapse. Near the current equilibrium point, the voltage stability sensitivity matrix provides the sensitivity of voltage decrease relative to the load change. It can be used to estimate the acceleration of the voltage decrease caused by the load increase, and thus predict the proximity of
187
the steep voltage depression that may lead to a voltage collapse. For a practically sized system, it is very difficult and unnecessary to monitor the whole sensitivity matrix. A practical solution is to identify the weak buses in the system and monitor their voltage sensitivities. The weak bus identification is determined not only by the power systern analysis, but also by the system engineering circumstances. An example of a weak bus is one where there is a large electrical distance (the transmission impedance weighted by the intermediate bus loading) between it and the nearest generating P V or swing bus. The transfer of power to a weak bus causes further transmission loss and voltage depression together with phase shift. This will also occur when a bus is at the end of a long transmission line between voltage control areas. The voltage stability sensitivity matrix indicates this case numerically by the largest diagonal element corresponding to the weakest bus. The second type of weak bus is that at which there is a sustained loading increase. The power system voltage problem and voltage control related to reactive power flow tends to be localized. Therefore, the voltage at a bus will be affected most directly if the bus sustains increasing loading. The voltage stability sensitivity matrix reflects this case by a fast growing diagonal element corresponding to the bus. Finally, the buses of importance or high priority should also be considered as weak buses and monitored. A power flow simulation proved that the sensitivity matrix SpQ predicts a sharp voltage drop by a jump of order five (i.e. equal to or larger than five times) in the rate of change of the monitored SpQ components corresponding to the weak buses. The steps of voltage collapse prediction using the voltage stability index dSpQ(i, i)/ dx can be summarized as follows. Step 1. Use (6c) to obtain SpQ(i, i) at the current state, where i is the row number relative to the weak bus. Step 2. Use (6c) to obtain SpQ(i, i) at a state with a disturbance such as a load increment in a specified pattern. Step 3. Obtain dSpQ(i, i)/dx, where x is the normalized load increment; if it is of the order of five times higher than its base-case value, the system is confronting a voltage collapse.
4.2. Maximum power transfer predictor The power balance equations are nonlinear and the P - V curves from power flow simulations have a parabolic shape. This observation leads to
an equivalent quadratic model P(V) = r2 V 2 + rl V -~- r0 where P is the local or system load level, and r0, rl, and r2 are the unknown coefficients of the quadratic function. If the quadratic model is correctly built, the system maximum transfer can be readily obtained as the maximum value of the quadratic function. To obtain the coefficients r0, rl, and r2 in the quadratic model, different curve fitting methods can be used. By including the load characteristics, dispatch strategy and reactive generation limits, the new sensitivity model provides a solid basis for the sensitivity analysis method. A new sensitivity approach is thus developed to solve the quadratic function and obtain the maximum transfer. The coefficients of the equivalent quadratic function, expressed in sensitivities, are 1 d2P r 2 - ~ dV 2'
dP rl= dV-2r2V
(7) ro = P -
r2V 2 - r l V
The predicted maximum transfer is rl 2
Pmax = ro -- - 4r2
(8)
Obviously, the manner of obtaining the sensitivity data, that is, dP/dV and d2P/dV 2, at an equilibrium state is a key factor in making a correct prediction of the maximum transfer. Here the sensitivities in the above formulas are obtained using the improved voltage stability sensitivity model developed in this research, which can be evaluated at a simulated system state or at an operational equilibrium state. For a single-load variance case, the sensitivity equations give directly AVL = SpQ(L, L) APL
(9)
where the sensitivity SpQ(L,L) is the diagonal element of the voltage sensitivity matrix SpQ corr e s p o n d i n g to the load bus. Then it follows that the first-order load sensitivity is d P L ,., A P a
1
d~L ~ AVL
SpQ(L, L)
(10a)
To obtain the second-order sensitivity, the power flows have to be solved for the same singleload case at two nearby loading levels, that is, p(~-1) and P(~). The results are used to calculate dP(~-I)/dVL and dP(~)/dVL, and then the secondorder sensitivity is obtained as
188
d2pL dP<~)/dV(~I)-dP(~L L
1)/dV(~
dYE 2
"
1)
(10b)
The maximum transfer can then be readily obtained using eqns. (7) and (8). For a system with a group of loads increasing by a similar a m o u n t x relative to the base case, the sensitivity equation gives A V = SpvPbase Ax
(11)
where Pbase is a column v e c t o r containing the base loadings of the load group. Selecting VL of a bus with a relatively low voltage as the independent variable of the equivalent q u a d r a t i c function and x as the dependent variable, then the first-order sensitivity is e v a l u a t e d by dx Ax = ~ AVL
transfer using eqn. (8) (or (8) with P replaced by x). The maximum transfer prediction method discussed above has been applied to several test systems. The predicted maximum transfers are very a c c u r a t e at different states, t h a t is, states a w a y from or a d j a c e n t to the simulated maximum transfer state. This means t h a t the maximum transfer and the corresponding critical voltage can be predicted by the data obtained at a couple of equilibrium states t h a t are not necessarily near the voltage collapse state or riding on the knee o f a P Vcurve.
