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Wide-area measurement-based modal decoupling for power system oscillation damping
Electric Power Systems Research 178 (2020) 106022
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Wide-area measurement-based modal decoupling for power system oscillation damping
T
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Rui Fan, Shaobu Wang, Renke Huang, Jianming Lian , Zhenyu Huang Pacific Northwest National Laboratory, 902 Battelle Blvd, Richland, WA 99354, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Decoupled control Power system stabilizer High-voltage direct current Power oscillation damping Small-signal stability
Inter-area oscillation modes usually fall within the same frequency range of 0.1–1.0 Hz; thus, power oscillation damping (POD) control methods should avoid unnecessary excitation of weakly damped modes. This paper presents a decoupled control strategy to damp inter-area oscillations in the power grid. Pure oscillation mode signals are extracted from wide-area measurements and used as feedback input for POD actuators such as power system stabilizers or high-voltage direct current transmission. The decoupled control strategy is able to increase the damping of the concerned oscillation modes without deteriorating others. This proposed strategy could be implemented independently, or serve as a complement and work along with conventional POD controllers to damp concerned inter-area modes. The proposed method has been tested on a modified MinniWECC system. Extensive results indicate that inter-area oscillations can be successfully and precisely damped using the proposed decoupled strategy.
1. Introduction Inter-area oscillation is one of the major concerns in modern largescale power systems. It is the interaction among different groups of machines that are connected by relatively weak transmissions [1]. The inter-area oscillations of typically 0.1–1.0 Hz potentially lead to unstable system operations or even grid blackouts. With the expansion of modern power systems and increase in area interconnections, inter-area oscillations have become a key challenge for large-scale interconnected grids [2]. Various techniques have been developed to study and damp interarea oscillations. A power system stabilizer (PSS) is a popular device that damps generator rotor angle swinging in a broad range of frequencies in the power system [3]. It adds damping to the rotor oscillations of the synchronous machine by providing supplementary control action through the generator excitation systems. The most popular power oscillation damping (POD) method using PSS is the linearizationbased eigenvalue analysis scheme [4–6]. This scheme performs an eigenvalue calculation on the basis of a state-space representation of the system that is linearized at a given operating point. With the expansion of the high-voltage DC (HVDC) transmission system, people have started to leverage its capability in modulating active power for interarea oscillation damping [7]. The transmitted power can be rapidly controlled through HVDC converters by changing the firing and
⁎
extinction angles. The most popular HVDC-based POD scheme uses the terminal frequency difference as feedback inputs to the converter modulation signals [8,9]. Except for PSS and HVDC, other actuators such as flexible AC transmission system (FACTS) and accumulated demand modulation have been also used for oscillation damping [10,11]. With the invention and wide-spread implementation of phasor measurement units (PMUs) in power systems, many advanced POD schemes have been designed for better performance, with a requirement of fast communication and accurate information exchanges. Various FACTS devices have been proposed to work with HVDC or PSS for better damping performance [12,13]. FACTS devices provide new options for damping oscillations, but are commonly accompanied by complex communication protocols, high maintenance costs, and other substantial burdens [14]. Loop-shaping and pole-placement methods have also been used to control HVDC or PSS and damp inter-area oscillations by bending the Nyquist curve away from the critical point [15–17]. In [15], a feedback compensator was designed using the loopshaping method to shape an HVDC loop transfer function for better performance. In [17], a linear matrix inequality based mixed outputfeedback control with regional pole placement was used to damp multiple oscillation modes. Carefully designed controllers using loopshaping theory had been found to offer several advantages, including more feedback in the system-dominant mode, the existence of actuator effort even on high-frequency content in the error signal, and the
Corresponding author. E-mail address: [email protected] (J. Lian).
https://doi.org/10.1016/j.epsr.2019.106022 Received 2 February 2019; Received in revised form 31 August 2019; Accepted 1 September 2019 Available online 25 September 2019 0378-7796/
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Section 4, case studies on a modified MinniWECC test system are presented. Section 5 presents a discussion of the case studies, and Section 6 presents conclusions.
