Coprime factorisation approach in designing multi-input stabiliser for damping electromechanical oscillations in power systems K.K. Anaparthi, B.C. Pal and H. El-Zobaidi Abstract: The application of loop shaping in HN control design through a dual-input power system stabiliser (PSS) in a benchmark power system model is demonstrated. The PSS stabilises two key system poles contributing to critical electromechanical oscillations involving turbine generators and electrical network. The IEEE First Benchmark Model has been considered in this study to demonstrate the effectiveness of the control. A linearised model of the system is pre- and post-compensated using loop-shaping approach. The problem of robust stabilisation of a normalised coprime factor plant description is formulated into a generalised HN problem and the solution is obtained analytically. The performance of the controller shows robustness to electromechanical modes, which were otherwise very poorly damped or undamped depending on operating conditions. The robustness is further validated in Simulink, including nonlinearities and device saturation limits.
1
Introduction
Power system stabilisers (PSSs) have been widely used in stabilising power systems, especially for post-fault damping to electromechanical oscillations [1, 2]. Modern robust control methodologies, like HN optimisations, have been used in recent years for designing stabilising controllers to address this issue [3, 4]. Fixed series capacitors have long been used in power systems as a cost-effective option for enhancing power transfer capability of HV transmission systems. However, series capacitance has caused shaft damage in multistage steam turbine systems on a couple of occasions through torsional oscillations of the turbine in a frequency range less than power frequency (50 or 60 Hz). This is known as subsynchronous resonance (SSR). SSR is an electric power system condition in which the electric network exchanges energy with a turbine generator at one or more of the natural frequencies of the combined system below the subsynchronous frequency of the system [5]. These natural frequencies are popularly known as electromechanical modes in power system stability literature. Generator rotor oscillations at a torsional mode frequency, fm, induce armature voltage components at frequencies (fem) given by fem ¼ f0 fm
ð1Þ
When the subsynchronous component fem is close to fer (electrical resonant frequency), the subsynchronous torques produced by subsynchronous voltage components can be sustained. This interplay between electrical and mechanical systems is termed as torsional oscillation [6]. The two shaft r IEE, 2005 IEE Proceedings online no. 20045102 doi:10.1049/ip-gtd:20045102 Paper first received 13th July 2004 and in revised form 1st February 2005 The authors are with the Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT, UK E-mail:
[email protected]
failures at Mohave generating station in western USA in December 1970 and October 1971 [6] were traced to torsional interactions (TI). Several countermeasures for damping SSR using static blocking filters, NGH damping, dynamic stabilisers and supplementary excitation damping controllers (SEDC) have been proposed over the years [6]. The present trend is to introduce thyristor controllers, such as thyristor controlled series capacitors (TCSC) and static synchronous series compensator (SSSC), to provide flexible and controlled series compensation. However, their higher costs make them a less attractive option and as a result usage of these advance devices in power systems is very few and far between. PSSs, so far have been deployed to dampen one mode, more commonly the local mode (1–2 Hz) that represents the electromechanical oscillations of the entire turbogenerator with respect to the electrical network. The PSSs basic function is to produce an electrical torque component in phase with the rotor speed variation and to add damping to the rotor oscillations by controlling its excitation through stabilising signal. It is equipped with torsional filter to prevent the PSS interacting with one or more of the torsional modes owing to the reduced gain margin of the PSS at torsional frequencies. In this paper, we damp both SSR and local modes by a single PSS, which has not been done so far. Speed input PSSs are generally added to excitation systems to enhance damping of the inertial modes of oscillations of turbine generators [7], but it was also established that they can destabilise torsional modes [8]. Fouad and Khu [9] were successful in damping torsional oscillations using generator-speed-based PSSs but the disturbances considered were small and only one mode could be controlled. Lee et al. [10] proposed a new scheme for PSSs using deviation in generator speed and output power as input signals. In this paper, we have chosen this combination of input signals for designing the PSS. The PSS controller is designed by a two-stage HN based design procedure, which uses a normalised coprime factor approach to robust stabilisation of the
