Such a station holds a power elecronic converter with active energy storage and can be viewed as a ... costly, HVDC station usually can be considered. The.
Damping of Electro-Mechanical Oscillations in a Multimachine System by Direct Load Control Olof Samuelsson
Bo Eliasson
IEA, Lund Institute of Technology Box 118, S-221 00 Lund Sweden
Sydkraft AB Box 50510, S-202 50 Malmö Sweden
Abstract—Utility controlled customer loads as actuators present new possibilities for power system control. The use of active loads controlled by local bus frequency is proposed for damping of electro-mechanical oscillations. The viability of the idea is studied for one load in a three machine system with a meshed network. Active power mode controllability and phase angle mode observability are determined from the eigenvectors of a differential algebraic description of the uncontrolled system. The geographical variations in the entire network of controllability and observability are shown to be identical. It is presented graphically on a 3D-view of the network topology and is used as a generalization of the term mass scaled electrical distance. System zeroes limit the maximum damping. An electro-mechanical mode pendulum analog is introduced that explains this. Time simulations verify the final controller design.
I. INTRODUCTION Control of active loads to improve transient and dynamic stability was suggested already in [1]. This idea has recently become realistic thanks to the advances in Distribution Automation and Demand Side Management. The installation of power line carrier communication systems and remote control facilities, have now turned loads usually classified as disturbances into actuators, that are available primarily for peak shaving, but also for other purposes. A very important property of these direct load control systems is the possibility to switch the controllable loads selectively in a non-disturbing manner – leaving critical loads uninterrupted. Typical loads that are well suited for utility control are thermal loads such as heaters and boilers, and other energy storages like future charging stations for electric vehicles. Such a station holds a power elecronic converter with active energy storage and can be viewed as a controlled, linear and bidirectional active power source, with little bus voltage dependence.
Considering damping of an inter-area mode with the controlled aggregated active loads of a distribution system is in many aspects similar to using HVDC for damping, in that they both modulate active power at a point in the transmission network. Important differences are that a large number of distributed controlled loads can be employed to a modest cost, whereas only one, immensely costly, HVDC station usually can be considered. The possibly large number of actuator groups (distribution systems) allows simultaneous damping of more electromechanical modes than the slowest inter-area mode, but also raises the question of undesired interaction. Inter-area mode damping by modulating active power in a longitudinal two machine system has been studied in [2] and [3]. As a step towards a damping system for large systems with many controlled loads, this paper aims at extending the intuitive understanding of active power controllability and phase angle observability to a multimachine system with a meshed network. II. POWER SYSTEM MODEL The three machine system of Fig. 1, from [4] is used as test system. It is the simplest possible multimode system with a meshed network, thus presenting a non trivial yet manageable topology. In [3] it is shown that modulation of active power has the greatest impact on a machine if it is done electrically close to it. Considering this, the chosen test system is a challenge as it features no obvious load bus as best candidate. A third order synchronous generator model is used together with a proportional AVR and constant mechanical power. All numerical data of network, impedance loads symbolizing distribution systems, and machines are given in [4] except the AVR gain that is 30. Load C
N2 N7 S2 H 6.40 s
N9 N3 S3 H 3.01 s
N8 N5
Load A
N6 Load B N4
H1
N1 H 23.64 s
Fig. 1. One-line diagram of the test system with inertia constants given for a 100 MVA system base
The device models of the power system simulator EUROSTAG [5] are used for convenience. EUROSTAG is intended for nonlinear time simulations but can, at any instant during a simulation, export a linearized system model. As all systems data are entered only once, the nonlinear and the linearized models are guaranteed to be consistent. To facilitate analysis and manipulation of the linearized model, a collection of Matlab functions or toolbox has been developed. The toolbox, called EuroMat, is described in Appendix 1 with examples from this study. EUROSTAG exports a matrix Differential Algebraic Equation (DAE). It holds explicit DAEs for the individual devices, including the network. For a generator this is
[
T x Gen = λ f V ref E fd Tm ω θ I q I d
e = diag([1 0 0 0 1 1 0 0 ]) 0 = cx Gen − yV Gen
∆i R =
(1)
where field flux linkage λf , rotor angular velocity ω and machine angle θ are the dynamic states. AVR setpoint Vref, stator EMF Efd, mechanical torque Tm and the q and d axis stator currents Iq and Id are all algebraic variables. The network is described by algebraic balance equations for (real and imaginary parts of) node current with (real and imaginary parts of) the bus voltages as state variables:
[
]
0 = ∆i − YV
(2)
∆i contains the current injections from the machines. The full system is then described by: x T = [ x Gen1 x Gen2 x Gen3 V ] e1 0 0 0
0 e2 0 0
0 0 e3 0
0 a1 0 0 ˙ 0 a2 x= 0 0 0 c c 0 1 2
0 0 a3 c3
An element in a left DAE eigenvector gives a measure of how active the equation on the corresponding row is during that mode [6]. The lower part of the system matrix contains the balance equations for the real and imaginary parts of the currents at each bus. The close relation to active power modulation is shown by (4), as derived in [7].
