COMPOSITE LOAD SENSITIVITY IN VOLTAGE STABILITY PROBLEM SOLVED BY UNIFIED POWER FLOW CONTROLLER

Nijaz Dizdarević Sejid Tešnjak Göran Andersson Energy Institute HRVOJE POŽAR Fac. of Elec. Engr. and Computing ETH Zürich Zagreb, Croatia University of Zagreb, Croatia Zürich, Switzerland www.eihp.hr/~ndizdar sejid.tesnjak@fer.hr andersson@eeh.ee.ethz.ch

Abstract - Alleviation of voltage stability problem by using Unified Power Flow Controller (UPFC) represents main concern of this paper. The UPFC is analysed as the device capable of giving voltage support and/or co-ordinating power flow control action. Voltage stability problem is viewed from longer-term time domain response of a composite load model. Sensitivity analysis is applied with respect to parameters of individual elements of the composite load model. Computational procedure reveals critical load parameters as well as critical time instant during time domain trajectory. The UPFC is accordingly initiated to simultaneously support bus voltage magnitudes and adjust power flow through the line in order to decrease total active power loss.

Keywords - voltage stability, composite load,

sensitivity, FACTS, UPFC

1 INTRODUCTION

The voltage stability problem with voltage collapse as its final consequence is an emerging phenomenon in planning and operation of modern power systems. The increase in utilisation of existing power systems may get the system operating closer to voltage stability boundaries making it subject to the risk of voltage collapse. Since the rapid development of power electronics has made it possible to design power electronic equipment of high rating for high voltage systems, the voltage stability problem resulting from transmission system may be, at least partly, improved by use of the FACTS-controllers.

Dynamic simulation of a power system is recognised as one of the most important tools of its assessment [1-3]. Load behaviour is denoted as one of the main driving forces of the voltage collapse. A composite load is used to exhibit longer-term recovery intention. Its response is dominated by dynamic behaviour of an LTC transformer, an induction motor, and a thermostatically controlled load. The other driving force is related to a limited generator reactive power. Over-excitation limiter is modelled in order to enable a decrease of generator excitation voltage. Analysis of a power system with embedded FACTS-controllers calls for development of adequate models. This paper is solely concerned with system wise aspects of the FACTS. In order to deal with the voltage stability problem, the solutions with FACTS-controllers must provide voltage support and/or appropriately co-ordinated control actions. Apart from several larger rating prototypes of Static Var Generator, no other FACTS application has

been considered to help solve the voltage stability problem. The UPFC is considered to be a new impetus.

In this paper, the impact of the UPFC on alleviation of voltage stability problem is investigated from combined dynamic and static aspects [4]. Combined model enables usage of different types of predictive indices. Indices based on singular value analysis do not only predict severity of the problem, but also open a way toward a general sensitivity analysis. A small-disturbance impact to the combined model results with a sensitivity of an operating point along time domain trajectory. The UPFC injection model is included in the overall system model. The control system of the injection model is proposed and the benefits within voltage stability problem are explored by analysing a multi-machine test system. During time-domain simulations, the extended Jacobian and state matrices are evaluated utilising linearised differential equations of generators and recovery loads as well as steady-state load flow equations. From a static approach, according to singular value decomposition of the matrices, the participation factors are computed resulting with the most critical components of the composite loads in the system. Sensitivity analysis of the matrices enables a critical point to be recognised denoting one of the possible moments for voltage support from the UPFC.

2 MODELLING 2.1 Multi-machine test system model

The benefits of the UPFC and its injection model with several types of control strategies are explored by analysing 6-machine, 400 kV test system (Figure 1). It is inspired by the one often used in CIGRÉ reports [5].

Figure 1: Multi-machine test system

The loads are either of the static ZIP type or the composite one. The composite loads are applied at buses 19 and 20. The UPFC is located in the middle of the line 26, which initially connects buses 19 and 20 with composite loads. During voltage collapse scenarios these loads are shown to have the largest contribution to instability due to their dynamics. The UPFC is located nearby these loads to solve voltage stability problem.

Dynamics of synchronous generator is represented by appropriate transient model [6]. The transient effects are accounted for, while the sub-transient ones are neglected. The differential equations which describe dynamics of synchronous generator are written for four generator state variables, namely δ, ω, Ed'gen, and Eq'gen. The algebraic equations that describe generator voltage/current relations are written for four generator algebraic variables, namely Id

gen, Iqgen, Vd

gen, and Vqgen.

