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An improved SSSC-based power flow controller design method and its impact on torsional interaction
Electrical Power and Energy Systems 43 (2012) 194–209
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An improved SSSC-based power flow controller design method and its impact on torsional interaction R. Pillay Carpanen ⇑, B.S. Rigby School of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, Howard College Campus, Durban 4041, South Africa
a r t i c l e
i n f o
Article history: Received 3 March 2010 Received in revised form 12 May 2012 Accepted 14 May 2012 Available online 19 June 2012 Keywords: Power flow control FACTS SSSC Subsynchronous resonance
a b s t r a c t This paper presents an improved method of designing a closed-loop AC power flow controller. The paper further examines the impact of this improved SSSC-based power flow control on the damping of the subsynchronous torsional modes of a turbine-generator. Both the constant power and constant angle modes of power flow control are considered; in each case the influence of the controller design on torsional mode damping is examined over a range of operating conditions. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The multi-inertia shafts of turbo-generators are susceptible to interaction with the electrical resonances of a transmission network compensated with conventional series capacitors and can lead to a form of instability known as subsynchronous resonance (SSR). This form of dynamic instability can cause mechanical failure of turbine shafts [1]. The emergence of Flexible AC Transmission Systems (FACTS) series compensators has led to a renewed interest in determining the SSR characteristics of these more advanced forms of series compensation, in particular the static synchronous series compensator (SSSC) [2,3]. Although, numerous applications of controllable FACTS series compensators have been considered in the literature, one particular application is the use of FACTS devices for closed-loop control of the flow of power down a particular line in an inter-connected transmission network [4,5]. Such closed-loop control of power flow in an AC system can provide a number of possible benefits: preventing unwanted loop flows in an interconnected system; allowing power to be directed along a ‘‘contract’’ path in a transmission system; preventing inadvertent overloading of lines already near their thermal limits [5]. However, more recently it has been shown [6] that a closed-loop power flow controller can affect the small-signal damping characteristics of a power system, and in particular that the nature and extent of these effects on the system’s stability depend on both the mode in which the power flow controller is operated and its controller response time. ⇑ Corresponding author. E-mail address: [email protected] (R. Pillay Carpanen). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.05.020
A logical question that then arises is whether the impact of closed-loop power flow control action on system damping reported in [6] could also influence the damping of any subsynchronous torsional modes in the shafts of the turbine-generators that feed the power system. However, in order to answer this question it is necessary to develop mathematical models of a representative study system in which both the torsional dynamics of the affected generators are adequately described, as well as the dynamic characteristics of the SSSC and its power flow controls. This paper begins by presenting the development of such a study system model, extended from earlier work described in [7]. In subsequent work [8], the results have indeed confirmed that a power flow controller can influence the damping of generator torsional modes and they further suggest that the influence of one particular mode of power flow control on torsional modes is operating point dependant. However, the design of SSSC-based power flow controller considered for study in [8] was relatively slow responding when compared to the response time of the power flow controller reported in work such as [9] which has shown that very fastresponse power flow controllers can be achieved and have been considered in the literature. This paper now uses the study system developed in [7] to examine the design of controller parameters for a SSSC-based power flow controller by using frequency-domain linearised transfer function analysis. In particular, the paper also uses the frequency-domain analysis to derive an understanding of how the design of the control gains of the power flow controller (and hence its speed of response) affects the resonant characteristics of a transmission network compensated by a SSSC. The paper further presents an improved design of the control gains of the
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power flow controller that enables a much faster controller bandwidth to be realised (comparable to that considered in [9]). Finally, the paper examines the impact of such fast responding power flow controllers on a generator’s torsional modes of oscillation, over a range of operating conditions. 2. Background theory 2.1. Subsynchronous resonance The shaft system of a turbine-generator has a complex mechanical structure comprising several predominant masses connected by shafts of finite stiffness [10]. Therefore such a multi-inertia system has several natural torsional frequencies fn of oscillation (torsional modes) usually in the subsynchronous frequency range. When conventional series capacitors are included in an RL transmission line, a series resonant RLC circuit is formed whose natural frequency fer is given by
1 pffiffiffiffiffiffi ¼ fo fer ¼ 2p LC
sffiffiffiffiffiffi XC XL
ð1Þ
where fo is the synchronous frequency of the system, XL = 2pfoL is the inductive line reactance, and X C ¼ 2p1f0 C is the capacitive compensating reactance. Such a network thus causes the transient component of any current in the line and armature to oscillate at a certain frequency which is closely related to the undamped natural frequency fer of the line. Thus, these oscillations produce rotor currents and torques at frequencies fo ± fer. If the frequency of the induced torques at the lower of these two frequencies (fo fer) coincides with one of the natural torsional mode frequencies fn, the shaft will be excited to oscillate at its natural frequency, sometimes with high amplitude. Such large shaft torques produce high shaft stresses, a reduction in shaft fatigue life and, possibly, shaft failure [10]. 2.2. Study system The system considered for study here is a modification of the IEEE First Benchmark Model (FBM) [10] for SSR analysis, altered to include two transmission lines in parallel as shown in Fig. 1. The study system, at present, consists of a 892.4 MVA turbine generator connected to a 500 kV transmission network via a step up transformer, where XT denotes the transformer reactance. The transmission network consists of two lines in parallel where RL denotes the resistance and XL the inductive reactance of the respective line. Line 1 is equipped with series compensation provided
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by an SSSC, where XSSSC denotes the effective capacitive reactance provided by the SSSC at the fundamental frequency. Although the model that has been developed for this study system also allows for the inclusion of series compensation in line 2, for the purpose of the investigations considered in this paper line 2 is left uncompensated. The transmission network is connected to the infinite bus via XS, the system reactance between the end of the transmission lines and the infinite busbar. The mechanical system comprises six lumped masses: a high pressure turbine (HP), an intermediate pressure turbine (LP), two low pressure turbines (LPA, LPB), the generator (GEN) and the exciter (EXC), which are mechanically coupled by shaft sections of known torsional elasticity. In the studies conducted thus far, the individual line parameters of this parallel-line system were changed (to twice their original resistance and inductive reactance) such that the impedance of the combined parallel-line system prior to compensation is equivalent to that of the single transmission line of the IEEE FBM model. 2.3. Power flow controllers In Fig. 1, the approximate expression for the active power transfer in line 1 is given by
PL1
jV S jjV R j sin dSR X L1 X SSSC
ð2Þ
where jVSjand jVRjare the magnitudes of the bus voltages VS and VR and dSR is the transmission angle between these bus voltages. Eq. (2) highlights the principle by which series compensation can be used to manipulate the net impedance of a particular transmission line and hence influence its power transfer. When the compensating reactance XSSSC is increased for a given transmission angle dSR, the net reactance XL1 XSSSC of the compensated line is reduced, thereby increasing the active power transfer. Hence, the magnitude of the active power transfer in the compensated line can be increased for a given transmission angle by increasing the amount of controllable compensation. Alternatively an increase in the amount of controllable compensation can be used to reduce the transmission angle required for a given active power transfer in the compensated line. These two observations underlie two distinct strategies that have been proposed for closed-loop control of line power flow [5]: the ‘‘constant power strategy’’ which keeps the power flow in the compensated line constant, and the ‘‘constant angle strategy’’ which ensures that the compensated line transfers any increase in dispatched power.
Fig. 1. Single line diagram of the electrical and mechanical parts of the study system.
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A feedback control system has been developed in order to implement closed-loop control of transmission line power flow in the benchmark system of Fig. 1; the structure of this control system has been devised in such a way that the power flow controller can be operated in either the constant power mode or the constant angle mode proposed in [5] simply by setting a user-selectable switch setting within the simulation model. Fig. 2 shows a simplified, conceptual layout of the structure of this feedback control system which is supplied at its inputs with the desired and measured values of active power in a particular line and the change in generator dispatch. The structure of the controller shown here is specifically intended for the case when the FACTS-based series compensator is implemented using a static synchronous series compensator (SSSC), for which the input control signal is the commanded magnitude of compensating reactance XSSSC. Specifically, in either constant power mode (position A in Fig. 2) or constant angle mode (position B) a feed-back loop compares a signal PL1 , representing the commanded value of power transfer in line 1, to the actual (measured) value of power transfer in line 1 (PL1) in order to generate an error signal ePL1; this error signal ePL1 is used to drive a proportional-integral (PI) controller which adjusts the value of XSSSC commanded from the SSSC around some set-point value XSSSC0 in order to force PL1 to follow the commanded value P L1 . For the case of constant power mode, PL1 ¼ PL1set where PL1set represents the set-point value for the power transfer in line 1. For the case of constant angle mode, PL1 ¼ PL1set þ DPt where DPt represents any change in generator dispatch. A more detailed discussion of the power flow controller can be found in [6,7].
2.5. Mathematical modelling
2.4. The SSSC
i ¼ ½id ; ifd ; ikd ; iq ; ikq1 ; ikq2 T
Fig. 3 shows the schematic block diagram of an SSSC and its main controls. The SSSC comprises a voltage-sourced inverter and a coupling transformer that is used to insert the ac output voltage of the inverter in series with the transmission line. In this paper, the internal control scheme that ensures the inverter presents the desired compensating reactance in the transmission line is that described in [12]. The theory of operation of this scheme and its design for dynamic performance have been treated in detail in [12], but the main features of its controls are shown in Fig. 3. These internal controls comprise two distinct parts that supply the required magnitude and phase angle of the ac compensating voltage to the inverter’s switching controls. The required magnitude of the compensating voltage to be supplied by the inverter is determined by calculating the amplitude jIj of the line current and multiplying this by the commanded compensating reactance value XSSSC. The required angle of the compensating voltage is maintained by a two-loop regulator within the angle control block that acts on the inverter dc voltage and the active power at the inverter ac terminals. This phase angle controller steers the phase of the injected voltages to be in quadrature with the line currents to ensure no active power is exchanged at the ac terminals of the inverter at steady state, and in so doing regulates the inverter’s dc voltage to a desired value [12].
and that of voltages by:
Fig. 3. Schematic diagram of an SSSC and its controls.
In order to analyse the SSR characteristics of the study system of Fig. 1 it is necessary to introduce a comprehensive mathematical model describing each system element. 2.5.1. Generator electrical model Based on the usual assumptions [13] relevant to the two-axis theory, the mathematical model of a synchronous generator, expressed in terms of currents and a rotor reference frame, takes the following state-space form:
pi ¼ ½L1 fv ð½R þ x½GÞig
ð3Þ
where x is the rotor speed, [G] is the rotational voltage inductance matrix, and [R] is the machine resistance matrix. The vector of the dq axis winding currents is given by:
v ¼ ½v d ; v fd ; 0; v q ; 0; 0T
ð4Þ
ð5Þ
The above model consists of one damper circuit on the d-axis and two damper circuits on the q-axis to represent the effects of the damper windings in the synchronous machine of the IEEE FBM model [11]. The axis currents from Eq. (3) are used to calculate the electrical torque Te as follows:
T e ¼ iT ½Gixo =2
ð6Þ
The mechanical torque Tm is related to Pm (the total output power onto the shaft by all the turbine stages) by the following relation:
T m ¼ P m xo = x
ð7Þ
The above Eqs. (3)–(7) are recast in a synchronously rotating two axis of reference frame in order to integrate them with the dynamic model of the transmission network. In this new reference frame, the generator is represented by an equivalent non-linear voltage source bg which does not neglect transformer voltage terms, in series with the stator phase resistance Ra and an equivalent inductance Lg as shown in Fig. 4 as part of an ac equivalent circuit of the transmission network [14].
Fig. 2. Power flow controller as it appears in the constant power mode of control.
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power electronic inverter is a near-ideal dc to ac converter, such that the instantaneous power Pi at the ac terminals of the SSSC inverter is available to charge or discharge its dc capacitor [12]. The dynamics of the inverter capacitor voltage in Fig. 3 are represented by
pV dc ¼ Fig. 4. Multi-machine model of a generator suitable for inclusion in the transmission network.