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301
system. This is based on the Glover–McFarlane HN loopshaping design procedure. The design technique involves loop shaping of the nominal plant to shape its singular values to give desired open-loop properties at frequencies of high and low loop gain and the normalised coprime factor robust stabilisation technique to stabilise the shaped plant [11]. This approach has distinct advantages over the loop transfer recovery (LTR) method in linear quadratic Gaussian (LQG) design. LTR cannot systematically deal with plants with RHP zeros [12] and is limited in that it can only guarantee performance and robust stability properties at either plant input or plant output. The open-loop shaping of nominal plant is also advantageous over standard HN design. This is another approach to MIMO feedback controller design, in which the design is specified on closedloop objectives in terms of requirements on the singular values of weighted closed-loop transfer functions. This HN design procedure produces undesirable controllers whose zeros cancel all the stable plant poles, which is unacceptable when the plant contains lightly damped modes. The control problem after formulation is solved by the normalised coprime factored HN optimisation technique used in [13, 14]. The performance of the controller is tested on the IEEE FBM SSR model, which comprises a single generator connected to an infinite bus through a series-compensated transmission line. Eigenvalue analysis and time-domain simulations are performed to confirm that the proposed PSS can dampen SSR at high values of series compensation. 2
Power system study model
2.1
Description
The IEEE First Benchmark Model (FBM) [15] for SSR analysis considered in this study is shown in Fig. 1. It consists of an 892.4 MVA synchronous generator connected to an infinite bus via a highly compensated 500 kV transmission line. The mechanical system consists of a four-stage turbine, the generator and a rotating exciter. The total system model can be derived by cascading the individual system models, which are explained in the next Sections.
XT
R
XL XC
Fig. 1
2.2
IEEE FBM model for SSR study
Mechanical system model
The mechanical system model is that of a multistage pressure-compounded turbine generator system. This consists of rotors of generator (g), exciter (e), LPB, LPA, IP and HP turbine shafts (see Fig. 2). The torques between the shafts are indicated by subscripts. The notations and symbols are taken from [6]. The nonlinear differential equations of the turbine generator model can be written in compact form as: X_ m ¼f ðXm ; Tm ; Te Þ Ym ¼gðXm Þ
ð2Þ
These equations after linearising about an operating point are then expressed as: D_xM ¼½AM DxM þ ½BM DTe DyM ¼½CM DxM 302
TIP,LPA
DHP
DIP
Fig. 2
TLPA,LPB
DLPA
TLPB,g
DLPB
Tg,e
DGEN
DEXC
Coupled multimass model for turbine generator system
where the state vector and output vectors are, respectively, given by: xtM ¼ dg Se Tg;e Sg TLPB:g SLPB TLPA;LPB SLPA TIP ;LPA SIP THP ;IP SHP t ¼ ½ dg Se yM where dg ¼ rotor angle corresponding to the generator rotor: Ss ¼ per unit slips of exciter, generator, LPB, LPA, IP and HP turbine masses; Tg,e ¼ torque in shaft section connecting generator and exciter; TLPB,g ¼ torque in shaft section connecting LPB and generator; TLPA,LPB ¼ torque in shaft section connecting LPA and LPB; TIP, LPA ¼ torque in shaft section connecting IP and LPA, and THP,IP ¼ torque in shaft section connecting HP and IP
2.3 Synchronous machine model with excitation system
XSYS
infinite bus
generator
THP,IP
The generator equations are modelled with two windings on the rotor, Model (1.1), i.e. one field winding on the d-axis and one damper winding on the q-axis [6]. A rotating AC exciter with thyristor-controlled rectifier is considered for the excitation system. The signal Vs from the designed PSS is the control input to the plant. The linearised electrical system (generator and excitation controller) equations are expressed as: D_xer ¼½Aer Dxer þ ½Ber2 Duer þ ½Br DVs þ ½Ber1 DyM Dyer ¼½Cer Dxer where
Ed0
Eq0
t yer ¼ ½ id
EFD
iq
and cd ; cq ¼ flux linkages along the d and q axes; Ed0 , Ed0 ¼ transient EMFs due to flux linkages along the d and q axes, respectively; EFD ¼ field voltage of excitation system; and id, iq ¼ d and q components of machine terminal currents.