]
ex˙ Gen = ax Gen + bV Gen
V iT = Re( v i ) Im ( v i )
Controllability therefore has the highest priority and is treated first. The control signal is continuously variable active power, that can be thought of as drawn by a bidirectional power electronic converter with an active energy storage.
b1 b2 b3 x Y bus
(3)
Note that the full bus admittance matrix enters the system matrix as the lower right block. This is an example of the structure preserving property of DAE models, that will prove very valuable in the following.
vR ∆P V2
∆i I =
vI ∆P V2
(4)
For light load the imaginary part of the bus voltage vI is negligible at all buses and the voltage magnitude V and the real part of the bus voltage vR are both close to unity. The mode controllability of active power can then directly be approximated by the real current ∆iR elements of the left DAE eigenvector. Mode controllability is thus determined at all buses just by computing the left eigenvectors of the uncontrolled system. This facilitates analysis, in particular for large systems, as no input matrix need be formed. One of the toolbox functions illustrates the geographical variation in the network of the mode controllability for the two electro-mechanical modes. As seen in Fig. 2, vertical bars are placed at the buses of a 3D-view of the network topology (thin solid line). This information can be used as a meshed network multi machine system generalization of mass-scaled electrical distance mentioned in [3]. Dashed lines between the buses are added for improved visualization. Fig.2 can be compared to [8] which uses a geographical map, but no network topology. For the slow mode Fig. 2a, indicates that among the load buses (N5, N6 and N8), N8 has the highest controllability. Considering the faster mode in Fig. 2b, N6 is superior, closely followed by N8. Taking both modes into account, N8 is the overall best location for a controlled active load. 1.3 Hz mode
0.025 0.02
III. ACTIVE POWER CONTROLLABILITY While the eigenvectors of a matrix ODE (Ordinary Differential Equation) only contain dynamic states, DAE eigenvectors also describe how the algebraic variables participate in a mode. In this case and in many FACTS applications, the damping system will be located in the network and is thus closely related to the algebraic states. The additional algebraic information of a DAE eigenvector is then of direct use when considering mode controllability and observability. Low controllability is more expensive to compensate for than low observability.
0.015 0.01 0.005 0
N2
N9
N8
N7
−0.005
N3
N6
N5 N4
−2 N1
−3 −4 −3
−2
−1
0
1
2
3
Fig. 2a. Geographical active power controllability of the slow mode superimposed on network topology (solid)
apply viscous damping i.e. let a force proportional to the mass velocity act on the mass. This is equivalent to controlling active power at the generators in proportion to the machine frequency deviation, which is known as governor action. This is, however, incompatible with the present location of the actuator, and alternatives are called for. Either the machine frequency can be transmitted to the controlled load, or the local frequency deviation at the transmission system bus can be used for feedback. As the intuitive understanding is based on the introduction of a viscous damper, the feedback must be local as in Fig. 3c.