This alternative format of the generator model nevertheless remains mathematically identical to that in the more familiar format of Eqs. (3)–(7). 2.5.2. Mechanical system The mechanical system of this steam turbine generator consists of a four stage turbine, a generator and an exciter. This system is represented in Fig. 1 as six inertias interconnected by five torsional springs (the connecting shafts), and can be described by a set of second order differential equations of the form:
½Jp2 d þ ½Dpd þ ½Kd þ T ¼ 0
ð8Þ
where [J] is a diagonal matrix of inertias, [D] is a diagonal matrix of damping coefficients, [K] is a symmetric matrix of shaft stiffness, T is a forcing torque vector and d is an angular position vector. 2.5.3. Transmission system model Each of the transmission lines is assumed to consist of a lumped series resistance RL and inductance LL and is compensated either by an SSSC (in line 1) or by a conventional series capacitor (line 2). The transmission system equations in Fig. 1 are represented in the dq reference frame as follows:
p½it dq ¼ a11 ½it dq þ a12 ½i1 dq þ a13 ½v SSSC dq þ a14 ½v C2 dq þ a15 ½V b dq 0 w0 ½it dq a15 ½bg dq ð9Þ w0 0 p½i1 dq ¼ a21 ½it dq þ a22 ½i1 dq þ a23 ½v SSSC dq þ a24 ½v C2 dq þ a25 ½V b dq 0 w0 a25 ½bg dq ð10Þ ½i1 dq w0 0 p½v C2 dq ¼ w0 X C2 ½it dq w0 X C2 ½i1 dq
0 w0
w0 ½v C2 dq 0
Pi CV dc
ð12Þ
where the real power Pi is given by
Pi ¼
3½v SSSCd i1d þ v SSSCq i1q 2
ð13Þ
The dynamics of the feedback regulator in Fig. 3 are described by the following equation:
puv ¼
K AI K VP V dc V dc P i 2 2 3=2 i1d þ i1q X SSSC
ð14Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 i1d þ i1q X SSSC sinðuv Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ¼ i1d þ i1q X SSSC cosðuv Þ
v SSSCd ¼
ð15Þ
v SSSCq
ð16Þ
and where uv is the phase angle at the output of the feedback regulator. 2.5.5. Power flow controller The dynamics of the power flow controller in Fig. 2 are represented by the following equations:
pX SSSC ¼ 1=ssssc X SSSC0 þ K p PL1 PL1 þ xpfc X SSSC pxpfc ¼ K I P L1 PL1
ð17Þ ð18Þ
where the output signal XSSSC from the power flow controller is the input control signal to the SSSC; the first order lag (time constant ssssc) is included to approximate the transport lag across the dc/ac inverter; and in so doing to prevent an algebraic loop in the dynamics of the system model as a whole. For time domain studies, these nonlinear differential equations are solved by iterative step-by-step numerical integration. For eigenvalue analysis these differential equations are linearised into a 26th-order state space representation of the system. 3. Eigenvalue analysis
ð11Þ
where the coefficients aij are made up of combinations of the parameters RL, LL, LS, Lt of the transmission network in Fig. 1 as well as the parameters Ra and Lg associated with the multi-machine form of the subsynchronous generator electrical model shown in Fig. 4. The term [bg]dq in Eqs. (9) and (10) represents the d- and q-axis components of the internal voltage bg of the synchronous generator in this multi-machine form of its electrical model shown in Fig. 4. The definitions of the coefficients aij in Eqs. (9) and (10) are shown in full in the appendix. A more-detailed treatment of the method by which the equations of a transmission network in the familiar abc phase-domain variables are recast into the dq reference frame can be found in [15] for passive transmission systems, and in [16] for transmission systems, such as Fig. 1, that contain a SSSC. A more-detailed treatment of the multi-machine form of the two-axis machine model used here (c.f. Fig. 4) can be found in [17,18]. 2.5.4. The SSSC model In the SSSC model used in this paper, the dynamic characteristics of the SSSC are represented with the assumption that the SSSC’s
3.1. Study system with line 2 uncompensated In the eigenvalue studies presented in this first subsection, in order to reduce the complexity of the SSR characteristics of the study system of Fig. 1 which has an SSSC as the series compensator in line 1, line 2 has been initially left uncompensated (i.e. XC2 = 0.0 pu). The results that follow show, firstly, the SSR characteristics of the study system with line 1 compensated by an SSSC but with no power flow controls around it. Secondly, an initial design and an improved design of the power flow controller are presented, and it is shown how the improved design enables much higher power flow controller bandwidths to be attained. The improved design of the power flow controller is then verified by means of time domain simulation. Finally the impact of the initial and improved power flow control designs on the torsional damping of the generator is shown. 3.1.1. Study system without power flow control This section initially investigates the SSR characteristics of the study system of Fig. 1 at different values of the series compensating reactance provided by the SSSC, with line 2 uncompensated,
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using the mathematical model of the study system without the power flow controller enabled. In the first case of the analysis, the value of series compensation in line 1 is varied from XSSSC = 0.10 pu to XSSSC = 0.95 pu corresponding to an increase in the percentage compensation of transmission line 1 from 10% to 95%. Fig. 5 shows the loci of the eigenvalues of this study system as the compensation of the line 1 is varied in this way. The loci of the individual eigenvalues of Fig. 5 are affected by different parts of the system in Fig. 1. Eigenvalues E2, E3 and E4 are related to the electrical transmission network (incorporating the SSSC, and generator stator circuits); in particular E2 represents the subsynchronous component of the air–gap flux; eigenvalues M0 to M5 represent the six torsional natural frequencies of the turbine-generator’s distributed–mass mechanical shaft; eigenvalue M is associated with the magnetic stability of the machine and it enters the right hand plane when the load angle of the generator exceeds 90°; eigenvalues R1, R2 and R3 are associated with the rotor circuits of the machine. The system of Fig. 5 has similar eigenvalues to a system compensated with conventional series capacitors, with the excep-
tion that eigenvalue E1 associated with the supersynchronous component of air–gap flux is re-placed by the eigenvalue P2 associated with the feed-back regulator dynamics of the SSSC. Fig. 5 shows that as is the case with conventional series capacitive compensation [7], the subsynchronous electrical frequency (eigenvalue E2) of the SSSC compensated system also decreases as XSSSC is increased. Torsional interaction occurs when this subsynchronous electrical frequency approaches the natural frequencies of the turbinegenerator’s mechanical shaft system. Torsional interaction in this study system occurs in the small rectangular area of the eigenvalue plot which is enlarged in the bottom half of Fig. 5. It can be observed that as the subsynchronous frequency of the electrical system (E2) approaches each natural frequency of the mechanical system (M1 to M4), strong torsional interaction between the mechanical and electrical system occurs, forcing the eigenvalue associated with that torsional shaft mode towards the right hand plane. Fig. 5 shows that the destabilising effect of this torsional interaction is sufficient to cause the eigenvalues associated with modes 4, 2 and 1 of this study system to enter the right hand plane,
Fig. 5. Loci of eigenvalues of the study system as the series compensation XSSSC is varied from 0.1 pu to 0.95 pu.
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with the exception of mode 3 which remains in the left plane at all the values of XSSSC. Fig. 5 therefore indicates that as XSSSC is increased, unstable mechanical oscillations in the turbine-generator shaft would be expected to occur in succession at the natural frequencies of modes 4, 2 and 1. The labels in Fig. 5 indicate the particular values of XSSSC at which each torsional shaft mode reaches its most unstable condition. It should also be noted, again, that the eigenvalue study in Fig. 5 indicates the SSR characteristics of the SSSC for a range of values of its compensating reactance without power flow control enabled around the SSSC. 3.1.2. Design of SSSC-based power flow controller This section initially examines the design of the power flow controller to be added around the SSSC by using frequency domain linearised transfer function analysis, in order to investigate the influence of the design of the gains of that controller on the resonant characteristics of a transmission network compensated by the SSSC. The design of the power flow controller gains KP and KI is first carried out using a simplified model of the full study system shown in Fig. 1 in which the generator is replaced by an ideal voltage source. The detailed SSR interaction studies are then carried
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out using the full system model once the range of the controller designs has been chosen. Each design of the power flow controller is carried out by adjusting its gains KP and KI to manipulate the positions of the first-order pole and zero in the transfer function DPL1 =DP L1 ðsÞ between the input and output of the controller in constant power mode (c.f. Fig. 2). Fig. 6 shows the small-signal transfer function DP L1 =DP L1 ðsÞ of the simplified study system when the values of KP and KI are together adjusted to increase the power flow controller bandwidth from 0.13 rad/s to 1.82 rad/s. Fig. 6a shows the transfer function in the form of its pole and zero locations in the complex plane whilst Fig. 6b shows the transfer function in the form of its magnitude and phase as a function of frequency at four selected bandwidth designs. As expected, it can be observed that as the controller’s bandwidth is increased, the real pole P0 associated with the closed-loop response of the controller moves further into the left-hand plane (6a). Fig. 6a also reveals that the design of the controller’s bandwidth influences the resonant characteristics of the transmission network. As the response of the controller is made faster, the eigenvalue E2 associated with the resonant frequency of the transmission network moves slightly closer to the
Fig. 6. Transfer function DP L1 =DP L1 ðsÞ with increasing controller bandwidth from 0.13 rad/s to 1.82 rad/s (sending end fed from ideal source).
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right-hand plane while also increasing in frequency. Fig. 6b also confirms that as bandwidth of the controller between 0 and 3 dB is increased, the underdamped resonant peak in the transfer function (associated with transmission line resonant frequency) increases slightly in frequency, but becomes significantly less damped. The above results demonstrate that when designing the power flow controller at different bandwidths in this way (fixing the position of its transfer function zero in the left hand plane at 10 s1, and moving the position of its pole) the choice of the controller bandwidth does in fact have an influence on the resonant characteristics of the transmission line, even at a single value of compensating reactance provided by the SSSC. Further studies into the design of the power flow controller gain parameters have shown that the position of zero in the transfer function DP L1 =DPL1 ðsÞ does also have an influence on the underdamped resonant peak in the transfer function when the controller’s bandwidth is increased. In order, to investigate this particular influence of the position of the zero in the power flow controller transfer function, the above study was repeated by fixing
the location of the zero of the transfer function at a different position, far to the left of the pole of the transfer function at 303 s1. Fig. 7 shows the small-signal transfer function DP L1 =DP L1 ðsÞ of the simplified study system when the values of KP and KI are together adjusted to increase the power flow controller bandwidth from 0.13 rad/s to 33.5 rad/s with the zero of the transfer function positioned at 303 s1. Fig. 7a shows the transfer function in the form of its pole and zero locations in the complex plane whilst Fig. 7b shows the transfer function in the form of its magnitude and phase as a function of frequency at four selected bandwidth designs. Fig. 7a again reveals that the design of the controller’s bandwidth influences the resonant characteristics of the transmission network. As the response of the controller is made faster, the eigenvalue E2 associated with the resonant frequency of the transmission network again moves slightly closer to the right-hand plane while also increasing in frequency: however, it can be observed that the extent to which this change in the resonant characteristics occurs is markedly reduced when the zero of the power
Fig. 7. Transfer function DP L1 =DP L1 ðsÞ with increasing controller bandwidth with its zero fixed at 303 1/s (sending end fed from ideal source).