The excitation system [1] is modelled in a very simplified way. Only one differential equation, written for its state variable EFD, describes corresponding dynamics. The over-excitation limiter (OEL) [1] is included, with only one differential equation being written for its state variable VOEL. Its algebraic equation concerns algebraic variable IFD.

The speed governor-turbine system [6] is represented according to its general model. It has four differential equations written for the state variables GOV1, GOV4, GOV5, and Tm. 2.2 Composite load model

Composite load model [1, 2] is discussed with respect to its individual elements (Figure 2). Looking from a high-voltage side (HV bus), first there is an LTC transformer. Tap changer is located at the HV primary side controlling the voltage magnitude of the secondary low-voltage side (LV bus). Local reactive power compensation is provided at the LV bus by a fixed shunt capacitor bank (SHUNT CAP). Equivalent impedance of a distribution feeder, Zeq, is located between the LV bus and a load one (LOAD bus). The LOAD bus has induction motor load (IM), thermostatically controlled load (TCL), and static load (SL) connected on itself.

Figure 2: Composite load model

Transformer is represented as a 4-pole element by using shunt injected powers. Its series susceptance has constant value, while the injected powers depend on the tap ratio tr. The LTC scheme is applied for voltage regulation of the secondary side [2].

In order to decrease reactive power flow through the high-voltage system, the shunt reactive compensation is provided locally at the LV bus. Fixed shunt capacitor is

used. Its susceptance, bSH, is positive for capacitive load, and negative for inductive one. The equivalent impedance of lumped distribution system, Zeq, is located between the LV bus and the LOAD one.

The induction motor is represented by using its transient model with additional internal bus [2]. At the internal bus, the transient voltage is set to adjust for dynamic behaviour of the motor. Its differential equations are written for three state variables, namely Eq'mot, Ed'mot, and ωm. Besides the differential equations, the transient model includes four algebraic ones describing motor voltage/current relations. They are written for four algebraic variables, namely Id

mot, Iqmot,

Vdmot, and Vq

mot. At the LOAD bus, the composite load model also

includes a thermostatically controlled load (TCL) [1]. The TCL represents aggregated thermal heating consumes supplied from the bus. The TCL active load power depends linearly on the conductance G, and quadratically on the bus voltage magnitude Vn. Generally, if the bus voltage magnitude is decreased, the conductance G is increased through temperature τH feedback control. The TCL model is described by using two differential equations written for its state variables τH, and G. It does not have any algebraic variables.

At the LOAD bus, there is also a part that is considered to be static in nature. Static load (SL) depends on the bus voltage magnitude Vn [1]. It is commonly known as a ZIP load model. 2.3 Injection model of the UPFC

The UPFC can provide simultaneous control of all basic power system parameters (transmission voltage, impedance and phase angle) and dynamic compensation of ac system. The controller can fulfil functions of reactive shunt compensation, series compensation and phase shifting meeting multiple control objectives.

From a functional perspective [7], the objectives are met by applying boosting transformer injected voltage and exciting transformer reactive current (Figure 3). The injected voltage is inserted by using series transformer. Its output value is added to the network bus voltage from the shunt side, and is controllable both in magnitude and angle. The reactive current is drawn or supplied by using shunt transformer.

Figure 3: The UPFC device circuit arrangement

Functional structure of the UPFC results with appropriate electric circuit arrangement [8]. The series converter output voltage is injected in series with the line and acts as an AC voltage source (Figure 4).

Figure 4: The UPFC electric circuit arrangement

The reactance xS describes a reactance seen from terminals of the series transformer. The voltage source exchanges active power with shunt converter (CONV1), whereas reactive power of the converters could be controlled independently.

The UPFC injection model is derived enabling three parameters to be simultaneously controlled [8]. They are the shunt reactive power Qconv1, and the magnitude r and angle γ of the injected series voltage SV . The bus power injections of the UPFC are formulated and included in the injection model (Figure 5). The model is defined by the constant series branch susceptance bS, which is included in the system bus admittance matrix, and the bus power injections PSi, QSi, PSj, and QSj.