2.4 ð3Þ
xter ¼ cd cq uter ¼½ vd vq
ð4Þ
Network model
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parameter p circuit. The symmetric nature of the network facilitates its representation in terms of a and b components whose configurations have been discussed in [6]. The differential equations of the RLC network are written and after transforming to synchronous D–Q reference frame are given by: v_ CD 0 oB vCD ¼ vCQ v_ CQ oB 0 ð5Þ iD oB xC 0 þ 0 o B xC i Q where vCD, vCD ¼ fixed capacitor voltages in D–Q reference frame iD, iQ; D–Q components of generator armature current and xC ¼ fixed series capacitive reactance, which is equal to the product of the series compensation level and the total external inductive reactance. The linearised state equations and output equations of the network model are derived as: D_xN ¼½AN DxN þ ½BN DUN DyN ¼½CN DxN
ð6Þ
where xtN ¼yNt ¼ ½ vCD vCQ UNt ¼½ iD iQ ¼ ½ id iq P t and
cosðdÞ P¼ sinðdÞ
2.5
sinðdÞ cosðdÞ
Complete electrical system equations
Combining (4) and (6) gives the total electrical system model (synchronous machine, excitation system and network model). The equations are written as follows: ð7Þ D_xE ¼ ½AE DxE þ ½BEM Dym þ ½Br DVs where xtE ¼ ½ xter
2.6
xtN
Combined system equations
Equations (3) and (7) can be combined to give the total state equations. DTe can be expressed as: " # Dcd Did DTe ¼ iqo ido þ cqo cdo Dcq Diq ð8Þ ¼½CME DxE The final system equation, when expressed in compact form, is: ð9Þ D_xT ¼ ½AT DxT þ ½BT DuT where xtT ¼½ xtE xtM uT ¼ Vs AE BEM CM AT ¼ and BT ¼ Br BM CME AM The objective is to design a PSS that would modulate exciter reference voltage through Vs. 3
Eigenvalue analysis
The IEEE FBM for computer simulation of SSR is considered as the study system with the parameters of the model taken from [6, 15]. Eigenanalysis is performed to analyse the interaction between network and torsional modes. The network mode varies with the level of series
compensation (see (5)). The FBM has five torsional modes near 16, 21, 25, 33 and 47 Hz. The degree of series compensation was varied in the range 0–90% to see the impact of the network mode on the torsional modes. It was found that the first four modes were unstable in the range of series compensation of 64–77%, 52–61%, 40–45% and 24– 31%. Mode 5 was found to be unaffected. It is the exchange of energy between network and torsional modes that excites the modes through resonance. Practical power systems exhibit instability through resonances. In the context of SSR, it is considered that the weak and strong resonances of the first order [16] are responsible for torsional instability. In the context of FBM system, for a compensation value very close to the inception of instability, a weak resonance condition exists. We have verified that the system matrix corresponding to that condition is diagonalisable (perfect decoupling), a mathematical condition for weak resonance. As series compensation is increased by a small amount, i.e., a perturbation is given around weak resonance, the system approaches a strong resonance point. Strong resonance results when two identical pair of stable complex eigenvalues collide with each other because of variation in system operating parameters. The eigenvalues are infinitely sensitive to parametric variations. This results in, at the collision, a sharp 901 turn of the eigenvalues. One becomes unstable, through Hopf bifurcation and the other moves further left in the left-half eigenplane. At strong resonance the matrix is non-diagonalisable [16]. However, in practice, with one parameter variation such as series compensation level, the power system will not experience an exact strong resonance but will pass close to such a resonance and the qualitative effects would be similar, i.e. the eigenvalues will move quickly and change direction as they interact. This will lead to oscillatory instability condition. Even in the vicinity of strong resonance, the torsional and network modes are highly sensitive to parametric variation. We have studied this phenomenon in the FBM model and results are shown in Fig. 3. The eigenvalues and dampings of the torsional modes of the power system study model are shown in Table 1. It is interesting to see that network mode becomes more stable as the torsional modes move to the right half plane. Physically this can be explained as a transfer of finite energy from the electrical network to the generator shaft, which appears as torsional force to various shaft sections. 4
HN based loop-shaping design approach
The objective of this approach, proposed by McFarlane and Glover [11, 13, 14, 17, 18], is to incorporate the simple performance/robustness tradeoff obtained in loop shaping, with the normalised left coprime factor (LCF) robust stabilisation method as a means of guaranteeing closed-loop stability. The design technique is a two stage procedure: 1 the nominal plant singular values are shaped using simple loop shaping to give desired open-loop properties of desired high or low gain at frequencies of interest; 2 the shaped plant is robustly stabilised using the normalised LCF approach of HN optimisation. The normalised coprime factorisation is formulated into a generalised HN problem and solved by ncfsyn, available with m-Analysis and Synthesis Toolbox in Matlab [19]. This procedure is briefly outlined in the following section [17].