suitability of the controller for the wider controller frequency bandwidth of the plant. However, the possibility of cancellation of the right half plane poles and zeros in the loop transfer functions might cause the system to be unstable [18]. Other advanced control methods, such as model predictive control (MPC), were also used to damp power system inter-area oscillations. MPC-based methods were well recognized and widely used in the process industry and in multiple research areas because of their advantages over conventional approaches [19]. The MPCbased method anticipated future events and modulated the control signal to minimize the oscillations, while respecting the HVDC constraints. However, the high computational burden and difficulties in application could result in unsuitable control schemes [20]. One challenge for inter-area oscillation damping is that oscillation modes usually fall in the same frequency ranges and they are physically coupled through system interaction. It is possible that a POD controller would damp certain modes while exciting others unnecessarily [21]. In [22], energy storage and large capacitor banks were used to mitigate the modal interactions, with an additional cost for the DC storage devices. Preece et al. [23] proposed a modal linear quadratic Gaussian (MLQG) control to damp targeted oscillation modes and achieved some good results. This approach requires the system model to be linear/ quadratic; however, the power system is naturally nonlinear. In addition, this approach requires detailed information about the system model, which could be difficult in the real world, especially when the system operation conditions change from time to time. Bjork et al. [24] proposed a solution that eliminated expensive storage devices, but it required two or more HVDC links. Zhang et al. [25] proposed a PSSbased modal decomposition approach by filtering mixed oscillation signals through a band-pass filter, but a few problems will need to be further addressed. Because the signal fed to the band-pass filter is from only one location in [25], the concerned mode may not be adequately observed. Another issue is that the frequencies of oscillation modes are so close (0.1–1.0 Hz) that it is very difficult to distinguish the oscillations through a band-pass filter. The performance of the band-pass filter depends on the input signals, thus, troublesome phase shift or unfiltered components would occur since the oscillation signals are time-variant and their frequency difference is too small. To resolve the above modal coupling challenge without an investment in additional storage devices or DC lines, this paper proposes a novel decoupled POD control strategy that can be implemented on either PSS or HVDC. The proposed method takes advantage of the development of the wide-area measurement system. The increasingly observable system information (bus voltage magnitudes, angles, and power flows) provides new options of designing POD controllers for specifically targeted oscillation modes. It is possible now to use the abundant wide-area data to extract the targeted pure oscillation modes using a decomposed matrix method. The extracted pure modal signal is further applied as a feedback signal to modulate the PSS output or the transferred power on HVDC lines. Because the feedback signal only relates to the corresponding system eigenvalue, it will have minimum negative impacts on the damping of other modes. The contribution of this paper comes from: (1) a novel modal decoupling approach and an efficient, high-quality pure mode extraction method from multi-location synchronized PMU data; (2) a novel decoupled POD approach that could damp a specific inter-area oscillation mode in a complex power system without affecting other oscillation modes. This decoupled strategy could be implemented independently to damp the concerned inter-area mode, or serve as complement and work along with conventional POD controllers for further improvements on system transient stability. The performance of the proposed method was tested and validated in a modified MinniWECC system [26]. Extensive results indicate that inter-area oscillations can be successfully and precisely damped using the proposed decoupled strategy. This paper is organized as follows. In Section 2, a pure mode signal as POD controller input is briefly introduced. In Section 3, the widearea measurement-based modal decomposition method is discussed. In
2. Pure mode signal as input of POD controller In this section, the underlying principles of using pure mode signal as POD controller input are briefly introduced. To proceed, consider the following single-input single-output system for oscillation damping,
⎧ x˙ (t ) = Ax (t ) + bu (t ), ⎨ ⎩ y (t ) = cx (t ),
(1)
where x ∈ ℝn is the state vector of the system, A ∈ ℝn × n is the system transition matrix, and u ∈ ℝ, y ∈ ℝ , b ∈ ℝn and c ∈ ℝ1 × n are the input, output, and input and output matrices, respectively. Applying the Laplace transform to (1) with zero initial conditions, i.e., x(0) = 0, we have
⎧ s X (s ) = AX (s ) + bU (s ), ⎨ ⎩Y (s ) = cX (s ).
(2)
For simplicity of the following derivation, we assume that A has n distinct eigenvalues; that is, A is diagonalizable. Let λi ∈ ℂ denote the ith eigenvalue of A, and let vi ∈ ℂn and wi ∈ ℂ1 × n denote the right and left eigenvectors associated with λi, respectively. Define T = [v1 ⋯ vn] and Λ = diag(λ1, …, λn). Then, it follows that
T−1AT = Λ, and
⎡ w1* ⎤ T−1 = ⎢ ⋮ ⎥, ⎢ w*⎥ ⎣ n⎦ where w i* denotes the complex conjugate of wi. To proceed, define the new state vector Z ∈ ℝn × n as
Z (s ) = T−1X (s ),
(3)
whose elements Zi(s), i = 1, …, n, are the pure modal signals. Substituting (3) into (2), it follows that −1 ⎧ s Z (s ) = ΛZ (s ) + T bU (s ), ⎨ ⎩Y (s ) = cTZ (s ).