Figure 5: The UPFC injection model with control system

If there is a control objective to be achieved, the bus power injections are modified through changes of the UPFC parameters r, γ, and Qconv1. Appropriate control system of the injection model is proposed to govern the system to a pre-defined operating point by changing the set of its parameters [4]. It is of de-coupled single-input single-output proportional-integral type. The selection of input/output signals depends on the predetermined control mode, which could be changed during the simulation.

In the model, the shunt side could be controlled only in the voltage mode, Vi↔Qconv1, emphasising that Qconv1 represents reactive power loading of the shunt converter. The series side could be controlled through the r⇔γ pair in several different modes. The variables are chosen to satisfy general V↔Q and Θ↔P de-

coupling. The UPFC participates in transmission voltage support at the shunt side and in power flow regulation at the series side. At the series side, bus voltage magnitude could be controlled as well, enabling voltage support at both sides of the UPFC during voltage emergency situations. Then, the angle γ, as the third parameter, could be used to fulfil a functional objective by simultaneous three-parameter control.

If the UPFC is included between buses i and j, four network algebraic equations (Pi, Pj, Qi, and Qj) are influenced. In the Newton-Raphson method, partial derivatives of the network equations with respect to the bus voltage angles Θ and magnitudes V are employed. Thus, there are at most 16 positions in the Jacobian matrix concerned due to the UPFC injection model. It is supposed that the UPFC has an impact to the differential-algebraic model only through its load flow part described by the network algebraic equations. Due to fast responses of its control loops, it is assumed that the UPFC differential equations should not be necessarily included in the model. 2.4 Differential-algebraic model

The voltage stability problem is analysed from two aspects, a large disturbance (dynamic) aspect and a small disturbance (static) one. The large disturbance aspect is orientated towards the appearance of a three-phase short circuit and addresses post-contingency system response. The small signal aspect investigates the stability of an operating point and needs development of a linearised model.

It is supposed that the system has following set of differential-algebraic equations [9, 10]

),,( pyxfdtdx

= , (1)

),,(0 pyxg= , (2) where x and y denote state and algebraic variables, p arbitrary parameters, f and g differential and algebraic equations.

From the equations, it is concluded that the vector of the state variables is defined as x = [ δ, ω, Eq

’gen, Ed’gen, EFD, GOV1, GOV4, GOV5,

Tm, VOEL, Eq’mot, Ed

’mot, ωm, τH, G]τ, (3) whereas the vector of the algebraic variables is given as y = [ Θ1…n, V1…n, Id

gen, Iqgen, Vd

gen, Vqgen, IFD, Id

mot, Iqmot,

Vdmot, Vq

mot]τ. (4) By linearising (1-2) about an operating point, the set

of the linearised equations appears in general form as

pfyfxfdtdx

pyx ∆+∆+∆=∆ , (5)

pgygxg pyx ∆+∆+∆=0 . (6) Linearisation procedure, given by (5-6), results with

formation of the extended Jacobian matrix JEXT as

=

4321

yyyy

x

yx

EXT

gggg

g

ff

J , (7)

where the ordinary Jacobian load flow matrix gy1 is a part of the major sub-matrix gy. The sub-matrix gy is composed of four minor sub-matrices due to the additional algebraic equations which bring additional generator/motor algebraic variables.

The extended Jacobian matrix JEXT is used in computations of the system, which is given as

∆

∆

=

∆

y

xggffdt

dx

yx

yx

0

. (8)

Besides on the extended Jacobian matrix, the analysis is carried out on the system state matrix as well. By eliminating the vector of algebraic variables ∆y following system is obtained

xAdtdx

S∆=∆ , (9)

where the state matrix AS is given as ( )[ ]xyyxS ggffA 1−−= . (10)

2.5 Sensitivity analysis

Sensitivity analysis is introduced to recognise characteristics of the matrices JEXT and AS aimed to show critical issues in the voltage stability problem. Singular value decomposition (SVD) of the matrices provides an approach to a voltage stability assessment. If applied to the matrix JEXT, SVD results with the right and left singular vectors that correspond to the minimum singular value. By correlating entries within these vectors to the steady state power flow algebraic equations, further dynamics insight is provided, which is valid for the algebraic variables. If the approach is applied to the matrix AS, the insight is given to the state variables. Minimum singular value presents an index of impended voltage instability, but also warns on voltage security level of the system.