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flow controller’s transfer function is fixed at 303 s1. A further benefit of this alternative location of the power flow controller’s zero is that it now allows the use of a much faster controller bandwidth of up to 33.5 rad/s as compared with the previous designs of Fig. 6 where the zero of the power flow controller’s transfer function was fixed at 10 s1. In the initial design shown earlier in Fig. 6, if the controller’s bandwidth is increased to a value above 1.82 rad/s, the magnitude of underdamped resonant peak in the transfer function DP L1 =DPL1 ðsÞ increases above the 0 dB line, hence controller’s bandwidth had to be limited to 1.82 rad/s in this design approach. By contrast, Fig. 7b shows that at similar bandwidths of 1.82 rad/s, the magnitude of the resonant peak is considerably reduced (29.2 dB) and it only approaches 0 dB for significantly larger bandwidth values of 33.5 rad/s. Fig. 7b shows that for this alternative design approach, as the bandwidth of the controller between 0 and 3 dB is increased, the underdamped resonant peak in the transfer function (associated with transmission line resonant frequency) still increases slightly in frequency, but it becomes significantly less damped only at much higher values of bandwidth. The above results demonstrate that when designing the power flow controller at different bandwidths via this second approach (fixing the position of its transfer function zero much further into the left hand plane at 301 s1, and moving the position of its pole) the choice of the controller bandwidth still has an influence on the resonant characteristics of the transmission line, even at a single value of compensating reactance provided by the SSSC. However, the choice of the position of the zero in the power flow controller’s transfer function influences the extent to which the change in the controller’s bandwidth affects the resonant characteristics of the transmission line. By positioning the zero of the transfer function much further to the left along the real axis, the extent of the influence of controller bandwidth on the transmission line resonances is markedly reduced, thus allowing a much faster power flow controller bandwidth to be achieved. The results so far have shown the SSR characteristics of the study system with and without power flow control enabled, and have shown how the controller’s design can be improved to achieve a relatively high controller bandwidth (comparable to fast-responding power flow controllers with settling times on the order of 60 ms as reported in [9]). Prior to examining the impact
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of such fast responding power flow controllers on the electromechanical modes of the study system, the improved power flow controller design is first verified by means of time domain simulation study. 3.1.3. Verification of the design the improved SSSC-based power flow controller The improved design of the power flow controller shown above (with bandwidth of 33.5 rad/s) was verified by means of a time domain simulation study by again using this simplified study system where the synchronous machine was replaced by an ideal source. Fig. 8 shows the response of this simplified study system following a step increase of 0.1 pu in P L1 . It can be observed that following this step increase in PL1 , the power flow controller responds by adjusting the commanded value of the SSSC reactance XSSSC to a new steady state value within 80 ms; this increase in the SSSC reactance increases the actual real power flow PL1 in line 1 to the desired value of PL1 again within 80 ms. Fig. 8 therefore confirms that the improved design of the power flow controller has allowed the value of the real power transmitted by line 1 to be adjusted to track a set point change within 4 ac cycles as specified by the dynamic performance specifications in the frequency domain design, and such a fast-responding controller is comparable to the speed of response of the power flow controller reported in [9]. 3.1.4. Impact of initial SSSC-based power flow controller design on electromechanical modes In order to investigate the influence of the power flow controller on the damping of the torsional modes of the study system of Fig. 1, the full model of the study system is used, that is with the transmission network now fed by a generator and multi-inertia turbine model at the sending end, and with the power flow controller enabled. The power flow controller designs considered in this case are those arrived at using the initial design approach shown in Fig. 6, by locating the zero of the power flow controller transfer function at 10 s1. Fig. 9 now shows the loci of the eigenvalues of the study system as the gains KP and KI of the power flow controller are together adjusted to increase the power flow controller bandwidth from
Fig. 8. Time domain response of the study system with sending end fed from an ideal source.
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Fig. 9. Loci of eigenvalues of the full study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 1.82 rad/s with controller set in constant power mode (initial design).
0.13 rad/s to 1.82 rad/s. As expected, it can be observed that as the bandwidth of the power flow controller is increased, the real eigenvalue P0, corresponding to the closed-loop pole in the power flow controller transfer function moves further into the left-hand plane. Fig. 9 also reveals that increasing the bandwidth of the power flow controller does influence the damping of the torsional modes of the study system. It can be observed that as the response time of the controller is made faster, the eigenvalues M1, M2, M3 and M4 associated with the torsional shaft modes, together with the eigenvalue M0 associated with the generator’s inertial swing mode, all move to the left. The result in Fig. 9 thus shows that in the constant power mode of operation, the effect of the power flow controller is to increase the damping of the inertial swing mode as well as the damping of the torsional modes; furthermore this positive influence on the electromechanical modes increases at increasing bandwidths of the controller. The above eigenvalue analysis was then repeated with the power flow controller set to operate in constant angle mode, with the results shown in Fig. 10. As expected, the real eigenvalue P0 associated with the response of the closed-loop pole of the power flow controller moves towards the left as the controller’s bandwidth is increased from 0.13 rad/s to 1.82 rad/s. Fig. 10 once again shows that in this mode of operation, the controller’s bandwidth design also affects the damping of the electromechanical modes of the study system. However, it can be observed that while the eigenvalues M0 and M4 move towards the right, and hence closer to the imaginary axis, the eigenvalues M1, M2 and M3 move towards the left. The results thus show that in this mode of power flow controller operation, the influence of increased controller bandwidth is to decrease the damping associated with the generator’s inertial swing mode as well as its torsional mode 4, whilst increasing the damping associated with torsional modes 1, 2 and 3, at least at this operating point. 3.1.5. Impact of improved SSSC-based power flow controller design on electromechanical modes In order to investigate the influence of the improved power flow controller design on the damping of the torsional modes of the study system of Fig. 1, the above study was repeated with the power flow controller gains determined using the improved approach shown in Fig. 7 which located the zero of the power flow
controller transfer function at 301 s1, thus allowing a much higher controller bandwidth to be achieved. Fig. 11 now shows the loci of the eigenvalues of the study system as the gains KP and KI of the power flow controller are together adjusted to increase the power flow controller bandwidth from 0.13 rad/s to 33.5 rad/s. As expected, it can be observed that as the bandwidth of the power flow controller is increased, the real eigenvalue P0, corresponding to the closed-loop pole in the power flow controller transfer function moves further into the left-hand plane. Fig. 11 also reveals that increasing the bandwidth of the power flow controller does influence the damping of the torsional modes of the study system. It can be observed that as the response time of the controller is made faster (up to a controller bandwidth of 9.6 rad/s), the eigenvalues M1, M2, M3 and M4 associated with the torsional shaft modes, together with the eigenvalue M0 associated with the generator’s inertial swing mode, all move to the left. However, when the controller’s bandwidth is increased beyond 9.6 rad/s, all the torsional mode eigenvalues M1, M2, M3 and M4 continue to move towards the left but the inertial mode eigenvalue M0 starts to move towards the right. The result in Fig. 11 thus shows that in the constant power mode of operation, the effect of the power flow controller designed in this manner is to increase the damping of the inertial swing mode as well as the damping of all of the torsional modes; furthermore this positive influence on the electromechanical modes increases at increasing bandwidths of the controller up to 9.6 rad/s. However, at bandwidths above 9.6 rad/s, there is still a net positive influence on the damping of all of the torsional modes, but there is less damping added to the inertial mode by the action of the power flow controller than is the case at lower controller bandwidths. The above eigenvalue analysis was then repeated with the power flow controller set to operate in constant angle mode, with the results shown in Fig. 12. As expected, the real eigenvalue P0 associated with the response of the closed-loop pole of the power flow controller moves towards the left as the controller’s bandwidth is increased from 0.13 rad/s to 33.5 rad/s. Fig. 12 once again shows that in this mode of operation, the controller’s bandwidth design also affects the damping of the electromechanical modes of the study system. However, it can be observed that the eigenvalues M1, M2, M3 and M4, together with the eigenvalue M0, all move towards the right, and hence closer to the imaginary axis.
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Fig. 10. Loci of eigenvalues of the full study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 1.82 rad/s with controller set in constant angle mode (initial design).
Fig. 11. Loci of eigenvalues of the full study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant power mode (improved design).
The results thus show that in this mode of power flow controller operation, the influence of increased controller bandwidth is to decrease the damping associated with both torsional and inertial modes. Furthermore, at controller bandwidths higher than 5.3 rad/s, the inertial mode M0 and the torsional mode M4 are unstable. The results in Figs. 11 and 12 have shown that although the design of the power flow controller’s bandwidth can be made very fast, the mode of operation chosen for the power flow controller does impose some limitations on its controller bandwidth design. At this particular operating point of the power flow controller, it was observed that in the constant power mode of operation, for bandwidths above 9.6 rad/s, there is less damping added to the inertial mode than is the case for lower bandwidth designs, although the net impact of the power flow controller on damping remained positive at all bandwidths considered. The results thus
show that if the aim is to optimise the damping of all the electromechanical modes, a bandwidth of 9.6 rad/s will be the best design of the controller for the constant power mode. However, if the aim is to achieve the fastest possible power flow controller response, a bandwidth of 33.5 rad/s can be used since all the electromechanical modes are still stable at this highest controller bandwidth. In the case of the constant angle mode of operation, it is not even possible to operate the power flow controller at bandwidths above 5.3 rad/s as some of the electromechanical modes become unstable. The results thus show that if the aim is to optimise the damping of all the modes, this mode of operation is not recommended at all since there is a detrimental impact on the damping of all the generator’s electromechanical modes. The results also show that if the aim is to achieve the fastest power flow controller response, this mode of operation is also not recommended since a maximum bandwidth of 5.3 rad/s can be used.
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Fig. 12. Loci of eigenvalues of the full study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant angle mode (improved design).
The results presented so far, have shown the influence of the power flow controller on torsional mode damping when line 2 in the study system of Fig. 1 is uncompensated. The following section considers the influence of adding conventional capacitive series compensation in line 2. 3.2. Study system with line 2 compensated The following eigenvalue studies aim to investigate the impact of the improved SSSC-based power flow controller design on the damping of the electromechanical modes of the study system in the presence of conventional series capacitors in line 2. However, in order to consider such a study, it is first necessary to identify a combination of values of SSSC-based compensation in line 1 and conventional compensation in line 2 for which the system is SSR stable prior to the addition of the power flow controller in line 1. In order to identify ranges of series compensation values over which all the electromechanical modes in a study system are stable, it is more useful to the present the results of the eigenvalue scans in an alternative format in which the real parts of the eigenvalue loci are plotted as a function the compensation value being varied; this format is adopted for the remainder of the eigenvalue results in this subsection. 3.2.1. Locating SSR-stable ranges of line 2 compensation In order to identify SSR stable ranges of the series compensation in line 2, the following study initially considers the SSR characteristics of the study system of Fig. 1 at different values of conventional series compensating reactance in line 2, while the series compensation provided by the SSSC in line 1 is held fixed at 0.1 pu, without the power flow controller enabled. The value of series compensation in line 2 is varied from XC2 = 0.10 pu to XC2 = 0.95 pu corresponding to an increase in the percentage compensation of transmission line 2 from 10% to 95%. Fig. 13 shows the real part of the eigenvalues associated with the torsional modes of this study system at each value of XC2 as the compensation of the line 2 is varied in this way. It can be observed that as the conventional series compensation in line 2 is increased, there is successive destabilisation of the mechanical torsional modes M4, M3, M2 and M1 as the real part of the eigenvalues associated with each of these torsional modes moves into the positive plane of Fig. 13. It can be observed that
there are certain ranges of series capacitive compensation in line 2 where the real part of all the torsional mode eigenvalues M1, M2, M3 and M4 are in the negative plane of Fig. 13 (that is all four torsional modes are simultaneously SSR stable). Within these stable ranges, two specific values of series compensating reactance were chosen for further study in the remaining eigenvalue scans: case A: XC2 = 0.23 pu and case B: XC2 = 0.36 pu. At each of these values of conventional series compensation in line 2, the SSR characteristics of the SSSC compensation in line 1 was then considered for a range of compensation values. 3.2.2. Locating SSR stable operating point for the SSSC in line 1 in the presence of conventional series capacitors in adjacent line 2 for two selected cases In order to identify the SSR stable operating ranges of the SSSC in line 1 for each of the above two values of series capacitive compensative compensation in line 2, the eigenvalues of the study system were calculated as the series compensation XSSSC provided by the SSSC is varied from 0.1 pu to 0.95 pu, without power flow control enabled around the SSSC, with series compensation XC2 in line 2 fixed, in turn, at each of the values chosen in case A and case B. Fig. 14 shows the real part of the eigenvalues associated with the torsional modes of this study system as the series compensation XSSSC in line 1 is varied in this way with the compensation in line 2 fixed at the value for case A: XC2 = 0.23 pu. It can be observed that as the series compensation provided by the SSSC in line 1 is increased there is successive destabilisation of the mechanical torsional modes M4 and M1 as the real part of the eigenvalues associated with these two torsional modes moves into the positive plane of Fig. 14. Fig. 14. shows that the SSR stable operating range of series compensation in line 1 is from XSSSC = 0.1 pu to a maximum value of XSSSC = 0.24 pu for the case A value of conventional compensation in line 2, since at all values of XSSSC above 0.24 pu, torsional mode M4 remains unstable. Fig. 15 shows the real part of the eigenvalues associated with the torsional modes of this study system as the series compensation of line 1 is again varied from 0.1 pu to 0.95 pu with the series compensation XC2 in line 2 fixed at the value for case B: XC2 = 0.36 pu. It can be observed that as the series compensation provided by the SSSC in line 1 is increased there is successive destabilisation of the torsional modes M4, M1 and M3 as the real part of the eigenvalues associated with these torsional modes
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Fig. 13. Real part of eigenvalues of the study system as the conventional series compensation XC2 is varied from 0.1 pu to 0.95 pu with XSSSC in line 1 fixed at 0.1 pu.