(4)
It follows from (4) that Y(s) can be expressed as a combination of different modal signals, n
Y (s ) = c ∑ vi Zi (s ) = i=1
n
∑ Yi (s ), i=1
where Yi(s) = cviZi(s). Now assume that Yi(s) (1 ≤ i ≤ n) can be perfectly extracted from Y(s) and is used as the feedback signal for the controller, that is,
U (s ) = P (s ) Yi (s ) = P (s ) cviZi (s ),
(5)
where P(s) represents the controller transfer function. Substituting (5) into (4), it follows that
s Z (s ) = ΛZ (s ) + T−1bU (s ) = ΛZ (s ) + T−1bP (s ) cviZi (s ) = ΛZ (s ) + T−1b [0 ⋯ P (s ) cvi ⋯ 0] Z (s ) = A c Z (s ), where 2
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⎡ λ1 ⎢0 ⎢ ⋮ Ac = ⎢ ⎢0 ⎢⋮ ⎢ ⎢0 ⎣
0 λ2 ⋮ 0 ⋮ 0
w1*bP (s ) cvi ⋯ w *2 bP (s ) cvi ⋯ ⋱ ⋮ ⋯ λi + w i*bP (s ) cvi ⋱ ⋮ w n*bP (s ) cvi ⋯
⋯ ⋯ ⋱ ⋯ ⋱ ⋯
0⎤ 0⎥ ⎥ ⋮⎥ . 0⎥ ⎥ ⋮⎥ λn ⎦ ⎥
4. Case studies In this section, we test the performance of the proposed decoupled control in enhancing the small-signal stability of a power system with two different active power actuators, the PSS and HVDC transmission. The test system is a modified MinniWECC system [26]. The MinniWECC test system, as shown in Fig. 2, is a reduced-order dynamic model of the Western Electricity Coordination Council (WECC) system in North America. It has 34 generators, and only three of them (Gen 13, 14, and 34) are equipped with PSSs. It also has 124 buses and 173 lines, with a HVDC line called the Pacific Direct Current Intertie (PDCI). The PDCI is a 3100 MW, ± 500 kV rated HVDC line that connects Bus 24 (in Oregon) and Bus 49 (in California). Most importantly, this test system contains seven oscillation modes shown in Table 1. The Alberta mode with 2.8% damping ratio, and British Columbia (BC) mode with 1.0% damping ratio are the most frequently observed dominant modes in the MinniWECC system. The MinniWECC system represents the overall inter-area modal properties and complexity of the full-size system for accurate evaluation and testing of potential system-wide damping control technologies. The simulation is conducted using the opensource Power System Toolbox [30].
(6)
It can be seen from (6) that the feedback control (5) based on the pure modal signal Yi(s) affects the ith mode λi only. In contrast, if the feedback control input contains two or more oscillation modal signals, more than one eigenvalue will be affected [27]. Therefore, if we would like to improve the damping of a specific oscillation mode without unnecessary excitation of other weakly damped modes, it is necessary to extract the corresponding pure oscillation modal signal as the feedback signal for decoupled oscillation damping. Note that the controllability of the oscillation mode would affect the proposed POD approach; thus, the power actuators should be placed wisely with adequate controllability of the concerned modal signals [28]. 3. Wide-area measurement-based modal decomposition In this paper, we propose a novel approach for extracting oscillation modal signals from real-time wide-area measurements at different locations to implement decoupled oscillation damping. The increasingly observable system information (bus voltage magnitudes, angles, and power flows) provides new options for designing POD controllers for specifically targeted oscillation modes. When power oscillation occurs in the power system, it is common that not all the oscillation modes are excited when the oscillation occurs. Those modes that are triggered are usually referred to as dominant modes. Assume that there are q (q < n/2) dominant modes. Let oj(t) = eαjt cos(ωjt + ϕj), j = 1, …, q, denote the time-domain pure oscillation modal signals corresponding to those dominant modes, where αj is the damping ratio, ωj is the angular oscillation frequency and ϕj is the phase shift. Let fi(t), i = 1, …, m, represent the ith wide-area measurement, where we assume that there are m measurements in the system and it is necessary that m ≥ q. Then, the wide-area measurements during oscillation can be approximately represented as the linear combination of those dominant oscillation modal signals, that is, q
fi (t ) ≈
First, we would like to show the results of using PSS with the proposed decoupled control to damp oscillations. In this case, the PSS on Gen 34 (Bus 118) is selected to damp the Alberta mode of 0.292 Hz (pure Alberta modal single is extracted for decoupled control). The conventional control approach has been well tuned following the method in [28]. At time t = 1.0 s, a pulse perturbation is added to the mechanical power of Gen 10 to simulate a small disturbance. The results of bus voltage oscillation and generator rotor speed deviation are shown in Fig. 3. It is noted that the oscillation magnitude is much smaller when the decoupled control is implemented. Therefore, the proposed decoupled control can successfully increase the inter-area oscillation damping effect. To explore which oscillation mode is exactly further damped, a spectrum analysis of the oscillation measurements with the two different control options (conventional and decoupled) is performed using the fast Fourier transform (FFT) [31]. The results are shown in Fig. 4. The Alberta mode and BC mode are the most dominant oscillation modes. When decoupled control is applied to damp the Alberta mode, only that specific mode is damped, and the BC mode is not aggravated. Therefore, the decoupled control method precisely damps the targeted mode without changing the other modes.