4.1
Loop shaping
The loop-shaping design involves using a prefilter W1 and/ or a postfilter W2 such that the nominal plant singular
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303
350
frequency of modes, rad/s
300
kc : percentage series compensation NM: network mode TM1: torsional mode 1 TM2: torsional mode 2 TM3: torsional mode 3 TM4: torsional mode 4
NM increase in kc
250
200
kc = 24%
TM4
kc = 39%
150
TM3 kc = 52%
100
TM2 TM1
kc = 64% kc = 77%
50 −8
−6
−4
NM
−2
0
2
4
6
real part of modes
Fig. 3
Interactions of network and torsional modes near strong resonances as percentage compensation varies
Table 1: Eigenvalues and damping ratios of torsional and local modes without PSS Modes
Eigenvalues
Damping (z)
Frequency f (Hz)
mode 5
1.850407298.17i
0.0062
47.455
mode 4
0.364587202.86i
0.0180
32.286
mode 3
0.644747160.46i
0.004
25.54
mode 2
0.060157126.86i
0.00047
20.19
mode 1
0.815237102.87i
0.008
16.372
mode 0 (local)
0.12165712.052i
0.01
2.046
values are shaped to give a desired open-loop shape. This approach is thus advantageous as it includes the performance considerations via loop shaping and still obtains an exact solution for the normalised LCF problem. The nominal plant G and the shaping filters W1 and W2 are combined to give the shaped plant, Gs, where Gs ¼ W2GW1 (Fig. 4). The controller K is designed by solving the robust stabilisation problem for the shaped plant Gs, as described later in the Section. The pre- and post-filters are to be carefully designed based on the systematic procedure outlined in [20]. This involves:
W1
Gs
W1
G
W1
W2
K
Keq
Fig. 4
Loop-shaping design procedure
G∆ ∆N
∆M
+ N
−
+
+
M −1
Robust stabilisation
The shaped plant Gs in the previous stage is factored into left and right normalised coprime factors as Gs (s) ¼ M1(s)N(s) such that M(s) M*(s)+N(s)N*(s) ¼ I is satisfied. The block diagram for normalised coprime factor robust stabilisation problem is shown in Fig. 5. Any perturbation of Gs(s) because of variation in operating conditions or unmodelled dynamics can also be expressed in the coprime factors symbolically as GD ¼ (M+DM)1(N+DN) where DM, DN are stable unknown 304
W2
G
K
The condition of the design problem is improved by scaling the plant inputs and outputs. The pre- and post-filters are selected in such a way that the shaped plant has high gain at low frequencies, rolloff rates of approximately 20 dB/decade at desired bandwidths, with higher rates at high frequencies. Integral action is also added at low frequencies.