Fig. 14. Real part of eigenvalues of the study system as the series compensation provided by the SSSC, XSSSC in line 1 is varied from 0.1 pu to 0.95 pu while the series capacitive reactance XC2 in line 2 is fixed at 0.23 pu (case A).
moves into the positive plane of Fig. 15. Fig. 15 shows that the SSR stable operating range of series compensation of the SSSC in line 1 for the case B value of conventional series compensation in line 2 is from XSSSC = 0.1 pu up to a maximum value of XSSSC = 0.29 pu after which, torsional mode M4 remains unstable. Figs. 14 and 15 have shown that the SSR-stable operating ranges of the series compensation XSSSC in line 1 at two different values of capacitive reactance in line 2 are quite limited. In order to study the effect of the power flow controller on torsional mode damping at these two different values of conventional series compensation in line 2, a common SSR-stable operating value for the SSSC’s reactance of XSSSC0 = 0.15 pu was used for the remaining eigenvalue studies, which consider the impact of the power flow controller added around the SSSC on the damping of the electromechanical modes. 3.2.3. Impact of improved SSSC-based power flow controller design on electromechanical modes with conventional series capacitors in line 2 In order to investigate the influence of the power flow controller on the damping of the torsional modes of the study system of
Fig. 1, Fig. 16 shows the loci of the eigenvalues of the study system as the gains KP and KI of the power flow controller are together adjusted to increase the power flow controller bandwidth from 0.13 rad/s to 33.5 rad/s with the power flow controller set to operate in constant power mode at a nominal XSSSC0 = 0.15 pu and with XC2 fixed at 0.23 pu. It can be observed that as the bandwidth of the power flow controller is increased, the real eigenvalue P0, corresponding to the closed-loop pole in the power flow controller transfer function moves further into the left-hand plane. Fig. 16 also reveals that increasing the bandwidth of the power flow controller does influence the damping of the torsional modes of the study system. It can be observed that as the response time of the controller is made faster (up to a controller bandwidth of 4.6 rad/s), the eigenvalues M1, M2, M3 and M4 associated with the torsional shaft modes, together with the eigenvalue M0 associated with the generator’s inertial swing mode, all move to the left. However, when the controller’s bandwidth is increased beyond 4.6 rad/s, all the torsional mode eigenvalues M1, M2, M3 and M4 continue to move towards
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Fig. 15. Real part of eigenvalues of the study system as the series compensation provided by the SSSC, XSSSC in line 1 is varied from 0.1 pu to 0.95 pu while the series capacitive reactance XC2 in line 2 is fixed at 0.36 pu (case B).
Fig. 16. Loci of eigenvalues of the study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant power mode, XSSSC0 = 0.15 pu and XC2 = 0.23 pu (case A).
the left, but the inertial mode eigenvalue M0 starts to move back towards the right hand plane. The result in Fig. 16 thus shows that in the constant power mode of operation, even with the presence of a fixed series capacitor in line 2, the effect of the power flow controller is to increase the damping of the inertial swing mode as well as the damping of all of the torsional modes; furthermore this positive influence on the electromechanical modes increases at increasing bandwidths of the controller up to 4.6 rad/s. However, at bandwidths that are higher than 4.6 rad/s, while there is still an increasing positive influence on the damping of the torsional modes, the influence on the inertial mode is less significant than at lower bandwidth (although still net positive). The above eigenvalue analysis was then repeated with the power flow controller set to operate in constant angle mode, with the results shown in Fig. 17. As expected, the real eigenvalue P0 associated with the response of the closed-loop pole of the power flow controller moves towards the left as the controller’s band-
width is increased from 0.13 rad/s to 33.5 rad/s. Fig. 17 shows that in this mode of operation the controller’s bandwidth design also affects the damping of the electromechanical modes of the study system. However, it can be observed that the eigenvalues M1, M2, M3 and M0 all move towards the right, and hence closer to the imaginary axis, whilst eigenvalue M4 moves to the left as the controller bandwidth is increased. The results thus show that in this mode of power flow controller operation, the influence of increased controller bandwidth is to decrease the damping associated with all the electromechanical modes other than mode 4. Furthermore, at controller bandwidths higher than 2.3 rad/s, the inertial mode M0 is unstable. The above study was repeated for the second value of compensation in line 2 at which the fixed series capacitive reactance XC2 was set at 0.36 pu. Fig. 18 shows the loci of the eigenvalues of the study system as the gains KP and KI of the power flow controller are together adjusted to increase the power flow controller
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Fig. 17. Loci of eigenvalues of the study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant angle mode, XSSSC0 = 0.15 pu and XC2 = 0.23 pu (case A).
Fig. 18. Loci of eigenvalues of the study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant power mode, XSSSC0 = 0.15 pu and XC2 = 0.36 pu (case B).
bandwidth from 0.13 rad/s to 33.5 rad/s with the power flow controller set to operate in constant power mode at a nominal XSSSC0 value of 0.15 pu and XC2 fixed at 0.36 pu. It can be observed that as the bandwidth of the power flow controller is increased, the real eigenvalue P0, corresponding to the closed-loop pole in the power flow controller transfer function moves further into the left-hand plane. Fig. 18 also reveals that increasing the bandwidth of the power flow controller does influence the damping of the torsional modes of the study system. It can be observed that as the response time of the controller is made faster (up to a controller bandwidth of 5.2 rad/s), the eigenvalues M1, M2, M3 and M4 associated with the torsional shaft modes, together with the eigenvalue M0 associated with the generator’s inertial swing mode, all move to the left. However, when the controller bandwidth is increased beyond 5.2 rad/s, all the torsional mode eigenvalues M1, M2, M3 and M4
continue to move towards the left while the inertial mode eigenvalue M0 starts to move back towards the right. The result in Fig. 18 thus shows that in the constant power mode of operation, even with the presence of a fixed series capacitor in line 2 at a different value of conventional series compensation XC2 = 0.36 pu in line 2, the effect of the power flow controller is still to increase the damping of the inertial swing mode as well as the damping of all of the torsional modes; furthermore this positive influence on the electromechanical modes increases at increasing bandwidths of the controller up to 5.2 rad/s. However, at bandwidths that are higher than 5.2 rad/s, there is still a positive influence on the damping of all of the torsional modes, but there is less positive damping added to the inertial mode than at a bandwidth of 5.2 rad/s. The above eigenvalue analysis was then repeated with the power flow controller set to operate in constant angle mode, with
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Fig. 19. Loci of eigenvalues of the study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant angle mode, XSSSC0 = 0.15 pu and XC2 = 0.36 pu (case B).
the results shown in Fig. 19. As expected, the real eigenvalue P0 associated with the response of the closed-loop pole of the power flow controller moves towards the left as the controller’s bandwidth is increased from 0.13 rad/s to 33.5 rad/s. Fig. 19 once again shows that in this mode of operation, the controller bandwidth design also affects the damping of the electromechanical modes of the study system. However, it can be observed that the eigenvalues M1, M2, M4, M0 all move towards the right, and hence closer to the imaginary axis, whilst eigenvalue M3 moves to the left as the controller bandwidth is increased. The results thus show that in this mode of power flow controller operation, the influence of increased controller bandwidth is to decrease the damping associated with all the electromechanical modes other than torsional mode M3. Furthermore, at controller bandwidths higher than 2.6 rad/s, the inertial mode M0 is unstable.
of generator torsional modes at risk, with the extent of the destabilisation being very sensitive to the system operating conditions and power flow controller design. Appendix A Coefficients aij in Eqs. (9) and (10) are defined as follows:
a11 ¼
Ra LLL1 ðRa þ RL2 Þ L2 LDEN1 LLL1 L2
a14 ¼ a22
a15 ¼
;
1 þ LLL1 L2
LDEN1 LDEN1 L1 LL2 RL1 RL2 LsRþL t þLg ¼ ; LDEN2 LL2 Ls þLt þLg
4. Conclusion
a25 ¼
Previous work has shown the dependence of torsional mode damping (and hence SSR stability) on both the mode of operation and the speed of response of a power flow controller designed for relatively slow-responding control action. This paper has investigated a method of redesigning the power flow controllers considered in earlier work to achieve higher-bandwidth controllers comparable to other fast-responding power flow controllers reported in the literature. In the process, it has been shown that an alternative method of pole-zero placement when designing the power flow controller’s closed-loop transfer function not only allows greater bandwidths to be attained, but, in the case of constant-power type controllers, allows significantly improved stabilisation of all generator electromechanical modes to be achieved. Extensive studies have been presented to confirm not only that a constant power mode controller adds stabilising damping to all torsional modes, and is able to do so even in the presence of conventional series compensation in an adjacent line, but that this stabilisation can be increased by a faster acting power flow controller. Similar studies in this paper support earlier work which has concluded that the alternative mode of operation of a power flow controller that has been proposed in the literature (constant angle) is not suitable for environments where SSR is a concern. In particular, this mode of control has been shown to destabilise a number
where
LDEN2
a12 ¼
;
RL1 LLL1 RL2 L2
a21 ¼
;
a23 ¼
; LDEN1 LL2 Ra RL2 Ls þLt þL g
LDEN2 1 Ls þLLL2t þL
g
LDEN2
;
a13 ¼
1 ; LDEN1
a24 ¼
1 ; LDEN2
;
;
LDEN1 ¼ Ls þ Lt þ Lg þ LL1 þ LDEN2 ¼ LL1 þ LL2 þ
LL1 Ls þ Lt þ Lg ; LL2
LL1 LL2 Ls þ Lt þ Lg
References [1] Ballance JW, Goldberg S. Subsynchronous resonance in series compensated transmission lines. IEEE Trans PAS 1973;PAS-92:1649–58. [2] Rigby BS, Harley RG. Resonant characteristics of inverter-based transmission line series compensators. In: Conference record of IEEE PESC. Charleston, USA; June 1999. p. 412–17. [3] Padiyar KR, Prabhu N. Analysis of SSR with three-level twelve pulse VSC-based inter-line power-flow controller. IEEE Trans PD 2007;22(3). [4] Martins N, Paserba JJ, Pinto JCP. Using a TCSC for line power scheduling and system oscillation damping – small signal and transient stability studies. In: Proc of IEEE PES winter meeting, vol. 2. Singapore; January 2000. p. 1455–61 [5] Larsen EV, Bowler CEJ, Damsky B, Nilsson S. Benefits of thyristor controlled series compensation. CIGRE Annual Meeting, Paper No. 14/37/38-04. Paris; 1992. [6] Ally A, Rigby BS. The impact of closed-loop power flow control strategies on the power system stability characteristics in a single generator system. SAIEE Afr Res J 2006;97(1):34–42. [7] Pillay Carpanen R, Rigby BS. A FACTS-based power flow controller model for the IEEE SSR first benchmark model. In: Proc of IEEE PES PowerAfrica conf. Johannesburg, South Africa; July 2007. ISBN 1-4244-1478-4.