q
∑ cij e αj t cos(ωj t + ϕj) = ∑ cij oj (t ), j=1
4.1. Using PSS
j=1
(7)
where cij is the amplitude of the jth dominant oscillation modal signal in the ith PMU measurement. For simplicity, (7) can be rewritten in the following vector form:
f (t ) = Co (t ),
4.2. Using HVDC transmission The proposed decoupled control can be also implemented on HVDC transmission for POD control, and it can work along with the conventional POD schemes to further damp inter-area oscillations. In this paper, a conventional POD method uses the terminal frequency difference as a negative feedback signal to modulate the real power change over the HVDC line [9], as shown in Fig. 5. The frequency difference is obtained from passing signals of electrical angle differences from PMUs through a derivative filter. The feedback signal is defined as
(8)
f(t ) = [f1 (t ) … fm (t )]⊤ ∈ ℝm , C = [cij] ∈ ℝm × q , and where ⊤ q o(t ) = [o1 (t ) … oq (t )] ∈ ℝ . The flowchart of the proposed wide-area measurement-based modal decomposition is shown in Fig. 1. Given the latest-historical PMU smalldisturbance measurements, we can apply Prony's algorithm [29] to determine the coefficient matrix C offline. When m ≥ q and C has a left inverse denoted by C−L, we can save the coefficient matrix C for future real-time application. When real-time disturbance occurs, we can extract the dominant oscillation modal signals o(t) from the PMU measurements by the following:
PΔDC = −K (frec − finv ),
(9)
where frec and finv are the frequencies on the HVDC rectifier and inverter side, K is the modulation gain, and PΔDC is the modulation (change) of real power over HVDC lines. The decoupled control serves as a complement to the conventional POD, by adding the pure oscillation modal signal to conventional feedback signals. In this case, the BC mode of 0.63 Hz is the targeted oscillation mode. Specifically, at time 1.0 s, a small pulse perturbation is added to the mechanical power of Machine 10 at Bus 24. Oscillations are triggered by this small transient event. Fig. 6 shows the extracted
o (t ) = C−Lf (t ). Remark 1. Note that the coefficient matrix C is the key for pure modal signal extraction, and it depends on the system topology and operating point. Therefore, it is recommended that C be determined or updated using the latest-historical PMU small-disturbance measurements. 3
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Fig. 1. Flowchart of the proposed decoupled damping control.
Fig. 2. Modified MinniWECC system.
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Table 1 Inter-area oscillation modes of the modified MinniWECC system. Mode
Frequency (Hz)
Damping (%)
N-S Alberta E-W south 1 Montana BC E-W south 2 Middle
0.134 0.292 0.529 0.548 0.632 0.694 0.714
35.8 2.8 8.6 5.4 1.0 5.3 6.1
oscillation signal using the proposed method, it is found the decoupled method can accurately separate the concerned BC mode, as the extracted signal is the very close to the pure mode. Three control options (no control, conventional control, and conventional plus decoupled control) using the PDCI are used to damp the inter-area oscillations. Here the parameters for conventional controller are fine-tuned and the feedback gain is K = 1500. The results of HVDC damping control are shown in Fig. 7. Fig. 7(a) shows the oscillation observed on the power flow in the AC transmission line from Bus 89 to Bus 38, which is the tie-line that parallels the PDCI; Fig. 7(b) shows the oscillation observed in voltage magnitude at Bus 24. As seen in the figure, the conventional POD controller increases the damping of the system compared to the base case where no POD controller exists. With the supplementary decoupled control approach, it provides even better damping than the conventional controller only. The oscillation magnitudes are dramatically decreased and the system returns to steady state much faster. A spectrum analysis is also performed using the FFT method on the oscillation measurements for the different control options. The results are shown in Fig. 8. Clearly, the Alberta mode and BC mode are the most dominant ones among the oscillation modes. Compared with the base case, the conventional POD controller increases the overall damping of all oscillation modes. When the decoupled controller is also applied, only the BC mode is further precisely damped and the other modes remain almost unchanged (compared to the conventional method). The results validate that only the corresponding decoupled eigenvalue is affected by the decoupled control approach, which complements the conventional POD method by further improving the
Fig. 4. Spectrum analysis of oscillation measurements associated with PSS case.
damping performance. 5. Discussion The case studies in the previous section demonstrate the advantage of using the proposed decoupled control to improve power system small signal stability, where pure modal signals can be extracted using the C matrix. However, the C matrix depends on the system topology and operating point. If large transient events occur we cannot linearize the system to get a stationary C matrix. In this section, we propose to test the feasibility of applying the proposed decoupled approach on large transients with the saved C matrix calculated using small-disturbance measurements. To test the performance of proposed approach, we have designed four typical large transient events that might happen in the system, and the results are presented as follows. 5.1. Large transient case one: three-phase fault In this case, a three-phase fault occurs on Line 86-87 (in parallel with the PDCI) at time 1.0 s. The faulted line is tripped two and a half cycles later, causing large transients in the system as well as a topology change. The PDCI and the three control options are used to damp the inter-area oscillations. The results of machine speed deviation, Bus 24
Fig. 3. Oscillations from small-signal stability studies using PSS. 5
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Fig. 5. Decoupled control work along with conventional HVDC POD control.
Fig. 6. Extracted BC mode using the proposed method.
Fig. 7. Oscillations from small-signal stability studies using HVDC.
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5.3. Large transient case three: switching, line loss In this case, one major transmission path Line 90-44, which carries 1970 MW real power, is tripped at time 1.0 s because of an open switching event. This loss of line changes the system topology and causes large transients in the system. The PDCI and the three control options are used to damp the inter-area oscillations. The results of machine speed deviation, Bus 24 voltage oscillation, and frequency difference of north-south areas are shown in Fig. 11. The results are again similar to small-signal stability studies: the conventional POD controller increases the damping of the system compared to the baseline, while with the supplementary decoupled control approach it provides even better damping performance. 5.4. Large transient case four: loss of large load Fig. 8. Spectrum analysis of oscillation measurements associated with HVDC case.