1.8 Hz mode
0.04 0.03 0.02 0.01 0
N9
N8
N7
N2
N3
N6
N5 N4
−2 N1
−3 −4
−1
−2
−3
1
0
2
3
Fig. 2b. Geographical active power controllability of the fast mode
IV. FEEDBACK SIGNAL Having chosen control variable, selecting a feedback signal is the most decisive design step. The freedom is much greater as both directly measurable quantities and synthesized signals can be considered. Intuitive understanding of the control problem is here very valuable, but is hard to acquire. In this context a pure mechanical analog to electro-mechanical oscillations has proven useful and is therefore outlined below. A. Pendulum analog The linearized equations of a single machine infinite bus system (Fig. 3a) have their exact analog in a linearized simple pendulum, provided the machine model is of second order. Adding a controlled active load at a bus between the machine and the infinite bus, is equal to letting a force act at a point x on the flexible string, in which the mass is suspended. The pendulum with a nonzero force is shown in Fig. 3b, and its transfer function (from force to angle α ) and that of the power system equivalent are given in Appendix 2. The validity can be extended to multimachine systems in a non-strict fashion by letting the pendulum illustrate one mode. It is intuitively obvious that, as x approaches the mass, the force will have more influence on the swinging mass. This is in full accordance with the results of [3] and section III. An intuitively effective way to damp the pendulum is to a)
b)
c)
An active load controlled by local bus frequency is attractive: The pendulum predicts damping to be improved at all damper locations except at swing nodes. This was shown for a longitudinal system in [3] and if it applies to a meshed multi mode system, topology is irrelevant. This is close to the desired properties of a damping system in [9]. Due to these potential advantages, the local frequency alternative will be used in the continued analysis. Note that the frequency at transmission system level is used even if loads in the distribution system are to be controlled. B. Phase angle observability Bus frequency is obtained by filtering phase angle, as in Fig. 4 which shows the resulting controller. s 1+sT
ϕ
∧ ω + –
Σ
∆ω
∆P
K
ωnom Fig. 4. Controller with T=10 ms and gain K to be determined
∆ϕ =
1 [ v R ∆v I − v I ∆v R ] V2
(5)
Phase angle variation ∆ϕ is extracted from bus voltage variation as shown in (5), which uses the notation of (4). With the above approximations, the elements of the right DAE eigenvector representing the imaginary part of bus voltages give the phase angle mode observability. Fig. 5, presents the geographical variability of phase angle observability for the two electro-mechanical modes. 1.3 Hz mode
0.1
V∞∠0 0.05
x
l2
∆P
α ∆F
x
V∠ϕ x’
0
N2
N9
N8
N7
N3
N6
N5
l1
N4
−2 N1
E∠δ
H
M
−3 −4
Fig. 3. a) Original single machine power system, b) Pendulum analog, c) Proposed damping system added
−3
−2
−1
0
1
2
3
Fig. 5a. Geographical phase angle observability of the slow mode
1.8 Hz mode
1.002 Undamped
0.06
1
0.04
0.998 0.02
N2
N3
N9
N8
0
0
1
2
3
4
5
1.002 Damped
N7 N5
N6
1
N1
0.998
N4
−2 −3 −4 −3
−2
−1
1
0
2
3
Fig. 5b. Geographical phase angle observability of the fast mode
With the pendulum in mind, the similarity between phase angle observability in Fig. 5, and active power controllability in Fig. 2, is not surprising yet very promising. Geographically equal observability and controllability means that the eigenvalue sensitivity, which results as the product, will have the same sign regardless of where in the network (and the mode) the damper with local feedback is placed. This is the most basic requirement in order to be able to damp several modes with the same control system. The similarity further makes the load bus with highest controllability to the one with the highest observability. The load bus N8 is obviously the most appropriate bus for using an active load controlled by local frequency to damp both modes. V. CLOSING THE LOOP The eigenvalue sensitivity analysis actually carried out above indicates that the gain K will move the eigenvalues of the electro-mechanical modes straight into the left halfplane. As with all sensitivity analysis, this is only valid at the point of zero gain. When studying larger gains, the root locus of Fig. 6, is more appropriate. It shows that the electro-mechanical eigenvalues pass a point of maximum damping, but end up at system zeroes close to the imaginary axis.
12
Imag [rad/s]
1
2 3 4 5 Time [s] Fig. 7a. Slow mode: Frequencies in p.u. of H1 (–), S2 (..) and S3 (–.) Undamped
1.001 1 0.999 0
1
2
3
4
5
Damped
1.001 1 0.999 0
1
2 3 4 5 Time [s] Fig. 7b. Fast mode: Frequencies in p.u. of H1 (–), S2 (..) and S3 (–.)
The zeroes are introduced by the chosen combination of input and output signals, and are exhibited also by the pendulum analog: as the gain, i.e. the viscous damping, is increased from zero, more swing energy is dissipated in the damper. For large gains, however, the swinging mass cannot move the piston. No energy is dissipated and the damping returns to zero. As the string suspending the mass is effectively shortened, the frequency is now increased. The point of maximum damping of the slow mode is used to determine the feedback gain to 540 MW/Hz. The damping of the faster mode is increased, but much less as expected from the sensitivity analysis above. Simulations in EUROSTAG of the nonlinear system verify the design. The modes are selectively excited by adding a sinewave with 1 % amplitude to the AVR setpoint of S2 (1.3 Hz) and S3 (1.8 Hz) respectively. At t=0 excitation ceases and the damper is engaged. Fig. 7 illustrates the performance with and without the proposed damping system for each mode. As predicted, the damping of the slow mode is improved considerably, while the damping of the fast mode increased very little.