R. Pillay Carpanen, B.S. Rigby / Electrical Power and Energy Systems 43 (2012) 194–209 [8] Pillay Carpanen R, Rigby BS. Impact of a SSSC-based power flow controller on SSR. In: Proc of the 16th power systems computation conf PSCC 2008. Glasgow, UK; 2008. ISBN 978-0-947649-28-9. [9] Mihalic R, Papic I. Static synchronous series compensator – a mean for dynamic power flow control in electric power systems. Electric Power Syst Res 1998;45:65–72. [10] Anderson PM, Agrawal BL, Van Ness JE. Subsynchronous resonance in power systems. New York: IEEE Press; 1990. [11] report IEEE Committee. First benchmark model for computer simulation of subsynchronous resonance. IEEE Trans PAS 1977;PAS-96:1565–72. [12] Rigby BS, Harley RG. An improved control scheme for a series-capacitive reactance compensator based on a voltage-source inverter. IEEE Trans IAS 1998;34(2):355–63. [13] Adkins B, Harley RG. The general theory of alternating current machines: applications to practical problems. London: Chapman and Hall; 1975. ISBN 0412-15560-5.
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[14] Jennings GD. Modelling and analysis of turbogenerators in single and multimachine subsynchronous resonance studies. PhD thesis, University of Natal, Durban, South Africa; 1987. [15] Limebeer DJN, Harley RG, Schuck SM. Subsynchronous resonance of the Koeberg turbo-generators and of a laboratory micro-alternator system. Trans SAIEE 1979;70(Part 11):278–97. [16] Harley RG, Rigby BS, Jennings GD. Design of a controlled converter which emulates a series capacitive compensator for long power lines. In: Proc of the international conf on power electronics and motion control PEMC’94. Warsaw, Poland; September 1994. p. 213–3. [17] Lahoud MA, Harley RG. Analysis techniques for dynamic behavior of a multigenerator system. Trans SAIEE 1983;74(Part 4):53–63. [18] Lahoud MA, Harley RG, Balda JC. Induction motor modelling for short-term transient studies in a multigenerator–multimotor power system. Electric Power Syst Res 1984;1:53–63.
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An improved SSSC-based power flow controller design method and its impact on torsional interaction R. Pillay Carpanen ⇑, B.S. Rigby School of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, Howard College Campus, Durban 4041, South Africa
a r t i c l e
i n f o
Article history: Received 3 March 2010 Received in revised form 12 May 2012 Accepted 14 May 2012 Available online 19 June 2012 Keywords: Power flow control FACTS SSSC Subsynchronous resonance
a b s t r a c t This paper presents an improved method of designing a closed-loop AC power flow controller. The paper further examines the impact of this improved SSSC-based power flow control on the damping of the subsynchronous torsional modes of a turbine-generator. Both the constant power and constant angle modes of power flow control are considered; in each case the influence of the controller design on torsional mode damping is examined over a range of operating conditions. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The multi-inertia shafts of turbo-generators are susceptible to interaction with the electrical resonances of a transmission network compensated with conventional series capacitors and can lead to a form of instability known as subsynchronous resonance (SSR). This form of dynamic instability can cause mechanical failure of turbine shafts [1]. The emergence of Flexible AC Transmission Systems (FACTS) series compensators has led to a renewed interest in determining the SSR characteristics of these more advanced forms of series compensation, in particular the static synchronous series compensator (SSSC) [2,3]. Although, numerous applications of controllable FACTS series compensators have been considered in the literature, one particular application is the use of FACTS devices for closed-loop control of the flow of power down a particular line in an inter-connected transmission network [4,5]. Such closed-loop control of power flow in an AC system can provide a number of possible benefits: preventing unwanted loop flows in an interconnected system; allowing power to be directed along a ‘‘contract’’ path in a transmission system; preventing inadvertent overloading of lines already near their thermal limits [5]. However, more recently it has been shown [6] that a closed-loop power flow controller can affect the small-signal damping characteristics of a power system, and in particular that the nature and extent of these effects on the system’s stability depend on both the mode in which the power flow controller is operated and its controller response time. ⇑ Corresponding author. E-mail address: [email protected] (R. Pillay Carpanen). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.05.020
A logical question that then arises is whether the impact of closed-loop power flow control action on system damping reported in [6] could also influence the damping of any subsynchronous torsional modes in the shafts of the turbine-generators that feed the power system. However, in order to answer this question it is necessary to develop mathematical models of a representative study system in which both the torsional dynamics of the affected generators are adequately described, as well as the dynamic characteristics of the SSSC and its power flow controls. This paper begins by presenting the development of such a study system model, extended from earlier work described in [7]. In subsequent work [8], the results have indeed confirmed that a power flow controller can influence the damping of generator torsional modes and they further suggest that the influence of one particular mode of power flow control on torsional modes is operating point dependant. However, the design of SSSC-based power flow controller considered for study in [8] was relatively slow responding when compared to the response time of the power flow controller reported in work such as [9] which has shown that very fastresponse power flow controllers can be achieved and have been considered in the literature. This paper now uses the study system developed in [7] to examine the design of controller parameters for a SSSC-based power flow controller by using frequency-domain linearised transfer function analysis. In particular, the paper also uses the frequency-domain analysis to derive an understanding of how the design of the control gains of the power flow controller (and hence its speed of response) affects the resonant characteristics of a transmission network compensated by a SSSC. The paper further presents an improved design of the control gains of the
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power flow controller that enables a much faster controller bandwidth to be realised (comparable to that considered in [9]). Finally, the paper examines the impact of such fast responding power flow controllers on a generator’s torsional modes of oscillation, over a range of operating conditions. 2. Background theory 2.1. Subsynchronous resonance The shaft system of a turbine-generator has a complex mechanical structure comprising several predominant masses connected by shafts of finite stiffness [10]. Therefore such a multi-inertia system has several natural torsional frequencies fn of oscillation (torsional modes) usually in the subsynchronous frequency range. When conventional series capacitors are included in an RL transmission line, a series resonant RLC circuit is formed whose natural frequency fer is given by
1 pffiffiffiffiffiffi ¼ fo fer ¼ 2p LC
sffiffiffiffiffiffi XC XL
ð1Þ
where fo is the synchronous frequency of the system, XL = 2pfoL is the inductive line reactance, and X C ¼ 2p1f0 C is the capacitive compensating reactance. Such a network thus causes the transient component of any current in the line and armature to oscillate at a certain frequency which is closely related to the undamped natural frequency fer of the line. Thus, these oscillations produce rotor currents and torques at frequencies fo ± fer. If the frequency of the induced torques at the lower of these two frequencies (fo fer) coincides with one of the natural torsional mode frequencies fn, the shaft will be excited to oscillate at its natural frequency, sometimes with high amplitude. Such large shaft torques produce high shaft stresses, a reduction in shaft fatigue life and, possibly, shaft failure [10]. 2.2. Study system The system considered for study here is a modification of the IEEE First Benchmark Model (FBM) [10] for SSR analysis, altered to include two transmission lines in parallel as shown in Fig. 1. The study system, at present, consists of a 892.4 MVA turbine generator connected to a 500 kV transmission network via a step up transformer, where XT denotes the transformer reactance. The transmission network consists of two lines in parallel where RL denotes the resistance and XL the inductive reactance of the respective line. Line 1 is equipped with series compensation provided
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by an SSSC, where XSSSC denotes the effective capacitive reactance provided by the SSSC at the fundamental frequency. Although the model that has been developed for this study system also allows for the inclusion of series compensation in line 2, for the purpose of the investigations considered in this paper line 2 is left uncompensated. The transmission network is connected to the infinite bus via XS, the system reactance between the end of the transmission lines and the infinite busbar. The mechanical system comprises six lumped masses: a high pressure turbine (HP), an intermediate pressure turbine (LP), two low pressure turbines (LPA, LPB), the generator (GEN) and the exciter (EXC), which are mechanically coupled by shaft sections of known torsional elasticity. In the studies conducted thus far, the individual line parameters of this parallel-line system were changed (to twice their original resistance and inductive reactance) such that the impedance of the combined parallel-line system prior to compensation is equivalent to that of the single transmission line of the IEEE FBM model. 2.3. Power flow controllers In Fig. 1, the approximate expression for the active power transfer in line 1 is given by
PL1
jV S jjV R j sin dSR X L1 X SSSC
ð2Þ
where jVSjand jVRjare the magnitudes of the bus voltages VS and VR and dSR is the transmission angle between these bus voltages. Eq. (2) highlights the principle by which series compensation can be used to manipulate the net impedance of a particular transmission line and hence influence its power transfer. When the compensating reactance XSSSC is increased for a given transmission angle dSR, the net reactance XL1 XSSSC of the compensated line is reduced, thereby increasing the active power transfer. Hence, the magnitude of the active power transfer in the compensated line can be increased for a given transmission angle by increasing the amount of controllable compensation. Alternatively an increase in the amount of controllable compensation can be used to reduce the transmission angle required for a given active power transfer in the compensated line. These two observations underlie two distinct strategies that have been proposed for closed-loop control of line power flow [5]: the ‘‘constant power strategy’’ which keeps the power flow in the compensated line constant, and the ‘‘constant angle strategy’’ which ensures that the compensated line transfers any increase in dispatched power.
Fig. 1. Single line diagram of the electrical and mechanical parts of the study system.
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A feedback control system has been developed in order to implement closed-loop control of transmission line power flow in the benchmark system of Fig. 1; the structure of this control system has been devised in such a way that the power flow controller can be operated in either the constant power mode or the constant angle mode proposed in [5] simply by setting a user-selectable switch setting within the simulation model. Fig. 2 shows a simplified, conceptual layout of the structure of this feedback control system which is supplied at its inputs with the desired and measured values of active power in a particular line and the change in generator dispatch. The structure of the controller shown here is specifically intended for the case when the FACTS-based series compensator is implemented using a static synchronous series compensator (SSSC), for which the input control signal is the commanded magnitude of compensating reactance XSSSC. Specifically, in either constant power mode (position A in Fig. 2) or constant angle mode (position B) a feed-back loop compares a signal PL1 , representing the commanded value of power transfer in line 1, to the actual (measured) value of power transfer in line 1 (PL1) in order to generate an error signal ePL1; this error signal ePL1 is used to drive a proportional-integral (PI) controller which adjusts the value of XSSSC commanded from the SSSC around some set-point value XSSSC0 in order to force PL1 to follow the commanded value P L1 . For the case of constant power mode, PL1 ¼ PL1set where PL1set represents the set-point value for the power transfer in line 1. For the case of constant angle mode, PL1 ¼ PL1set þ DPt where DPt represents any change in generator dispatch. A more detailed discussion of the power flow controller can be found in [6,7].
2.5. Mathematical modelling
2.4. The SSSC
i ¼ ½id ; ifd ; ikd ; iq ; ikq1 ; ikq2 T
Fig. 3 shows the schematic block diagram of an SSSC and its main controls. The SSSC comprises a voltage-sourced inverter and a coupling transformer that is used to insert the ac output voltage of the inverter in series with the transmission line. In this paper, the internal control scheme that ensures the inverter presents the desired compensating reactance in the transmission line is that described in [12]. The theory of operation of this scheme and its design for dynamic performance have been treated in detail in [12], but the main features of its controls are shown in Fig. 3. These internal controls comprise two distinct parts that supply the required magnitude and phase angle of the ac compensating voltage to the inverter’s switching controls. The required magnitude of the compensating voltage to be supplied by the inverter is determined by calculating the amplitude jIj of the line current and multiplying this by the commanded compensating reactance value XSSSC. The required angle of the compensating voltage is maintained by a two-loop regulator within the angle control block that acts on the inverter dc voltage and the active power at the inverter ac terminals. This phase angle controller steers the phase of the injected voltages to be in quadrature with the line currents to ensure no active power is exchanged at the ac terminals of the inverter at steady state, and in so doing regulates the inverter’s dc voltage to a desired value [12].
and that of voltages by:
Fig. 3. Schematic diagram of an SSSC and its controls.