In this case, one very large load on Bus 29 is suddenly tripped at time 1.0 s, losing 1570 MW load on that bus. This loss of large load changes the system topology and causes large transients in the system. The PDCI and the three control options are used to damp the inter-area oscillations. The results of machine speed deviation, Bus 24 voltage oscillation, and frequency difference between north and south areas are shown in Fig. 12. The results are again similar to small-signal stability studies: the conventional POD controller increases the damping of the system compared to the baseline, while with the supplementary decoupled control approach it provides even better damping performance.
voltage oscillation, and frequency difference of north-south areas are shown in Fig. 9. The results are similar to small-signal stability studies: the conventional POD controller increases the damping of the system compared to the baseline, while with the supplementary decoupled control approach it provides even better damping performance.
5.2. Large transient case two: generator trip In this case, Gen 11 on Bus 25 is suddenly tripped at time 1.0 s, losing 2150 MW generation. This sudden generator trip changes the system topology and causes large transients in the system. The PDCI and the three control options are used to damp the inter-area oscillations. The results of machine speed deviation, Bus 24 voltage oscillation, and frequency difference of north-south areas are shown in Fig. 10. The results are again similar to small-signal stability studies: the conventional POD controller increases the damping of the system compared to the baseline, while with the supplementary decoupled control approach it provides even better damping performance.
5.5. Discussion summary The four large transient events have something in common: the conventional POD controller can damp the inter-area oscillations quite well, while the best performance is achieved when it works along with the proposed decoupled control approach. To have a better view of how much the decoupled control contributes to oscillation damping, we used the Oscillation Baselining and Analysis Tool (OBAT) [32] developed at Pacific Northwest National Laboratory to examine the damping ratios of dominant modes in each large transient event. The results are summarized in Table 2. The dominant modes for baseline oscillations are still the Alberta and BC mode. The conventional POD control
Fig. 9. Results of large transient case one: three-phase fault. 7
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Fig. 10. Results of large transient case two: generator trip.
Fig. 11. Results of large transient case three: switching, line loss.
6. Conclusion and future work
sufficiently damps the Alberta modes, and slightly improves the damping ratio of the BC mode. After decoupled control is also implemented, the BC mode is much further damped. Therefore, the proposed decoupled control enhances the transient stability (precisely on the BC mode) in the typical four large transient events, where the C matrix might be changed. We could not guarantee the performance of decoupled control on all other large transient events, but at least we are sure that it works well for small-signal transients as well as the above four typical large transient events in the system.
A decoupled control strategy has been proposed to damp inter-area oscillations in power systems. The extracted pure oscillation mode is leveraged in the feedback control of PSS, HVDC, or other actuators that help damp a specific mode without affecting other modes. This decoupled approach can be implemented independently or along with other POD controllers. The decoupled control strategy is powerful for providing precise damping of targeted inter-area oscillation modes without deteriorating other modes unnecessarily. Extensive results of studies of both small-signal transients and four typical large transients have demonstrated the excellent performance of the proposed 8
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Fig. 12. Results of large transient case four: loss of large load. Table 2 Oscillation analysis of large-transient events. Event
Alberta damping ratio (%)
BC damping ratio (%)
Base
Conv.
Conv. + Decouple
Base
Conv.
Conv. + Decouple
2.8 2.7 2.7 3.0
> 10.0 > 10.0 > 10.0 > 10.0
> 10.0 > 10.0 > 10.0 > 10.0
0.9 0.9 1.0 1.1
2.1 1.8 2.2 1.9
3.9 4.2 4.3 3.1
[9]
[10] One Two Three Four
[11]
[12]
decoupled approach. In future work, we plan to study how to efficiently update the modal decomposition matrix. We also plan to investigate the impact of communication delay and measurement noises on the proposed decoupled POD controller and design compensations for these network effects.
[13]
[14] [15]
Conflict of interest [16]
None declared.