13
11
0
o: zero (infinite K) x: K=0 +: K=135, 270, 405, 540 MW/Hz
10 9
VI. CONCLUSIONS 8 7 −6
−5
−4
−3 −2 Real [1/s]
−1
0
Fig. 6. Root locus of the electro-mechanical modes
1
Active distribution loads, modelled as an active transmission load, controlled by the transmission system bus frequency, are shown to simultaneously increase damping of both electro-mechanical modes. It seems likely that this applies to all operating points and
topologies, which would be a major advantage over many other damping systems. The maximum damping is, however, limited by the zeroes introduced by the used combination of input and output. Estimation of modal angle by additional local measurements considerably improves damping, by moving these zeroes. This is shown for a longitudinal system in [10], but generalization to a meshed network still remains. Using the described frequency signal, a practical system would switch many distribution system loads remotely, thereby controlling the aggregated power drawn from the transmission system in a quasi-linear fashion. The use of more distribution systems for damping requires studies on their interaction and a procedure for coordinated gain selection. In any case administration of controlled loads is important. A pendulum analog to an electro-mechanical mode is outlined. It fully explains the root locus of the system. The use of DAEs has proven convenient during the modelling phase. It provides structural information, which is useful, especially when graphically presented along with the network topology. This applies in particular for larger systems, that are easily handled using the Matlab toolbox presented in Appendix 1. VII. ACKNOWLEDGEMENTS The work reported in this paper is supported by Sydkraft AB and NUTEK, which is gratefully acknowledged. VIII. REFERENCES [1]
R. H. Park, "Improved Reliability of Bulk Power Supply by Fast Load Control," Proceedings of the 1968 American Power Conference, pp 445-457
[2]
O. Samuelsson, B. Eliasson, G. Olsson, "Power Oscillation Damping with Controlled Active Loads," IEEE/KTH Stockholm Power Tech Conference, Stockholm, Sweden, June 18-22, 1995, pp 274-279
[3]
T. Smed, G. Andersson, "Utilising HVDC to Damp Power Oscillations," IEEE T-PWRD, Vol. 8, No. 2, April 1993, pp 620-627
[4]
P. M. Anderson, A. A. Fouad, Power System Control and Stability, IEEE, 1993
[5]
EUROSTAG user's manual - Release 2.3, Tractebel Electricité de France, April 1995
[6]
T. Smed, "Feasible Eigenvalue Sensitivity for Large Power Systems," IEEE T-PWRS, Vol. 8, No. 2, May 1993, pp 555-563
[7]
P. Kundur, Power System Stability and Control, McGraw-Hill, New York, 1994
[8]
B. Eliasson, "Damping Structure and Sensitivity in the Nordel Power System," IEEE T-PWRS, Vol. 7, No. 1, February 1992, pp 97-105
[9]
J. F. Gronquist, W. A. Sethares, F. L. Alvarado, R.
H. Lasseter, "Power Oscillation Damping Control Strategies for FACTS Devices using Locally Measurable Quantities," IEEE T-PWRS, Vol. 10, No. 3, August 1995, pp 1598-1605 [10] E. V. Larsen, J. J. Sanchez-Gasca, J. H. Chow, "Concepts for Design of FACTS Controllers to Damp Power Swings," IEEE T-PWRS, Vol. 10, No. 2, May 1995, pp 948-956 [11] H. Othman, R. Vedam, J. Finney, L. Ängquist, "Robust Supplementary Damping Controllers," IEEE/KTH Stockholm Power Tech Conference, Stockholm, Sweden, June 18-22, 1995, pp 244-249 Olof Samuelsson was born in Västerås, Sweden in 1966. He received his M.Sc. degree in Electrical Engineering from Lund Institute of Technology in 1989. During 1990 he was research engineer at the University Hospital of Lund. Since 1991 Olof Samuelsson is a postgraduate student at the Department of Industrial Electrical Engineering and Automation. Bo Eliasson was born in Åre, Sweden in 1947. He obtained his M.Sc. degree in Mathematics and Physics at the University of Lund. In 1976 he received his M.Sc. degree in Electrical Engineering from Lund Institute of Technology. After six years at ASEA, he joined Sydkraft AB in 1982 for employment as systems engineer. 1990 Bo Eliasson received the Ph.D. degree in Automatic Control at Lund Institute of Technology. He is currently working with Energy Management Systems at Sydkraft AB.