4.2
W2
G
K
Fig. 5
Normalised coprime factor robust stabilisation problem
transfer functions that represent the uncertainty in the nominal plant model Gs and D ¼ [DN, DM];77D77Noe takes care of all the uncertainties in the plant model. IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 3, May 2005
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It has been shown by Glover–McFarlane [14] that the control design problem is to maximise the uncertainty measure ‘‘or calculate emax such that GD can be stabilised by a single controller K, where I 1 1 ¼ inf M ð10Þ ð I GK Þ e1 max k k stabilising 1 More e means that more uncertainty in Gs is allowed and hence greater stability and performance margins is achieved. Mathematically, the control design is stated as minimising the cost function: I 1 1 ðI GK Þ M ð11Þ min K K2S
where xk, represents the controller states. u is the controller output that links with DVS in (9). DSe and DPe are the perturbations of the plant outputs (yp) Se and Pe around the nominal operating point. The disturbance (w) in the system can be represented by change in input torque (Tm), bus voltage (Eb) or reference voltage (Vref) at the voltage regulator etc. The exogenous output z is torque at different turbine sections, speed, line power etc. The transfer matrix between z and w is given by: Tzw ðsÞ ¼ Cd ðsI Ad Þ1 Bd þ Dd where
1
1
where S ¼ (IGK)1 is the sensitivity and S is the set of all stabilising controllers. Thus, the generalised HN problem for the normalised coprime factor HN robust stabilisation problem can be written as: S SG ð12Þ min KS KSG K2S 1
where S ¼ parametric perturbation of the plant; SG ¼ additive perturbation of the controller, KS ¼ additive perturbation of the plant; and KSG ¼ input multiplicative perturbation of the plant. Hence, minimising (12) maximises the amount of allowable perturbations with guaranteed stability. The generalised regulator description of (12) is given by: 2 3 A B 0 B 6C 0 I 0 7 7 P ¼6 ð13Þ 40 0 0 I 5 C 0 I 0 The state representation corresponding to the generalised plant P is given by: 32 3 2 3 2 x x_ A B1 B2 4 z 5 ¼ 4 C1 D11 D12 54 w 5 ð14Þ C2 D21 0 u y where B1 ¼ ½ B 0 , B2 ¼ ½ B, C 1 ¼ ½ C 0 T , 0 I D11 ¼ , D ¼ ½ 0 I T and D12 ¼ ½ 0 I . The 0 0 12 AK BK controller K ¼ can be obtained by solving the CK DK HN optimisation problem given in (12). All along we have designed controller K for shaped plant Gs ¼ W2GW1. Hence the shaping filters must be absorbed in K such that the actual controller Keq for G is Keq ¼ W1KW2 (see Fig. 4). The statespace representation of the controller is given by: x_ k ¼ Ak xk þ Bk y ð15Þ u ¼ Ck xk þ Dk y
ð16Þ
Ad ¼
A þ B2 Dk C2
B2 Ck
Bk C 2 Ak B1 þ B2 Sk D21 Bd ¼ Bk D21
Cd ¼ ½ C1 þ D12 Dk C2 D12 Ck Dd ¼D11 þ D12 Dk D21 The design in (12) is now made robust so that the plant tolerates as much HN coprime factor uncertainty as possible. This is done with the MATLAB’s m-analysis toolbox using the command ncfsyn. This paper extends the methodology to dampen SSR modes and a local mode through a single PSS. 5
Dual-input PSS design
A two-input, one-output controller was designed to stabilise the torsional and local modes. The machine operates at full load power (P ¼ 1.0 pu) for a series capacitor compensation of 65%. Mechanical dampings are considered with excitation system parameters KA ¼ 200 and TA ¼ 0.025 s. Taking rotor speed and electric power as output variables of the power system and DVs of the PSS as the input to the system, the output equation can be written as: DyT ¼ ½CT DxT
ð18Þ
where yTt ¼ ½ Se
Pe
Loop shaping: Inspection of the nominal plant (see Fig. 6) indicates that the open-loop cross-over frequency at C0.2 rad/s implies a very low closed-loop bandwidth. Hence high gain was added in the low-frequency range. Integral action is added by adding poles at zero radians per second so that the shaped plant has high gain at lower singular values
150
nominal plant shaped plant
100 50 gain, dB
As M and N are normalised left coprime factors of G, the cost function can be written as: I 1 1 K ðI GK Þ M 1 I 1 ðI GK Þ M 1 ½M N ¼ K 1 I ðI GK Þ1 ½I G ¼ K 1 S SG ¼ KS KSG
ð17Þ
0 −50 −100 −150 10−2
Fig. 6
10−1
100
101 102 frequency, rad/s
103
104
Nominal plant and shaped plant open-loop gain
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305
25
singular values
150
20
output 1 of shaped plant output 2 of shaped plant
100
15
full controller reduced controller
50 gain, dB
gain, dB
10 0
−50
0 −5
−100
−10
−150 −200 10−2
5
10−1
100
101
102
103
104
−15 101
102 frequency, rad/s
frequency, rad/s
Fig. 7
Singular value response of two outputs of shaped plant