[17]
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Oscillation damping control using multiple high voltage dc transmission lines: controllability exploration, 2018 IEEE/PES Transmission and Distribution Conference and Exposition (T&D), IEEE, 2018, pp. 1–9. D. Trudnowski, D. Kosterev, J. Undrill, PDCI damping control analysis for the western north American power system, Power and Energy Society General Meeting (PES), 2013 IEEE, IEEE, 2013, pp. 1–5. J. Lian, Q. Zhang, L.D. Marinovici, R. Fan, J. Hansen, Wide-area demand-side control for inter-area oscillation mitigation in power systems, 2018 IEEE/PES Transmission and Distribution Conference and Exposition (T&D), IEEE, 2018, https://doi.org/10.1109/tdc.2018.8440257. M. Khaji, M. Aghamohammadi, Online emergency damping controller to suppress power system inter-area oscillation using load-generation tripping, Electr. Power Syst. Res. 140 (2016) 806–820. Y. Li, C. Rehtanz, S. Ruberg, L. Luo, Y. Cao, Wide-area robust coordination approach of HVDC and facts controllers for damping multiple interarea oscillations, IEEE Trans. Power Deliv. 27 (3) (2012) 1096–1105. R. Fan, L. Sun, Z. Tan, Linear quadratic control of SSSC to increase power oscillations damping of HVDC-ac power system, Power & Energy Society General Meeting, 2015 IEEE, IEEE, 2015, pp. 1–5. N. Khatoon, S. Shaik, A survey on different types of flexible ac transmission systems (facts) controllers, Int. J. Eng. Dev. Res. 4 (5) (2017) 796–814. D. Roberson, J.F. OBrien, Loop shaping of a wide-area damping controller using HVDC, IEEE Trans. Power Syst. 32 (3) (2017) 2354–2361. R. Eriksson, A new control structure for multiterminal dc grids to damp interarea oscillations, IEEE Trans. Power Deliv. 31 (3) (2016) 990–998. Y. Zhang, A. Bose, Design of wide-area damping controllers for interarea oscillations, IEEE Trans. Power Syst. 23 (3) (2008) 1136–1143, https://doi.org/10.1109/ tpwrs.2008.926718. K. Zhou, J.C. Doyle, K. Glover, et al., Robust and Optimal Control, vol. 40, Prentice Hall, New Jersey, 1996. S.P. Azad, R. Iravani, J.E. Tate, Damping inter-area oscillations based on a model predictive control (MPC) HVDC supplementary controller, IEEE Trans. Power Syst. 28 (3) (2013) 3174–3183. W.-H. Chen, D.J. Ballance, J. O’Reilly, Model predictive control of nonlinear systems: computational burden and stability, IEE Proc.-Control Theory Appl. 147 (4) (2000) 387–394. W. Du, Q. Fu, H. Wang, Strong dynamic interactions between multi-terminal DC network and AC power systems caused by open-loop modal coupling, IET Gen. Transm. Distrib. 11 (9) (2017) 2362–2374, https://doi.org/10.1049/iet-gtd.2016. 1920. Y. Li, Z. Xu, J. Ostergaard, D.J. Hill, Coordinated control strategies for offshore wind farm integration via VSC-HVDC for system frequency support, IEEE Trans. Energy Convers. 32 (3) (2017) 843–856, https://doi.org/10.1109/tec.2017. 2663664. R. Preece, J.V. Milanović, A.M. Almutairi, O. Marjanovic, Damping of inter-area oscillations in mixed ac/dc networks using WAMS based supplementary controller, IEEE Trans. Power Syst. 28 (2) (2012) 1160–1169. J. Björk, K.H. Johansson, L. Harnefors, R. Eriksson, Analysis of Coordinated HVDC Control for Power Oscillation Damping, (2018). J. Zhang, C. Chung, Y. Han, A novel modal decomposition control and its application to PSS design for damping interarea oscillations in power systems, IEEE
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Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Wide-area measurement-based modal decoupling for power system oscillation damping
T
⁎
Rui Fan, Shaobu Wang, Renke Huang, Jianming Lian , Zhenyu Huang Pacific Northwest National Laboratory, 902 Battelle Blvd, Richland, WA 99354, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Decoupled control Power system stabilizer High-voltage direct current Power oscillation damping Small-signal stability
Inter-area oscillation modes usually fall within the same frequency range of 0.1–1.0 Hz; thus, power oscillation damping (POD) control methods should avoid unnecessary excitation of weakly damped modes. This paper presents a decoupled control strategy to damp inter-area oscillations in the power grid. Pure oscillation mode signals are extracted from wide-area measurements and used as feedback input for POD actuators such as power system stabilizers or high-voltage direct current transmission. The decoupled control strategy is able to increase the damping of the concerned oscillation modes without deteriorating others. This proposed strategy could be implemented independently, or serve as a complement and work along with conventional POD controllers to damp concerned inter-area modes. The proposed method has been tested on a modified MinniWECC system. Extensive results indicate that inter-area oscillations can be successfully and precisely damped using the proposed decoupled strategy.