5. T e s t r e s u l t s
(12a)
where the sensitivity coefficient is dx Ax 1 (12b) d VL "" AVE [SpQ(L, ")]gbase with [8pQ(L, ')] as the row of Spv relative to VL and Phase. In the same w a y as for the single-load case, the power flows have to be solved with the same load group at two n e a r b y loading levels, x~ -1) and x~ ). The results are used to calculate dx(i-1)/dVL and dx(i)/dVL, and then the secondorder loading sensitivity is e v a l u a t e d by d2x dx(i)/dV(~) -- d x ( i - 1 ) / d V ( ~ -1) dVL2~ V~ ) _ _ V ~ i _ 1) (12c) The predicted apex value of x can be readily obtained using eqns. (7) and (8) with the dependent variable P c h a n g e d to x. The maximum loading at each load bus of the load group is Xm~x times its base-case value. In the case of a group of loads increasing correlatively, the maximum transfer can be obtained with the same m e t h o d o l o g y as illustrated above for the groupload increase case. As a s u m m a r y of the above discussion, the steps for the application of the quadratic model in a maximum transfer prediction are listed below. Step 1. For a single-load increase case (or a group-load increase case), solve the sensitivity data needed for eqn. (10) (or (12)) at the c u r r e n t state or a projected state. Step 2. For a single-load increase case (or a group-load increase case), solve for the coefficients in the equivalent function using eqns. (10) and (7) (or (12) and (7) with P replaced by x). Step 3. For a single-load increase case (or a group-load increase case), o b t a i n the m a x i m u m
The test results from the I E E E 30-bus test system are presented in this section to illustrate the performance of the improved sensitivity approach. The l o a d - v o l t a g e characteristics and s h u n t c a p a c i t o r b a n k s were added to the test system to extend the simulation cases. The simulation data provided the P - V curves for bus voltage versus load level. As an example, the system load increase P - V curves for a relatively w e a k bus, bus 30, which is furthest a w a y from the generation, and a relatively strong bus, bus 7, which is located in the center of the generation area, are given in Fig. 1. The curves of bus voltage phase angle versus system load level are given in Fig. 2. In Fig. 1, at the knee or the steep voltage drop of the P V curve, the voltage m a g n i t u d e of bus 7, 177, is still quite high, a b o u t 0.9 p.u., and the phase angle of bus 7 in Fig. 2 is far below 90 °, the
l.I l.O
0.9 A v >
0.8 0.7 V30
0.5 0.5 0.5
V7
I
I
I
I
!