In order to analyse the SSR characteristics of the study system of Fig. 1 it is necessary to introduce a comprehensive mathematical model describing each system element. 2.5.1. Generator electrical model Based on the usual assumptions [13] relevant to the two-axis theory, the mathematical model of a synchronous generator, expressed in terms of currents and a rotor reference frame, takes the following state-space form:
pi ¼ ½L1 fv ð½R þ x½GÞig
ð3Þ
where x is the rotor speed, [G] is the rotational voltage inductance matrix, and [R] is the machine resistance matrix. The vector of the dq axis winding currents is given by:
v ¼ ½v d ; v fd ; 0; v q ; 0; 0T
ð4Þ
ð5Þ
The above model consists of one damper circuit on the d-axis and two damper circuits on the q-axis to represent the effects of the damper windings in the synchronous machine of the IEEE FBM model [11]. The axis currents from Eq. (3) are used to calculate the electrical torque Te as follows:
T e ¼ iT ½Gixo =2
ð6Þ
The mechanical torque Tm is related to Pm (the total output power onto the shaft by all the turbine stages) by the following relation:
T m ¼ P m xo = x
ð7Þ
The above Eqs. (3)–(7) are recast in a synchronously rotating two axis of reference frame in order to integrate them with the dynamic model of the transmission network. In this new reference frame, the generator is represented by an equivalent non-linear voltage source bg which does not neglect transformer voltage terms, in series with the stator phase resistance Ra and an equivalent inductance Lg as shown in Fig. 4 as part of an ac equivalent circuit of the transmission network [14].
Fig. 2. Power flow controller as it appears in the constant power mode of control.
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power electronic inverter is a near-ideal dc to ac converter, such that the instantaneous power Pi at the ac terminals of the SSSC inverter is available to charge or discharge its dc capacitor [12]. The dynamics of the inverter capacitor voltage in Fig. 3 are represented by
pV dc ¼ Fig. 4. Multi-machine model of a generator suitable for inclusion in the transmission network.
This alternative format of the generator model nevertheless remains mathematically identical to that in the more familiar format of Eqs. (3)–(7). 2.5.2. Mechanical system The mechanical system of this steam turbine generator consists of a four stage turbine, a generator and an exciter. This system is represented in Fig. 1 as six inertias interconnected by five torsional springs (the connecting shafts), and can be described by a set of second order differential equations of the form:
½Jp2 d þ ½Dpd þ ½Kd þ T ¼ 0
ð8Þ
where [J] is a diagonal matrix of inertias, [D] is a diagonal matrix of damping coefficients, [K] is a symmetric matrix of shaft stiffness, T is a forcing torque vector and d is an angular position vector. 2.5.3. Transmission system model Each of the transmission lines is assumed to consist of a lumped series resistance RL and inductance LL and is compensated either by an SSSC (in line 1) or by a conventional series capacitor (line 2). The transmission system equations in Fig. 1 are represented in the dq reference frame as follows:
p½it dq ¼ a11 ½it dq þ a12 ½i1 dq þ a13 ½v SSSC dq þ a14 ½v C2 dq þ a15 ½V b dq 0 w0 ½it dq a15 ½bg dq ð9Þ w0 0 p½i1 dq ¼ a21 ½it dq þ a22 ½i1 dq þ a23 ½v SSSC dq þ a24 ½v C2 dq þ a25 ½V b dq 0 w0 a25 ½bg dq ð10Þ ½i1 dq w0 0 p½v C2 dq ¼ w0 X C2 ½it dq w0 X C2 ½i1 dq
0 w0
w0 ½v C2 dq 0
Pi CV dc
ð12Þ
where the real power Pi is given by
Pi ¼
3½v SSSCd i1d þ v SSSCq i1q 2
ð13Þ
The dynamics of the feedback regulator in Fig. 3 are described by the following equation:
puv ¼
K AI K VP V dc V dc P i 2 2 3=2 i1d þ i1q X SSSC
ð14Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 i1d þ i1q X SSSC sinðuv Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ¼ i1d þ i1q X SSSC cosðuv Þ
v SSSCd ¼
ð15Þ
v SSSCq
ð16Þ
and where uv is the phase angle at the output of the feedback regulator. 2.5.5. Power flow controller The dynamics of the power flow controller in Fig. 2 are represented by the following equations:
pX SSSC ¼ 1=ssssc X SSSC0 þ K p PL1 PL1 þ xpfc X SSSC pxpfc ¼ K I P L1 PL1
ð17Þ ð18Þ
where the output signal XSSSC from the power flow controller is the input control signal to the SSSC; the first order lag (time constant ssssc) is included to approximate the transport lag across the dc/ac inverter; and in so doing to prevent an algebraic loop in the dynamics of the system model as a whole. For time domain studies, these nonlinear differential equations are solved by iterative step-by-step numerical integration. For eigenvalue analysis these differential equations are linearised into a 26th-order state space representation of the system. 3. Eigenvalue analysis
ð11Þ
where the coefficients aij are made up of combinations of the parameters RL, LL, LS, Lt of the transmission network in Fig. 1 as well as the parameters Ra and Lg associated with the multi-machine form of the subsynchronous generator electrical model shown in Fig. 4. The term [bg]dq in Eqs. (9) and (10) represents the d- and q-axis components of the internal voltage bg of the synchronous generator in this multi-machine form of its electrical model shown in Fig. 4. The definitions of the coefficients aij in Eqs. (9) and (10) are shown in full in the appendix. A more-detailed treatment of the method by which the equations of a transmission network in the familiar abc phase-domain variables are recast into the dq reference frame can be found in [15] for passive transmission systems, and in [16] for transmission systems, such as Fig. 1, that contain a SSSC. A more-detailed treatment of the multi-machine form of the two-axis machine model used here (c.f. Fig. 4) can be found in [17,18]. 2.5.4. The SSSC model In the SSSC model used in this paper, the dynamic characteristics of the SSSC are represented with the assumption that the SSSC’s
3.1. Study system with line 2 uncompensated In the eigenvalue studies presented in this first subsection, in order to reduce the complexity of the SSR characteristics of the study system of Fig. 1 which has an SSSC as the series compensator in line 1, line 2 has been initially left uncompensated (i.e. XC2 = 0.0 pu). The results that follow show, firstly, the SSR characteristics of the study system with line 1 compensated by an SSSC but with no power flow controls around it. Secondly, an initial design and an improved design of the power flow controller are presented, and it is shown how the improved design enables much higher power flow controller bandwidths to be attained. The improved design of the power flow controller is then verified by means of time domain simulation. Finally the impact of the initial and improved power flow control designs on the torsional damping of the generator is shown. 3.1.1. Study system without power flow control This section initially investigates the SSR characteristics of the study system of Fig. 1 at different values of the series compensating reactance provided by the SSSC, with line 2 uncompensated,
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using the mathematical model of the study system without the power flow controller enabled. In the first case of the analysis, the value of series compensation in line 1 is varied from XSSSC = 0.10 pu to XSSSC = 0.95 pu corresponding to an increase in the percentage compensation of transmission line 1 from 10% to 95%. Fig. 5 shows the loci of the eigenvalues of this study system as the compensation of the line 1 is varied in this way. The loci of the individual eigenvalues of Fig. 5 are affected by different parts of the system in Fig. 1. Eigenvalues E2, E3 and E4 are related to the electrical transmission network (incorporating the SSSC, and generator stator circuits); in particular E2 represents the subsynchronous component of the air–gap flux; eigenvalues M0 to M5 represent the six torsional natural frequencies of the turbine-generator’s distributed–mass mechanical shaft; eigenvalue M is associated with the magnetic stability of the machine and it enters the right hand plane when the load angle of the generator exceeds 90°; eigenvalues R1, R2 and R3 are associated with the rotor circuits of the machine. The system of Fig. 5 has similar eigenvalues to a system compensated with conventional series capacitors, with the excep-
tion that eigenvalue E1 associated with the supersynchronous component of air–gap flux is re-placed by the eigenvalue P2 associated with the feed-back regulator dynamics of the SSSC. Fig. 5 shows that as is the case with conventional series capacitive compensation [7], the subsynchronous electrical frequency (eigenvalue E2) of the SSSC compensated system also decreases as XSSSC is increased. Torsional interaction occurs when this subsynchronous electrical frequency approaches the natural frequencies of the turbinegenerator’s mechanical shaft system. Torsional interaction in this study system occurs in the small rectangular area of the eigenvalue plot which is enlarged in the bottom half of Fig. 5. It can be observed that as the subsynchronous frequency of the electrical system (E2) approaches each natural frequency of the mechanical system (M1 to M4), strong torsional interaction between the mechanical and electrical system occurs, forcing the eigenvalue associated with that torsional shaft mode towards the right hand plane. Fig. 5 shows that the destabilising effect of this torsional interaction is sufficient to cause the eigenvalues associated with modes 4, 2 and 1 of this study system to enter the right hand plane,
Fig. 5. Loci of eigenvalues of the study system as the series compensation XSSSC is varied from 0.1 pu to 0.95 pu.
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with the exception of mode 3 which remains in the left plane at all the values of XSSSC. Fig. 5 therefore indicates that as XSSSC is increased, unstable mechanical oscillations in the turbine-generator shaft would be expected to occur in succession at the natural frequencies of modes 4, 2 and 1. The labels in Fig. 5 indicate the particular values of XSSSC at which each torsional shaft mode reaches its most unstable condition. It should also be noted, again, that the eigenvalue study in Fig. 5 indicates the SSR characteristics of the SSSC for a range of values of its compensating reactance without power flow control enabled around the SSSC. 3.1.2. Design of SSSC-based power flow controller This section initially examines the design of the power flow controller to be added around the SSSC by using frequency domain linearised transfer function analysis, in order to investigate the influence of the design of the gains of that controller on the resonant characteristics of a transmission network compensated by the SSSC. The design of the power flow controller gains KP and KI is first carried out using a simplified model of the full study system shown in Fig. 1 in which the generator is replaced by an ideal voltage source. The detailed SSR interaction studies are then carried
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out using the full system model once the range of the controller designs has been chosen. Each design of the power flow controller is carried out by adjusting its gains KP and KI to manipulate the positions of the first-order pole and zero in the transfer function DPL1 =DP L1 ðsÞ between the input and output of the controller in constant power mode (c.f. Fig. 2). Fig. 6 shows the small-signal transfer function DP L1 =DP L1 ðsÞ of the simplified study system when the values of KP and KI are together adjusted to increase the power flow controller bandwidth from 0.13 rad/s to 1.82 rad/s. Fig. 6a shows the transfer function in the form of its pole and zero locations in the complex plane whilst Fig. 6b shows the transfer function in the form of its magnitude and phase as a function of frequency at four selected bandwidth designs. As expected, it can be observed that as the controller’s bandwidth is increased, the real pole P0 associated with the closed-loop response of the controller moves further into the left-hand plane (6a). Fig. 6a also reveals that the design of the controller’s bandwidth influences the resonant characteristics of the transmission network. As the response of the controller is made faster, the eigenvalue E2 associated with the resonant frequency of the transmission network moves slightly closer to the
Fig. 6. Transfer function DP L1 =DP L1 ðsÞ with increasing controller bandwidth from 0.13 rad/s to 1.82 rad/s (sending end fed from ideal source).
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right-hand plane while also increasing in frequency. Fig. 6b also confirms that as bandwidth of the controller between 0 and 3 dB is increased, the underdamped resonant peak in the transfer function (associated with transmission line resonant frequency) increases slightly in frequency, but becomes significantly less damped. The above results demonstrate that when designing the power flow controller at different bandwidths in this way (fixing the position of its transfer function zero in the left hand plane at 10 s1, and moving the position of its pole) the choice of the controller bandwidth does in fact have an influence on the resonant characteristics of the transmission line, even at a single value of compensating reactance provided by the SSSC. Further studies into the design of the power flow controller gain parameters have shown that the position of zero in the transfer function DP L1 =DPL1 ðsÞ does also have an influence on the underdamped resonant peak in the transfer function when the controller’s bandwidth is increased. In order, to investigate this particular influence of the position of the zero in the power flow controller transfer function, the above study was repeated by fixing
the location of the zero of the transfer function at a different position, far to the left of the pole of the transfer function at 303 s1. Fig. 7 shows the small-signal transfer function DP L1 =DP L1 ðsÞ of the simplified study system when the values of KP and KI are together adjusted to increase the power flow controller bandwidth from 0.13 rad/s to 33.5 rad/s with the zero of the transfer function positioned at 303 s1. Fig. 7a shows the transfer function in the form of its pole and zero locations in the complex plane whilst Fig. 7b shows the transfer function in the form of its magnitude and phase as a function of frequency at four selected bandwidth designs. Fig. 7a again reveals that the design of the controller’s bandwidth influences the resonant characteristics of the transmission network. As the response of the controller is made faster, the eigenvalue E2 associated with the resonant frequency of the transmission network again moves slightly closer to the right-hand plane while also increasing in frequency: however, it can be observed that the extent to which this change in the resonant characteristics occurs is markedly reduced when the zero of the power
Fig. 7. Transfer function DP L1 =DP L1 ðsÞ with increasing controller bandwidth with its zero fixed at 303 1/s (sending end fed from ideal source).