frequencies. The zeros are added at the required frequencies to ensure that at desired bandwidths the rolloff rate is approximately 20 dB/decade with higher rates at high frequencies. The post-filter W2 is chosen as unity as W1 itself provides all the shaping requirements needed. The pre-filter W1 thus obtained is: ðs þ 1Þðs þ 1:1Þ W1 ¼ 400 ð19Þ s3 So, the shaped plant Gs is of the form Gs ¼ GW1. The resulting openloop gain is shown in Fig. 6. The frequency response of the two outputs of the shaped plant is given in Fig. 7. Robust stabilisation: A controller KN is synthesised for the nominal plant to achieve robust stability with an optimal stability margin of e ¼ 0.234 indicating that deterioration in the high- and low-frequency loop shapes is minimal, and acceptable robust stability properties in the crossover frequency region are expected. The final controller is then obtained by cascading W1 with K to form the final controller Keq. The ‘robustification’ of this design, so that the shaped plant tolerates as much HN coprime factor uncertainty as possible, is done with the MATLAB’s m-Analysis and Synthesis Toolbox [19] using the arguments of the function ncfsyn. This is the analytical method of obtaining the controller. The controller obtained from the loop shaping HN optimisation was of high order (23 states), which was reduced to 16th order by the Hankel norm approximation model reduction method. Even a 16th-order controller is quite large and any further reduction was seen to deteriorate the closed-loop damping performance of the system. The controller’s singular value response is shown in Fig. 8 for both full order and reduced order. It is seen that the reduced order controller retains the frequency response characteristics. The system with the proposed PSS becomes stable as all the modes have positive and adequate damping. The proposed PSS has improved the damping of all the modes as shown in Table 2. 6
Results and discussions
Eigenvalue analyses were performed on the FBM without and with the controller for a series compensation level of 65%. As seen from Table 1, which gives the eigenvalues and dampings of torsional modes of the power system model, without PSS, mode 1 and the local mode have negative damping, indicating that the system is unstable. Table 2 shows the closed-loop eigenvalues and damping for electromechanical modes when the PSS is included into 306
103
Fig. 8 Singular value response of full-order and reduced-order controller
Table 2: Eigenvalues and damping ratios of torsional modes with PSS Mode
Eigenvalues
Damping (z) Frequency, f (Hz)
mode 5
1.84887298.17i
0.0062
47.456
mode 4
2.37317202.7i
0.0170
32.308
mode 3
2.197907160.65i
0.0137
25.624
mode 2
0.704167124.82i
0.00564
19.84
mode 1
11.102773.511i
0.1490
11.825
the control loop. This shows good damping experienced by the SSR mode 1 with the local mode shifted far deep into the left half eigenplane. Another interesting point to note is that the damping of all the electromechanical modes has considerably increased. It is seen that the damping of mode 1 is improved significantly with little effect on modes 2–5. The controller was designed to focus on local mode and mode 1 only. It is mode 1 that is very critical, especially as the series compensation level normally ranges between 60 and 70% for maximum utilisation of transmission capacity. The frequencies of modes 2 to 4 are high, and although damping appears small, they are good enough to settle the oscillations of those frequencies within 5–10 s. However, this is for a particular operating condition (the design was based on system operating at a power output of 1.0 p.u or real power at unity power factor). The power output of the generator is varied to examine the robustness of the controller. Variation in power output causes changes in initial operating values of the system, namely armature current, flux linkages, transient voltages etc., resulting in eigenvalues getting shifted to different locations. The results given in Table 3 show that the controller maintained adequate damping for a range of power outputs. The nonlinear power system model was simulated using MATLAB’s Simulink to investigate whether the proposed controller can effectively damp out the SSR modes. The system parameters are the same as in the eigenvalue analysis with all the system nonlinearities and component saturation incorporated in detail. All the independent models of the system have been modelled in Simulink. The system was tuned to excite the first torsional mode of oscillation and the local mode. The variables that have been plotted are: DiL: change in line current; DTe: change in electrical torque acting on generator rotor: DThi: change in HP-LP IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 3, May 2005
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Table 3: Eigenvalues and damping ratios of mode 1 without and with controller for various operating conditions Operating condition
Without controller
With controller
Eigenvalue
Damping (z)
Eigenvalue
Damping (z)
P ¼ 0.95, Q ¼ 0
0.769397102.82i
0.0075
11.303773.321i
0.152
P ¼ 1.05, Q ¼ 0
0.864197102.92i
0.0084
10.942773.680i
0.147
P ¼ 0.90, Q ¼ 0
0.726937102.76i
0.0071
11.555773.102i
0.156
P ¼ 1.10, Q ¼ 0
0.915987102.97i
0.0090
10.819773.830i
0.145
P ¼ 0.95, Q ¼ 0.312
0.769397102.82i
0.0075
11.303773.321i
0.152
P ¼ 0.89, Q ¼ 0.456
0.718877102.75i
0.007
11.612773.054i
0.157
30
300 electrical torque, p.u.