1. Introduction Inter-area oscillation is one of the major concerns in modern largescale power systems. It is the interaction among different groups of machines that are connected by relatively weak transmissions [1]. The inter-area oscillations of typically 0.1–1.0 Hz potentially lead to unstable system operations or even grid blackouts. With the expansion of modern power systems and increase in area interconnections, inter-area oscillations have become a key challenge for large-scale interconnected grids [2]. Various techniques have been developed to study and damp interarea oscillations. A power system stabilizer (PSS) is a popular device that damps generator rotor angle swinging in a broad range of frequencies in the power system [3]. It adds damping to the rotor oscillations of the synchronous machine by providing supplementary control action through the generator excitation systems. The most popular power oscillation damping (POD) method using PSS is the linearizationbased eigenvalue analysis scheme [4–6]. This scheme performs an eigenvalue calculation on the basis of a state-space representation of the system that is linearized at a given operating point. With the expansion of the high-voltage DC (HVDC) transmission system, people have started to leverage its capability in modulating active power for interarea oscillation damping [7]. The transmitted power can be rapidly controlled through HVDC converters by changing the firing and
⁎
extinction angles. The most popular HVDC-based POD scheme uses the terminal frequency difference as feedback inputs to the converter modulation signals [8,9]. Except for PSS and HVDC, other actuators such as flexible AC transmission system (FACTS) and accumulated demand modulation have been also used for oscillation damping [10,11]. With the invention and wide-spread implementation of phasor measurement units (PMUs) in power systems, many advanced POD schemes have been designed for better performance, with a requirement of fast communication and accurate information exchanges. Various FACTS devices have been proposed to work with HVDC or PSS for better damping performance [12,13]. FACTS devices provide new options for damping oscillations, but are commonly accompanied by complex communication protocols, high maintenance costs, and other substantial burdens [14]. Loop-shaping and pole-placement methods have also been used to control HVDC or PSS and damp inter-area oscillations by bending the Nyquist curve away from the critical point [15–17]. In [15], a feedback compensator was designed using the loopshaping method to shape an HVDC loop transfer function for better performance. In [17], a linear matrix inequality based mixed outputfeedback control with regional pole placement was used to damp multiple oscillation modes. Carefully designed controllers using loopshaping theory had been found to offer several advantages, including more feedback in the system-dominant mode, the existence of actuator effort even on high-frequency content in the error signal, and the
Corresponding author. E-mail address: [email protected] (J. Lian).
https://doi.org/10.1016/j.epsr.2019.106022 Received 2 February 2019; Received in revised form 31 August 2019; Accepted 1 September 2019 Available online 25 September 2019 0378-7796/
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Section 4, case studies on a modified MinniWECC test system are presented. Section 5 presents a discussion of the case studies, and Section 6 presents conclusions.
suitability of the controller for the wider controller frequency bandwidth of the plant. However, the possibility of cancellation of the right half plane poles and zeros in the loop transfer functions might cause the system to be unstable [18]. Other advanced control methods, such as model predictive control (MPC), were also used to damp power system inter-area oscillations. MPC-based methods were well recognized and widely used in the process industry and in multiple research areas because of their advantages over conventional approaches [19]. The MPCbased method anticipated future events and modulated the control signal to minimize the oscillations, while respecting the HVDC constraints. However, the high computational burden and difficulties in application could result in unsuitable control schemes [20]. One challenge for inter-area oscillation damping is that oscillation modes usually fall in the same frequency ranges and they are physically coupled through system interaction. It is possible that a POD controller would damp certain modes while exciting others unnecessarily [21]. In [22], energy storage and large capacitor banks were used to mitigate the modal interactions, with an additional cost for the DC storage devices. Preece et al. [23] proposed a modal linear quadratic Gaussian (MLQG) control to damp targeted oscillation modes and achieved some good results. This approach requires the system model to be linear/ quadratic; however, the power system is naturally nonlinear. In addition, this approach requires detailed information about the system model, which could be difficult in the real world, especially when the system operation conditions change from time to time. Bjork et al. [24] proposed a solution that eliminated expensive storage devices, but it required two or more HVDC links. Zhang et al. [25] proposed a PSSbased modal decomposition approach by filtering mixed oscillation signals through a band-pass filter, but a few problems will need to be further addressed. Because the signal fed to the band-pass filter is from only one location in [25], the concerned mode may not be adequately observed. Another issue is that the frequencies of oscillation modes are so close (0.1–1.0 Hz) that it is very difficult to distinguish the oscillations through a band-pass filter. The performance of the band-pass filter depends on the input signals, thus, troublesome phase shift or unfiltered components would occur since the oscillation signals are time-variant and their frequency difference is too small. To resolve the above modal coupling challenge without an investment in additional storage devices or DC lines, this paper proposes a novel decoupled POD control strategy that can be implemented on either PSS or HVDC. The proposed method takes advantage of the development of the wide-area measurement system. The increasingly observable system information (bus voltage magnitudes, angles, and power flows) provides new options of designing POD controllers for specifically targeted oscillation modes. It is possible now to use the abundant wide-area data to extract the targeted pure oscillation modes using a decomposed matrix method. The extracted pure modal signal is further applied as a feedback signal to modulate the PSS output or the transferred power on HVDC lines. Because the feedback signal only relates to the corresponding system eigenvalue, it will have minimum negative impacts on the damping of other modes. The contribution of this paper comes from: (1) a novel modal decoupling approach and an efficient, high-quality pure mode extraction method from multi-location synchronized PMU data; (2) a novel decoupled POD approach that could damp a specific inter-area oscillation mode in a complex power system without affecting other oscillation modes. This decoupled strategy could be implemented independently to damp the concerned inter-area mode, or serve as complement and work along with conventional POD controllers for further improvements on system transient stability. The performance of the proposed method was tested and validated in a modified MinniWECC system [26]. Extensive results indicate that inter-area oscillations can be successfully and precisely damped using the proposed decoupled strategy. This paper is organized as follows. In Section 2, a pure mode signal as POD controller input is briefly introduced. In Section 3, the widearea measurement-based modal decomposition method is discussed. In
2. Pure mode signal as input of POD controller In this section, the underlying principles of using pure mode signal as POD controller input are briefly introduced. To proceed, consider the following single-input single-output system for oscillation damping,
⎧ x˙ (t ) = Ax (t ) + bu (t ), ⎨ ⎩ y (t ) = cx (t ),
(1)
where x ∈ ℝn is the state vector of the system, A ∈ ℝn × n is the system transition matrix, and u ∈ ℝ, y ∈ ℝ , b ∈ ℝn and c ∈ ℝ1 × n are the input, output, and input and output matrices, respectively. Applying the Laplace transform to (1) with zero initial conditions, i.e., x(0) = 0, we have
⎧ s X (s ) = AX (s ) + bU (s ), ⎨ ⎩Y (s ) = cX (s ).