At an equilibrium point, the system (1-2) becomes ( )pyxf ,,0 = , (11)

),,(0 pyxg= . (12) Its linearisation at an equilibrium point results with

pfyfxf pyx ∆+∆+∆=0 , (13)

pgygxg pyx ∆+∆+∆=0 . (14) Upon mathematical elaboration, parametric

sensitivity of state and algebraic variables with respect to arbitrary parameters [11] are defined as

( )ppyyS fggfApx

−= −− 11∂∂ , (15)

( )[ ]pppyySxy gfggfAggpy

+−−= −−− 111∂∂ . (16)

Matrices fx, fy, gx, and gy are parts of the differential-algebraic model (8). In addition, matrices fp and gp are defined with respect to arbitrary parameters p. Parameters of the individual components of the composite load are considered. Thereby, following set of parameters appears given as p = [Tm0, τREF, tr, bshu, (PL

0)SL, (QL0)SL]τ, where Tm0 denotes value of the motor

load torque at the speed ωm0, τREF referent temperature of the TCL model, tr transformer ratio, bshu susceptance of the shunt capacitor, and (PL

0)SL, (QL0)SL initial active

and reactive power demands of the static loads. Besides parametric sensitivity, the functional

sensitivity is evaluated at snapshots of time domain simulations as well [3]. It is defined as a sensitivity of a function with respect to arbitrary parameters. Total generator reactive power production QG

Σ is considered as an appropriate scalar quantity η(x,y). Its sensitivity with respect to parameters p is obtained as a system-wide indicator, given by

( )

∂∂

∂∂

−= −

p

pEXT g

fJ

yxp1

ττ ηη∂η∂ . (17)

In the differential-algebraic system model, the total reactive power production of the generators is given as

( )∑∑==

Σ −==GENGEN n

i

gendi

genqi

genqi

gendi

n

iGiG IVIVQQ

11. (18)

From (18), it is seen that the entries of ∂η/∂x and ∂η/∂y for the total reactive power production are

0=∂∂

xη , (19)

( )gendi

genqi

gendi

genqi IIVV

y−−=

∂∂ ;;;η . (20)

Initial active (PL0)SL and reactive (QL

0)SL power demands of the static loads serve as arbitrary parameters p. The static loads are given as

( ) ( )β00 / iiLSL VVPVP = , (21)

( ) ( )β00 / iiLSL VVQVQ = . (22) The sensitivities (15-17) are helpful in detection of a

critical point. Its appearance represents a reliable sign of impended voltage unstable situation and could trigger voltage support from the UPFC.

3 NUMERICAL RESULTS

Numerical results are given with respect to an initial voltage collapse scenario. The scenario is based on a slow dynamics response of the system. It is initiated by a three-phase short circuit at bus 15 that is neutralised by outage of line 17. Due to initial voltage drop, the restoration of the composite load consumption followed in a longer-term. Generators 5 and 6 become OEL restricted, which leads the system towards the collapse.

In order to avoid the collapse, the UPFC voltage support is applied at several different time instants with respect to the critical point detection. The time instants are equal to t=250 s, 310 s, 340 s, and 380 s. Initial voltage collapse scenario (without the UPFC support) is also depicted. The UPFC two-parameter support is first activated. Step changes are applied to the referent values of the shunt and series bus voltage magnitudes of the UPFC. The support is activated by switching on the regulation modes Vi↔Qconv1 and Vj↔r through equal step changes in their references set at 0.95 pu.

The UPFC bus voltage magnitudes become equalised in the final part of each successful case (Figure 6). Initially unstable case collapses at t=589 s, whereas in the unsuccessful case of voltage support (t=380 s) the collapse appears at t=781 s with splitting of the voltage magnitudes at the UPFC sides. In successful cases, the stable operating point is established with more or less dynamics involved. For the successful case applied at t=340 s, dynamics is rather significant. Bus voltage magnitude Vj is regulated significantly faster. If the UPFC voltage support is applied earlier, the voltage stability problem is alleviated in successful cases.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400 500 600 700 800 900 1000

Bus

Vol

tage

Mag

nitu

des

V_i

and

V_j

(pu

)