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flow controller’s transfer function is fixed at 303 s1. A further benefit of this alternative location of the power flow controller’s zero is that it now allows the use of a much faster controller bandwidth of up to 33.5 rad/s as compared with the previous designs of Fig. 6 where the zero of the power flow controller’s transfer function was fixed at 10 s1. In the initial design shown earlier in Fig. 6, if the controller’s bandwidth is increased to a value above 1.82 rad/s, the magnitude of underdamped resonant peak in the transfer function DP L1 =DPL1 ðsÞ increases above the 0 dB line, hence controller’s bandwidth had to be limited to 1.82 rad/s in this design approach. By contrast, Fig. 7b shows that at similar bandwidths of 1.82 rad/s, the magnitude of the resonant peak is considerably reduced (29.2 dB) and it only approaches 0 dB for significantly larger bandwidth values of 33.5 rad/s. Fig. 7b shows that for this alternative design approach, as the bandwidth of the controller between 0 and 3 dB is increased, the underdamped resonant peak in the transfer function (associated with transmission line resonant frequency) still increases slightly in frequency, but it becomes significantly less damped only at much higher values of bandwidth. The above results demonstrate that when designing the power flow controller at different bandwidths via this second approach (fixing the position of its transfer function zero much further into the left hand plane at 301 s1, and moving the position of its pole) the choice of the controller bandwidth still has an influence on the resonant characteristics of the transmission line, even at a single value of compensating reactance provided by the SSSC. However, the choice of the position of the zero in the power flow controller’s transfer function influences the extent to which the change in the controller’s bandwidth affects the resonant characteristics of the transmission line. By positioning the zero of the transfer function much further to the left along the real axis, the extent of the influence of controller bandwidth on the transmission line resonances is markedly reduced, thus allowing a much faster power flow controller bandwidth to be achieved. The results so far have shown the SSR characteristics of the study system with and without power flow control enabled, and have shown how the controller’s design can be improved to achieve a relatively high controller bandwidth (comparable to fast-responding power flow controllers with settling times on the order of 60 ms as reported in [9]). Prior to examining the impact
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of such fast responding power flow controllers on the electromechanical modes of the study system, the improved power flow controller design is first verified by means of time domain simulation study. 3.1.3. Verification of the design the improved SSSC-based power flow controller The improved design of the power flow controller shown above (with bandwidth of 33.5 rad/s) was verified by means of a time domain simulation study by again using this simplified study system where the synchronous machine was replaced by an ideal source. Fig. 8 shows the response of this simplified study system following a step increase of 0.1 pu in P L1 . It can be observed that following this step increase in PL1 , the power flow controller responds by adjusting the commanded value of the SSSC reactance XSSSC to a new steady state value within 80 ms; this increase in the SSSC reactance increases the actual real power flow PL1 in line 1 to the desired value of PL1 again within 80 ms. Fig. 8 therefore confirms that the improved design of the power flow controller has allowed the value of the real power transmitted by line 1 to be adjusted to track a set point change within 4 ac cycles as specified by the dynamic performance specifications in the frequency domain design, and such a fast-responding controller is comparable to the speed of response of the power flow controller reported in [9]. 3.1.4. Impact of initial SSSC-based power flow controller design on electromechanical modes In order to investigate the influence of the power flow controller on the damping of the torsional modes of the study system of Fig. 1, the full model of the study system is used, that is with the transmission network now fed by a generator and multi-inertia turbine model at the sending end, and with the power flow controller enabled. The power flow controller designs considered in this case are those arrived at using the initial design approach shown in Fig. 6, by locating the zero of the power flow controller transfer function at 10 s1. Fig. 9 now shows the loci of the eigenvalues of the study system as the gains KP and KI of the power flow controller are together adjusted to increase the power flow controller bandwidth from
Fig. 8. Time domain response of the study system with sending end fed from an ideal source.
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Fig. 9. Loci of eigenvalues of the full study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 1.82 rad/s with controller set in constant power mode (initial design).
0.13 rad/s to 1.82 rad/s. As expected, it can be observed that as the bandwidth of the power flow controller is increased, the real eigenvalue P0, corresponding to the closed-loop pole in the power flow controller transfer function moves further into the left-hand plane. Fig. 9 also reveals that increasing the bandwidth of the power flow controller does influence the damping of the torsional modes of the study system. It can be observed that as the response time of the controller is made faster, the eigenvalues M1, M2, M3 and M4 associated with the torsional shaft modes, together with the eigenvalue M0 associated with the generator’s inertial swing mode, all move to the left. The result in Fig. 9 thus shows that in the constant power mode of operation, the effect of the power flow controller is to increase the damping of the inertial swing mode as well as the damping of the torsional modes; furthermore this positive influence on the electromechanical modes increases at increasing bandwidths of the controller. The above eigenvalue analysis was then repeated with the power flow controller set to operate in constant angle mode, with the results shown in Fig. 10. As expected, the real eigenvalue P0 associated with the response of the closed-loop pole of the power flow controller moves towards the left as the controller’s bandwidth is increased from 0.13 rad/s to 1.82 rad/s. Fig. 10 once again shows that in this mode of operation, the controller’s bandwidth design also affects the damping of the electromechanical modes of the study system. However, it can be observed that while the eigenvalues M0 and M4 move towards the right, and hence closer to the imaginary axis, the eigenvalues M1, M2 and M3 move towards the left. The results thus show that in this mode of power flow controller operation, the influence of increased controller bandwidth is to decrease the damping associated with the generator’s inertial swing mode as well as its torsional mode 4, whilst increasing the damping associated with torsional modes 1, 2 and 3, at least at this operating point. 3.1.5. Impact of improved SSSC-based power flow controller design on electromechanical modes In order to investigate the influence of the improved power flow controller design on the damping of the torsional modes of the study system of Fig. 1, the above study was repeated with the power flow controller gains determined using the improved approach shown in Fig. 7 which located the zero of the power flow
controller transfer function at 301 s1, thus allowing a much higher controller bandwidth to be achieved. Fig. 11 now shows the loci of the eigenvalues of the study system as the gains KP and KI of the power flow controller are together adjusted to increase the power flow controller bandwidth from 0.13 rad/s to 33.5 rad/s. As expected, it can be observed that as the bandwidth of the power flow controller is increased, the real eigenvalue P0, corresponding to the closed-loop pole in the power flow controller transfer function moves further into the left-hand plane. Fig. 11 also reveals that increasing the bandwidth of the power flow controller does influence the damping of the torsional modes of the study system. It can be observed that as the response time of the controller is made faster (up to a controller bandwidth of 9.6 rad/s), the eigenvalues M1, M2, M3 and M4 associated with the torsional shaft modes, together with the eigenvalue M0 associated with the generator’s inertial swing mode, all move to the left. However, when the controller’s bandwidth is increased beyond 9.6 rad/s, all the torsional mode eigenvalues M1, M2, M3 and M4 continue to move towards the left but the inertial mode eigenvalue M0 starts to move towards the right. The result in Fig. 11 thus shows that in the constant power mode of operation, the effect of the power flow controller designed in this manner is to increase the damping of the inertial swing mode as well as the damping of all of the torsional modes; furthermore this positive influence on the electromechanical modes increases at increasing bandwidths of the controller up to 9.6 rad/s. However, at bandwidths above 9.6 rad/s, there is still a net positive influence on the damping of all of the torsional modes, but there is less damping added to the inertial mode by the action of the power flow controller than is the case at lower controller bandwidths. The above eigenvalue analysis was then repeated with the power flow controller set to operate in constant angle mode, with the results shown in Fig. 12. As expected, the real eigenvalue P0 associated with the response of the closed-loop pole of the power flow controller moves towards the left as the controller’s bandwidth is increased from 0.13 rad/s to 33.5 rad/s. Fig. 12 once again shows that in this mode of operation, the controller’s bandwidth design also affects the damping of the electromechanical modes of the study system. However, it can be observed that the eigenvalues M1, M2, M3 and M4, together with the eigenvalue M0, all move towards the right, and hence closer to the imaginary axis.
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Fig. 10. Loci of eigenvalues of the full study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 1.82 rad/s with controller set in constant angle mode (initial design).
Fig. 11. Loci of eigenvalues of the full study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant power mode (improved design).
The results thus show that in this mode of power flow controller operation, the influence of increased controller bandwidth is to decrease the damping associated with both torsional and inertial modes. Furthermore, at controller bandwidths higher than 5.3 rad/s, the inertial mode M0 and the torsional mode M4 are unstable. The results in Figs. 11 and 12 have shown that although the design of the power flow controller’s bandwidth can be made very fast, the mode of operation chosen for the power flow controller does impose some limitations on its controller bandwidth design. At this particular operating point of the power flow controller, it was observed that in the constant power mode of operation, for bandwidths above 9.6 rad/s, there is less damping added to the inertial mode than is the case for lower bandwidth designs, although the net impact of the power flow controller on damping remained positive at all bandwidths considered. The results thus
show that if the aim is to optimise the damping of all the electromechanical modes, a bandwidth of 9.6 rad/s will be the best design of the controller for the constant power mode. However, if the aim is to achieve the fastest possible power flow controller response, a bandwidth of 33.5 rad/s can be used since all the electromechanical modes are still stable at this highest controller bandwidth. In the case of the constant angle mode of operation, it is not even possible to operate the power flow controller at bandwidths above 5.3 rad/s as some of the electromechanical modes become unstable. The results thus show that if the aim is to optimise the damping of all the modes, this mode of operation is not recommended at all since there is a detrimental impact on the damping of all the generator’s electromechanical modes. The results also show that if the aim is to achieve the fastest power flow controller response, this mode of operation is also not recommended since a maximum bandwidth of 5.3 rad/s can be used.
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Fig. 12. Loci of eigenvalues of the full study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant angle mode (improved design).