line current, p.u.
25 20 15 10
−100 −200
40
200
20
0 −20
0
2
4 6 time, s
8
150 100 50 0 −50 −100 −150
10
0
2
4
6 time, s
8
10
1.25
1.20
1.20
electrical torque, p.u.
1.25
1.15 1.10 1.05 1.00
1.15 1.10 1.05 1.00
0.95
0.95
0.90
0.90
0.40
1.50 torque of LPB-GEN, p.u.
torque of HP-IP, p.u.
line current, p.u.
Time-domain simulations without controller
0.35 0.30 0.25 0.20
0
2
4
6
8
10
time, s
Fig. 10
0
−300
−40
Fig. 9
100
0
torque of LPB-GEN, p.u.
torque of HP-IP, p.u.
5
200
1.40 1.30 1.20 1.10 1.00 0.90 0.80
0
2
4
6
8
10
time, s
Time-domain simulations with proposed controller
shaft torque; and DTbg: change in LPB-Gen shaft torque (all in p.u.). The series-capacitor compensated system without the PSS controller at a compensation level of 65% was first
simulated and the disturbance applied was a 20% increase in the input torque for 100 ms. The system response is unstable, as shown in Fig. 9. It can be observed that the oscillations grow very quickly, leading to instability. The
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307
1.06
1.04
1.04
electrical torque, p.u.
line current, p.u.
1.06
1.02 1.00 0.98 0.96
0.31 0.30 0.29 0.28
1.10 1.05 1.00 0.95 0.90
2
4
6
8
10
0
2
4
6
8
10
time, s
Time-domain simulations with proposed controller for disturbance in infinite bus voltage
controller is now included in the system and time-domain simulations were carried out. As seen from Fig. 10, the controller has been able to damp the SSR oscillations in about 10 s. Although the oscillations of the turbine torques still exist after the disturbance, they dampen out eventually with time. To validate the robustness of the controller, another disturbance of 2% decrease in infinite bus voltage, Eb for 200 ms was applied to the system. The controller was successful in damping out the oscillations in about 5 s (see Fig. 11), thus demonstrating the robustness of the HN loop-shaping controller. Conclusions
A dual-input PSS has been designed using GloverMcFarlane’s HN loop-shaping procedure to mitigate the SSR and the power system local modes that may occur in a series compensated power system. Proper loop shaping weighting functions in the form of pre- and postcompensators have been selected to stabilise the SSR and local modes simultaneously. Our study considered IEEE FBM. The HN control design strategy on loop-shaping methods was set- up through loop-shaping methods and solved by ncfsyn available with m-Analysis and Synthesis Toolbox in Matlab. The performance robustness has been tested in both the frequency and time domain for a range of power outputs. Nonlinear simulations demonstrate the good damping performance of the controller. The controller has also improved the damping of all other electromechanical modes. References
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308
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time, s
8
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0
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Fig. 11
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torque of LPB-GEN, p.u.
torque of HP-IP, p.u.
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IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 3, May 2005
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