Time (s)

initial unsuccessful

successful

Figure 6: The UPFC bus voltage magnitudes Vi and Vj While establishing the voltage collapse scenario, the TCL restorative intention is detected to be most influential within the composite load. Components of the singular vector product vn x un related to the state matrix AS, indicate sensitive state variables (Figure 7). Multiplied components of the critical singular vectors combine overall participation. Maximum number of the state variables is equal to 65 (1-55 for generators, 56-61 for induction motors, and 62-65 for thermostatically controlled loads). As the most sensitive state variables, the ones that belong to the TCLs are recognised. Especially, the TCL conductances G are detected to have the largest participation. In addition to the TCL state variables, the participation of the generators 6 and 5 appear due to the OEL activation.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10 20 30 40 50 60Com

pone

nts

of th

e S

ingu

lar

Vec

tors

Pro

duct

u_n

x v

_n

State Variables Numberings (max 65) Figure 7: Singular vector product vn x un of the state matrix AS

The TCL recovery intention appears due to temperature control. In each successful case, the TCL control schemes keep the temperature of the heated area τH at the pre-specified referent value τREF (Figure 8). In unsuccessful case, voltages collapse together with the temperatures. From the initial case, it is seen that around t=300 s the temperatures start falling significantly.

16

17

18

19

20

21

22

23

0 200 400 600 800 1000

TC

L T

empe

ratu

re t_

H a

t BU

S24

and

BU

S26

(de

g C

)

Time (s)

initial unsuccessful

referent

successful

Figure 8: Thermostatically controlled loads temperature τH

The UPFC voltage support has proven impact to the behaviour of the indices aimed for critical point recognition. Critical point is recognised at the moment when minimum singular value hits zero (Figure 9). Preferably, the support should be given upon its recognition or otherwise overall level of system dynamics becomes more pronounced. SVD of the state matrix AS exhibits sensitive responses upon the voltage support activation. Minimum singular value becomes largely increased in each successful case as the voltage support is timely provided. In the unsuccessful case, the support is provided too late. The voltages collapse and the motors lose stable operating points.

0

2e-06

4e-06

6e-06

8e-06

1e-05

1.2e-05

0 200 400 600 800 1000

Min

imum

Sin

gula

r V

alue

of S

tate

Mat

rix

Time (s)

successful unsuccessful

initial

Figure 9: Minimum singular value of the state matrix AS

Besides critical singular values, the results comprise the sensitivity ones as well. To serve illustrative purposes, the parametric sensitivities, ∂x/∂p and ∂y/∂p, are provided here only for the TCLs. The TCL conductance G and the bus voltage magnitude V are taken as state and algebraic variables. The TCL referent temperature τREF is taken as an arbitrary parameter. Thereby, parametric sensitivities ∂G/∂τREF and ∂V/∂τREF are evaluated along pre-established scenario.

Responses of the parametric sensitivity ∂G/∂τREF are provided (Figure 10). Around an initial stable operating point, the increase of the temperature τREF is achieved by the increase of the conductance G, and vice versa. The sensitivity ∂G/∂τREF is positive then. This sensitivity involves the inverse of the state matrix. In the initial case, as the singularity point is approached, the sensitivity becomes very large. At the singularity point, it abruptly changes the sign to the opposite one and moves towards zero within unstable region. The singularity point is considered as the critical one. It appears at the same moment when minimum singular value hits zero. As the support is timely provided in successful cases, the sensitivity ∂G/∂τREF is brought to a small positive value that is close to the initial one. In the unsuccessful case, the system eventually collapses.

Responses of the parametric sensitivity ∂V/∂τREF also denote stability (Figure 11). Around a stable point, increase of the temperature τREF is achieved by increase of the conductance G, which increases active power load, and therefore bus magnitude drops. Thus, it is negative around a stable operating point. In successful cases, it is brought to a small negative value that is close to the initial one. By giving the support later on, new steady state becomes harder to achieve. In the unsuccessful case, the system collapses in oscillations.