APPENDIX 1 EuroMat — a EUROSTAG toolbox in Matlab Anytime during simulation in EUROSTAG, the full power system model can be linearized and exported in two files. One file contains the matrix differential-algebraic equation. The other file contains the state vector element labels, assigned partly by the program and partly by the user. Starting from these files the Matlab toolbox offers a number of functions for manipulation and analysis of the linearized system, automatically handling the labels and maintaining consistency with the full nonlinear model. The menu functions of the graphical user interface, are described below, exemplified by the test system of Fig. 1. Load file Having read the two files into Matlab, system matrices are created together with variables, such as number of generators or network nodes. Using the matrices of the autonomous system (A1), inputs and outputs are identified. E x˙ = Ax
(A1)
Input and output matrices of the DAE system can then be formed (A2), as well as all the four ODE matrices of (A3). E' x˙ ' = A' x' +B' u y = C' x' x˙ " = A " x" +B" u y = C" x" +D" u
(A2)
(A3)
H1
H1
LAMBDAF H1 VREF H1 EFD H1 CM H1 OMEGA H1 TETA H1 IQ H1 ID S2 LAMBDAF S2 VREF S2 EFD S2 CM S2 OMEGA S2 TETA S2 IQ S2 ID S3 LAMBDAF S3 VREF S3 EFD S3 CM S3 OMEGA S3 TETA S3 IQ S3 ID N1 VR N1 VI N2 VR N2 VI N3 VR N3 VI N4 VR N4 VI N5 VR N5 VI N6 VR N6 VI N7 VR N7 VI N8 VR N8 VI N9 VR N9 VI
S2
S3
N1 N2 N3 N4 N5 N6 N7 N8
−0.6
N9
−0.4
−0.2
0
0.2
0.4
0.6
Fig. A3. Animation of the full right eigenvector for the 1.3 Hz mode Fig. A1. Sparsity of A and block matrix structure
H1
LAMBDAF
H1 OMEGA H1 TETA S2
LAMBDAF
S2 OMEGA S2 TETA S3
LAMBDAF
S3 OMEGA S3 TETA
3D-visualization of DAE eigenvector information Geographical variation of controllability and observability can be presented in a 3D-view of the network topology as in Fig.2 and Fig. 5. The network structure can be extracted from the system matrix A, but the user is required to specify bus locations by pointing and clicking. Multi-Modal Decomposition The toolbox is prepared for analysis of joint operation of several damping controllers, in which the Controller Interaction and the Maximum Damping indices of [11] are valuable. The indices are based on partial Multi-Modal Decomposition [10]. Functions for this and evaluation of the indices for two controllers are included. APPENDIX 2 Transfer functions
Fig. A2. Sparsity of A' and B' with dynamic states indicated
To verify the automatic conversion, input, output and state vector labels can be listed. Sparsity plots The sparsity of the system matrices is easily visualized. The label information can be used for finding block matrix borders between individual devices as in Fig. A1, cmp (3). Alternatively bus voltage or dynamic states are identified as in Fig. A2, which shows that V ref of each AVR has been classified as an input, leading to a B matrix. Modal analysis When a system is loaded into Matlab, finite eigenvalues and their left and right eigenvectors are computed for the full DAE system. The mode shape of the machine angle states, can be illustrated using a bar graph. Furthermore, the full right eigenvector information can be presented as an animated horizontal bar graph, where the user can select to include any subset of states. Fig. A3, shows a snapshot from an animation of the 1.3 Hz mode.
The single machine power system with active power as input and local phase angle as output has the following transfer function (ωR is nominal angular velocity in rad/s): ∆ϕ (s ) s 2 + ω z2 1 =− where ∆P(s) k 1 + k 2 s 2 + ω p2
ωR ω k 1k 2 k 1 ; ω p2 = R 2H 2H k 1 + k 2 VV ∞ EV k1 = cos(δ 0 − ϕ 0 ); k 2 = cos ϕ 0 x x'
ω z2 =
The flexible pendulum with force ∆F as input and angle ∆α as output has a similar transfer function: ∆α (s ) l1 s 2 + ω z2 1 =− where Mg l1 + l 2 s 2 + ω p2 ∆F(s)
ω z2 =
g g ; ω p2 = l1 l1 + l 2