The results presented so far, have shown the influence of the power flow controller on torsional mode damping when line 2 in the study system of Fig. 1 is uncompensated. The following section considers the influence of adding conventional capacitive series compensation in line 2. 3.2. Study system with line 2 compensated The following eigenvalue studies aim to investigate the impact of the improved SSSC-based power flow controller design on the damping of the electromechanical modes of the study system in the presence of conventional series capacitors in line 2. However, in order to consider such a study, it is first necessary to identify a combination of values of SSSC-based compensation in line 1 and conventional compensation in line 2 for which the system is SSR stable prior to the addition of the power flow controller in line 1. In order to identify ranges of series compensation values over which all the electromechanical modes in a study system are stable, it is more useful to the present the results of the eigenvalue scans in an alternative format in which the real parts of the eigenvalue loci are plotted as a function the compensation value being varied; this format is adopted for the remainder of the eigenvalue results in this subsection. 3.2.1. Locating SSR-stable ranges of line 2 compensation In order to identify SSR stable ranges of the series compensation in line 2, the following study initially considers the SSR characteristics of the study system of Fig. 1 at different values of conventional series compensating reactance in line 2, while the series compensation provided by the SSSC in line 1 is held fixed at 0.1 pu, without the power flow controller enabled. The value of series compensation in line 2 is varied from XC2 = 0.10 pu to XC2 = 0.95 pu corresponding to an increase in the percentage compensation of transmission line 2 from 10% to 95%. Fig. 13 shows the real part of the eigenvalues associated with the torsional modes of this study system at each value of XC2 as the compensation of the line 2 is varied in this way. It can be observed that as the conventional series compensation in line 2 is increased, there is successive destabilisation of the mechanical torsional modes M4, M3, M2 and M1 as the real part of the eigenvalues associated with each of these torsional modes moves into the positive plane of Fig. 13. It can be observed that
there are certain ranges of series capacitive compensation in line 2 where the real part of all the torsional mode eigenvalues M1, M2, M3 and M4 are in the negative plane of Fig. 13 (that is all four torsional modes are simultaneously SSR stable). Within these stable ranges, two specific values of series compensating reactance were chosen for further study in the remaining eigenvalue scans: case A: XC2 = 0.23 pu and case B: XC2 = 0.36 pu. At each of these values of conventional series compensation in line 2, the SSR characteristics of the SSSC compensation in line 1 was then considered for a range of compensation values. 3.2.2. Locating SSR stable operating point for the SSSC in line 1 in the presence of conventional series capacitors in adjacent line 2 for two selected cases In order to identify the SSR stable operating ranges of the SSSC in line 1 for each of the above two values of series capacitive compensative compensation in line 2, the eigenvalues of the study system were calculated as the series compensation XSSSC provided by the SSSC is varied from 0.1 pu to 0.95 pu, without power flow control enabled around the SSSC, with series compensation XC2 in line 2 fixed, in turn, at each of the values chosen in case A and case B. Fig. 14 shows the real part of the eigenvalues associated with the torsional modes of this study system as the series compensation XSSSC in line 1 is varied in this way with the compensation in line 2 fixed at the value for case A: XC2 = 0.23 pu. It can be observed that as the series compensation provided by the SSSC in line 1 is increased there is successive destabilisation of the mechanical torsional modes M4 and M1 as the real part of the eigenvalues associated with these two torsional modes moves into the positive plane of Fig. 14. Fig. 14. shows that the SSR stable operating range of series compensation in line 1 is from XSSSC = 0.1 pu to a maximum value of XSSSC = 0.24 pu for the case A value of conventional compensation in line 2, since at all values of XSSSC above 0.24 pu, torsional mode M4 remains unstable. Fig. 15 shows the real part of the eigenvalues associated with the torsional modes of this study system as the series compensation of line 1 is again varied from 0.1 pu to 0.95 pu with the series compensation XC2 in line 2 fixed at the value for case B: XC2 = 0.36 pu. It can be observed that as the series compensation provided by the SSSC in line 1 is increased there is successive destabilisation of the torsional modes M4, M1 and M3 as the real part of the eigenvalues associated with these torsional modes
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Fig. 13. Real part of eigenvalues of the study system as the conventional series compensation XC2 is varied from 0.1 pu to 0.95 pu with XSSSC in line 1 fixed at 0.1 pu.
Fig. 14. Real part of eigenvalues of the study system as the series compensation provided by the SSSC, XSSSC in line 1 is varied from 0.1 pu to 0.95 pu while the series capacitive reactance XC2 in line 2 is fixed at 0.23 pu (case A).
moves into the positive plane of Fig. 15. Fig. 15 shows that the SSR stable operating range of series compensation of the SSSC in line 1 for the case B value of conventional series compensation in line 2 is from XSSSC = 0.1 pu up to a maximum value of XSSSC = 0.29 pu after which, torsional mode M4 remains unstable. Figs. 14 and 15 have shown that the SSR-stable operating ranges of the series compensation XSSSC in line 1 at two different values of capacitive reactance in line 2 are quite limited. In order to study the effect of the power flow controller on torsional mode damping at these two different values of conventional series compensation in line 2, a common SSR-stable operating value for the SSSC’s reactance of XSSSC0 = 0.15 pu was used for the remaining eigenvalue studies, which consider the impact of the power flow controller added around the SSSC on the damping of the electromechanical modes. 3.2.3. Impact of improved SSSC-based power flow controller design on electromechanical modes with conventional series capacitors in line 2 In order to investigate the influence of the power flow controller on the damping of the torsional modes of the study system of
Fig. 1, Fig. 16 shows the loci of the eigenvalues of the study system as the gains KP and KI of the power flow controller are together adjusted to increase the power flow controller bandwidth from 0.13 rad/s to 33.5 rad/s with the power flow controller set to operate in constant power mode at a nominal XSSSC0 = 0.15 pu and with XC2 fixed at 0.23 pu. It can be observed that as the bandwidth of the power flow controller is increased, the real eigenvalue P0, corresponding to the closed-loop pole in the power flow controller transfer function moves further into the left-hand plane. Fig. 16 also reveals that increasing the bandwidth of the power flow controller does influence the damping of the torsional modes of the study system. It can be observed that as the response time of the controller is made faster (up to a controller bandwidth of 4.6 rad/s), the eigenvalues M1, M2, M3 and M4 associated with the torsional shaft modes, together with the eigenvalue M0 associated with the generator’s inertial swing mode, all move to the left. However, when the controller’s bandwidth is increased beyond 4.6 rad/s, all the torsional mode eigenvalues M1, M2, M3 and M4 continue to move towards
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Fig. 15. Real part of eigenvalues of the study system as the series compensation provided by the SSSC, XSSSC in line 1 is varied from 0.1 pu to 0.95 pu while the series capacitive reactance XC2 in line 2 is fixed at 0.36 pu (case B).
Fig. 16. Loci of eigenvalues of the study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant power mode, XSSSC0 = 0.15 pu and XC2 = 0.23 pu (case A).
the left, but the inertial mode eigenvalue M0 starts to move back towards the right hand plane. The result in Fig. 16 thus shows that in the constant power mode of operation, even with the presence of a fixed series capacitor in line 2, the effect of the power flow controller is to increase the damping of the inertial swing mode as well as the damping of all of the torsional modes; furthermore this positive influence on the electromechanical modes increases at increasing bandwidths of the controller up to 4.6 rad/s. However, at bandwidths that are higher than 4.6 rad/s, while there is still an increasing positive influence on the damping of the torsional modes, the influence on the inertial mode is less significant than at lower bandwidth (although still net positive). The above eigenvalue analysis was then repeated with the power flow controller set to operate in constant angle mode, with the results shown in Fig. 17. As expected, the real eigenvalue P0 associated with the response of the closed-loop pole of the power flow controller moves towards the left as the controller’s band-
width is increased from 0.13 rad/s to 33.5 rad/s. Fig. 17 shows that in this mode of operation the controller’s bandwidth design also affects the damping of the electromechanical modes of the study system. However, it can be observed that the eigenvalues M1, M2, M3 and M0 all move towards the right, and hence closer to the imaginary axis, whilst eigenvalue M4 moves to the left as the controller bandwidth is increased. The results thus show that in this mode of power flow controller operation, the influence of increased controller bandwidth is to decrease the damping associated with all the electromechanical modes other than mode 4. Furthermore, at controller bandwidths higher than 2.3 rad/s, the inertial mode M0 is unstable. The above study was repeated for the second value of compensation in line 2 at which the fixed series capacitive reactance XC2 was set at 0.36 pu. Fig. 18 shows the loci of the eigenvalues of the study system as the gains KP and KI of the power flow controller are together adjusted to increase the power flow controller
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Fig. 17. Loci of eigenvalues of the study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant angle mode, XSSSC0 = 0.15 pu and XC2 = 0.23 pu (case A).
Fig. 18. Loci of eigenvalues of the study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant power mode, XSSSC0 = 0.15 pu and XC2 = 0.36 pu (case B).
bandwidth from 0.13 rad/s to 33.5 rad/s with the power flow controller set to operate in constant power mode at a nominal XSSSC0 value of 0.15 pu and XC2 fixed at 0.36 pu. It can be observed that as the bandwidth of the power flow controller is increased, the real eigenvalue P0, corresponding to the closed-loop pole in the power flow controller transfer function moves further into the left-hand plane. Fig. 18 also reveals that increasing the bandwidth of the power flow controller does influence the damping of the torsional modes of the study system. It can be observed that as the response time of the controller is made faster (up to a controller bandwidth of 5.2 rad/s), the eigenvalues M1, M2, M3 and M4 associated with the torsional shaft modes, together with the eigenvalue M0 associated with the generator’s inertial swing mode, all move to the left. However, when the controller bandwidth is increased beyond 5.2 rad/s, all the torsional mode eigenvalues M1, M2, M3 and M4
continue to move towards the left while the inertial mode eigenvalue M0 starts to move back towards the right. The result in Fig. 18 thus shows that in the constant power mode of operation, even with the presence of a fixed series capacitor in line 2 at a different value of conventional series compensation XC2 = 0.36 pu in line 2, the effect of the power flow controller is still to increase the damping of the inertial swing mode as well as the damping of all of the torsional modes; furthermore this positive influence on the electromechanical modes increases at increasing bandwidths of the controller up to 5.2 rad/s. However, at bandwidths that are higher than 5.2 rad/s, there is still a positive influence on the damping of all of the torsional modes, but there is less positive damping added to the inertial mode than at a bandwidth of 5.2 rad/s. The above eigenvalue analysis was then repeated with the power flow controller set to operate in constant angle mode, with
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Fig. 19. Loci of eigenvalues of the study system as the power flow controller’s bandwidth is varied from 0.13 rad/s to 33.5 rad/s with controller set in constant angle mode, XSSSC0 = 0.15 pu and XC2 = 0.36 pu (case B).
the results shown in Fig. 19. As expected, the real eigenvalue P0 associated with the response of the closed-loop pole of the power flow controller moves towards the left as the controller’s bandwidth is increased from 0.13 rad/s to 33.5 rad/s. Fig. 19 once again shows that in this mode of operation, the controller bandwidth design also affects the damping of the electromechanical modes of the study system. However, it can be observed that the eigenvalues M1, M2, M4, M0 all move towards the right, and hence closer to the imaginary axis, whilst eigenvalue M3 moves to the left as the controller bandwidth is increased. The results thus show that in this mode of power flow controller operation, the influence of increased controller bandwidth is to decrease the damping associated with all the electromechanical modes other than torsional mode M3. Furthermore, at controller bandwidths higher than 2.6 rad/s, the inertial mode M0 is unstable.
of generator torsional modes at risk, with the extent of the destabilisation being very sensitive to the system operating conditions and power flow controller design. Appendix A Coefficients aij in Eqs. (9) and (10) are defined as follows:
a11 ¼
Ra LLL1 ðRa þ RL2 Þ L2 LDEN1 LLL1 L2
a14 ¼ a22
a15 ¼
;
1 þ LLL1 L2
LDEN1 LDEN1 L1 LL2 RL1 RL2 LsRþL t þLg ¼ ; LDEN2 LL2 Ls þLt þLg
4. Conclusion
a25 ¼
Previous work has shown the dependence of torsional mode damping (and hence SSR stability) on both the mode of operation and the speed of response of a power flow controller designed for relatively slow-responding control action. This paper has investigated a method of redesigning the power flow controllers considered in earlier work to achieve higher-bandwidth controllers comparable to other fast-responding power flow controllers reported in the literature. In the process, it has been shown that an alternative method of pole-zero placement when designing the power flow controller’s closed-loop transfer function not only allows greater bandwidths to be attained, but, in the case of constant-power type controllers, allows significantly improved stabilisation of all generator electromechanical modes to be achieved. Extensive studies have been presented to confirm not only that a constant power mode controller adds stabilising damping to all torsional modes, and is able to do so even in the presence of conventional series compensation in an adjacent line, but that this stabilisation can be increased by a faster acting power flow controller. Similar studies in this paper support earlier work which has concluded that the alternative mode of operation of a power flow controller that has been proposed in the literature (constant angle) is not suitable for environments where SSR is a concern. In particular, this mode of control has been shown to destabilise a number
where
LDEN2
a12 ¼
;
RL1 LLL1 RL2 L2
a21 ¼
;
a23 ¼
; LDEN1 LL2 Ra RL2 Ls þLt þL g
LDEN2 1 Ls þLLL2t þL
g
LDEN2
;
a13 ¼
1 ; LDEN1
a24 ¼
1 ; LDEN2
;
;
LDEN1 ¼ Ls þ Lt þ Lg þ LL1 þ LDEN2 ¼ LL1 þ LL2 þ
LL1 Ls þ Lt þ Lg ; LL2
LL1 LL2 Ls þ Lt þ Lg
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