-30

-20

-10

0

10

20

30

0 200 400 600 800 1000

Par

amet

ric S

ensi

tivity

d G

/ d

t_R

EF

Time (s)

initial unsuccessfulsuccessful

Figure 10: Parametric sensitivity of TCL loads, ∂G/∂τREF

-3

-2

-1

0

1

2

3

0 200 400 600 800 1000

Par

amet

ric S

ensi

tivity

d V

/ d

t_R

EF

Time (s)

initial unsuccessful

successful

Figure 11: Parametric sensitivity of TCL loads, ∂V/∂τREF

The parametric sensitivity ∂y/∂p is also illustrated for the LTC transformer at bus 19. Responses of the sensitivity ∂V/∂ tr are provided for HV and LV bus

voltage magnitudes (Figure 12). Around a stable point, decrease of the tap ratio tr causes increase of the LV bus voltage magnitude. Then, the sensitivity ∂V/∂ tr is negative for the LV bus. The sensitivity for the HV bus is much smaller. It is close to zero and even has the opposite sign. In successful cases, the ∂V/∂ tr for the LV bus is brought to a negative value that is close to the initial one. The sensitivity for the HV bus is brought close to zero with the opposite sign. Final values of the sensitivity are the same for each successful case. By giving the support too late, the system collapses with induced oscillations as in the unsuccessful case.

-10

-5

0

5

10

0 200 400 600 800 1000

Par

amet

ric S

ensi

tivity

d V

/ d

t_r

Time (s)

initial unsuccessful

successful

Figure 12: Parametric sensitivity of LTC transformer, ∂ V/∂ tr

The functional sensitivity ∂η/∂p is also illustrated in presence of the UPFC support (17-22). It is computed with respect to each of the static reactive power loads at the LOAD buses. Responses of the QG

Σ are provided first (Figure 13). In successful cases, the QG

Σ becomes decreased. It is brought down approximately to a value prior to the first LTCs tap changes. In the unsuccessful case, the system ends up in the collapse.

18

20

22

24

26

0 200 400 600 800 1000

Tot

al R

eact

ive

Pow

er G

ener

atio

n (p

u)

Time (s)

successful

initial unsuccessful

Figure 13: Total reactive power production of generators QG

Σ

Responses of the functional sensitivities reveal stability (Figure 14). Around a stable operating point, if the load consumption is increased, the total reactive power production is increased as well. They are positive then. In successful cases of support, these sensitivities are brought to small positive values that are close to the initial ones. In the unsuccessful case, the support is given too late.

-30

-20

-10

0

10

20

30

0 200 400 600 800 1000

Fun

ctio

nal S

ensi

tivity

d Q

_G /

d Q

_L

Time (s)

initialunsuccessful

successful

Figure 14: Functional sensitivity ∂QG

Σ/∂Q at LOAD buses

As the last case, the UPFC third-parameter control is simultaneously activated. The successful case of two-parameter voltage support applied at t=250 s is retained for comparison. It denotes the case without LF control. The third-parameter control of the angle γ is applied additionally at t=750 s, while keeping the UPFC shunt and series side bus voltage magnitudes regulated. Its referent value is changed from 0° to 80°, 85°, and 87.5°. These changes increase the series side active power flow Pj2 from -1.84 pu to -1.91 pu, -1.97 pu, and -2.05 pu. The UPFC bus voltage magnitudes are successfully controlled at 0.95 pu (Figure 15). The third parameter γ control slightly decreases the total active power loss in the network (Figure 16).

0.948

0.9485

0.949

0.9495

0.95

0.9505

0.951

750 800 850 900 950 1000

Bus

Vol

tage

Mag

nitu

des

V_i

and

V_j

(pu

)

Time (s)

V_i

V_j

Figure 15: The UPFC bus voltage magnitudes Vi and Vj

1.553

1.5535

1.554

1.5545

1.555

1.5555

1.556

750 800 850 900 950 1000

Tot

al A

ctiv

e P

ower

Los

s P

_los

s (p

u)

Time (s)

w/o LF control

with LF control, 80deg

with LF control, 85degwith LF control, 87.5deg

Figure 16: The total active power loss Ploss

4 CONCLUSIONS

Benefits of the UPFC in voltage stability problem are explored from a viewpoint of the composite load. Differential-algebraic model of multi-machine power system is developed in order to study dynamic and static phenomena of longer-term periods with decreased voltage security. The UPFC injection model with adequate control system is included in the overall system. Upon recognition of the critical point through the sensitivity analysis, the UPFC is initiated to provide voltage support. It is shown that the UPFC is capable to provide enough support to avoid voltage collapse.

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