1.0
1.5
2.0
2.5
3.0
3.5
x (P / Po) Fig. 1. P - V b u s 7.
c u r v e s ( s y s t e m l o a d i n c r e a s e c a s e ) for b u s 30 a n d
189
.,0
-20
•~ -30
-40
I*
-50 0.5
,
,
1.0
1.5
delta'/
,
2.0 X (P / Po)
,
,
2.5
3.0
3.5
Fig. 2. B u s 30 a n d b u s 7 v o l t a g e p h a s e a n g l e s (delta) v e r s u s s y s t e m loading.
angular limit. It is clear that neither the voltage magnitude nor the phase angle indicates the voltage collapse directly. With Fig. 1 as a reference, let us first look at the test result of the voltage stability index. The monitored weak bus is bus 30, whose corresponding voltage stability index, dSpQ(30,30)/dx, that is, the rate of change of SpQ(30, 30), versus system loading level x (P/Po) is shown in Fig. 3. The index value near x = 3, which is at the knee of the P - V curve in Fig. 1, is extremely high (about 30) and is not shown in the Figure. In Fig. 3, at a load level greater than 2.3, the voltage stability index of bus 30, dSpQ(30, 30)/dx, rises to about five times its base-case value. This
index value indicates that the system is confronting the knee of the P - V curve, that is, the sudden drop in voltage level. This prediction is proved by the simulated case shown in Fig. 1. In Fig. 3, the voltage stability index values of bus 30 for all the states from Fig. 1 clearly indicate that the voltage stability index increases when the system states move nearer to the knee of the P - V curve. At the knee it jumps to an extremely large value, which shows that the voltage stability index is very sensitive to voltage collapse. Next, with bus 30 as the monitored weak bus, the test results of the maximum transfer predictor are demonstrated. The adjustment of the voltage regulator has a significant effect on the system maximum transfer and voltage stability. As the voltage problems arise, the voltage regulators and compensators are fully adjusted and thus their limit settings are used in the following discussion. With Fig. 1 as a reference, the predictions are performed at states both away from and near the knee of the P - V curve, with both high and low voltage levels. The predicted maximum transfers and the corresponding critical voltages are very consistent. The predicted P - V curve for a state at a high voltage level near the base case is given in Fig. 4. Also in Fig. 4, for comparison with the predicted P - V curve, a simulated case is shown with the regulators and compensators set to their limit settings, taken from the simulated knee point of the P - V curve in Fig. 1. In Fig. 4, it can be seen that the prediction is very accurate, that is, there is only a 2% error in the predicted Xmax and only 5% in the corresponding critical voltage. The maximum transfer
10
1.2 state where prediction is made 1.0
~" 6
A
t:~ 4"
V o 0.8 o~ ;>
0.6
0 0.5
1.5
2.5
3.5
x (P / Po) Fig. 3. V o l t a g e s t a b i l i t y i n d e x dSpQ(30, 30)/dx v e r s u s s y s t e m l o a d i n g level x (P/Po).
¢
0.4 0.5
Xpredict~l
i
!
1.5
2.5
3.5
X (P / PO) Fig. 4. C o m p a r i s o n of t h e s i m u l a t e d a n d t h e p r e d i c t e d P - V curves.
190
predictions demonstrated in the current literature have not achieved this kind of accuracy at states away from the knee of the P - V curve. The accuracy of the predictor from this research results from the correct modeling of the system physical conditions and the careful development of the predictor modeling technique. The improved sensitivity approach was tested with various coefficients of the load-voltage characteristics. It predicted the voltage instability and maximum transfer in agreement with the corresponding power flow simulation results. Test results from different cases, for example, single-load increase and group-load increase, also proved that the voltage stability index is sensitive and accurate for voltage collapse prediction, and the maximum transfer predictor provides consistently accurate results. These results indicate that the improved sensitivity approach is promising for voltage collapse analysis and prediction, and it is hoped that its application will lead to further refinements.
6. C o n c l u s i o n s An improved sensitivity approach has been developed. By explicitly including load characteristics and generation dispatch strategy, the sensitivity approach is made more pertinent to voltage collapse prediction and suggestions for corrective action. The sensitivity model can be calculated using the present power flow results, which can be easily obtained from the N e w t o n Raphson power flow program. The accurate solution provided by the method discussed above can be used with more approximate treatments in voltage collapse prediction, margin estimation and reactive resource planning.
Acknowledgements The authors would like to thank H. P. Nguyen, J. M. Adams and C. W. Fromen of Houston Lighting and Power Company for their collaboration during this research.
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