(2)
For simplicity of the following derivation, we assume that A has n distinct eigenvalues; that is, A is diagonalizable. Let λi ∈ ℂ denote the ith eigenvalue of A, and let vi ∈ ℂn and wi ∈ ℂ1 × n denote the right and left eigenvectors associated with λi, respectively. Define T = [v1 ⋯ vn] and Λ = diag(λ1, …, λn). Then, it follows that
T−1AT = Λ, and
⎡ w1* ⎤ T−1 = ⎢ ⋮ ⎥, ⎢ w*⎥ ⎣ n⎦ where w i* denotes the complex conjugate of wi. To proceed, define the new state vector Z ∈ ℝn × n as
Z (s ) = T−1X (s ),
(3)
whose elements Zi(s), i = 1, …, n, are the pure modal signals. Substituting (3) into (2), it follows that −1 ⎧ s Z (s ) = ΛZ (s ) + T bU (s ), ⎨ ⎩Y (s ) = cTZ (s ).
(4)
It follows from (4) that Y(s) can be expressed as a combination of different modal signals, n
Y (s ) = c ∑ vi Zi (s ) = i=1
n
∑ Yi (s ), i=1
where Yi(s) = cviZi(s). Now assume that Yi(s) (1 ≤ i ≤ n) can be perfectly extracted from Y(s) and is used as the feedback signal for the controller, that is,
U (s ) = P (s ) Yi (s ) = P (s ) cviZi (s ),
(5)
where P(s) represents the controller transfer function. Substituting (5) into (4), it follows that
s Z (s ) = ΛZ (s ) + T−1bU (s ) = ΛZ (s ) + T−1bP (s ) cviZi (s ) = ΛZ (s ) + T−1b [0 ⋯ P (s ) cvi ⋯ 0] Z (s ) = A c Z (s ), where 2
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⎡ λ1 ⎢0 ⎢ ⋮ Ac = ⎢ ⎢0 ⎢⋮ ⎢ ⎢0 ⎣
0 λ2 ⋮ 0 ⋮ 0
w1*bP (s ) cvi ⋯ w *2 bP (s ) cvi ⋯ ⋱ ⋮ ⋯ λi + w i*bP (s ) cvi ⋱ ⋮ w n*bP (s ) cvi ⋯
⋯ ⋯ ⋱ ⋯ ⋱ ⋯
0⎤ 0⎥ ⎥ ⋮⎥ . 0⎥ ⎥ ⋮⎥ λn ⎦ ⎥
4. Case studies In this section, we test the performance of the proposed decoupled control in enhancing the small-signal stability of a power system with two different active power actuators, the PSS and HVDC transmission. The test system is a modified MinniWECC system [26]. The MinniWECC test system, as shown in Fig. 2, is a reduced-order dynamic model of the Western Electricity Coordination Council (WECC) system in North America. It has 34 generators, and only three of them (Gen 13, 14, and 34) are equipped with PSSs. It also has 124 buses and 173 lines, with a HVDC line called the Pacific Direct Current Intertie (PDCI). The PDCI is a 3100 MW, ± 500 kV rated HVDC line that connects Bus 24 (in Oregon) and Bus 49 (in California). Most importantly, this test system contains seven oscillation modes shown in Table 1. The Alberta mode with 2.8% damping ratio, and British Columbia (BC) mode with 1.0% damping ratio are the most frequently observed dominant modes in the MinniWECC system. The MinniWECC system represents the overall inter-area modal properties and complexity of the full-size system for accurate evaluation and testing of potential system-wide damping control technologies. The simulation is conducted using the opensource Power System Toolbox [30].
(6)
It can be seen from (6) that the feedback control (5) based on the pure modal signal Yi(s) affects the ith mode λi only. In contrast, if the feedback control input contains two or more oscillation modal signals, more than one eigenvalue will be affected [27]. Therefore, if we would like to improve the damping of a specific oscillation mode without unnecessary excitation of other weakly damped modes, it is necessary to extract the corresponding pure oscillation modal signal as the feedback signal for decoupled oscillation damping. Note that the controllability of the oscillation mode would affect the proposed POD approach; thus, the power actuators should be placed wisely with adequate controllability of the concerned modal signals [28]. 3. Wide-area measurement-based modal decomposition In this paper, we propose a novel approach for extracting oscillation modal signals from real-time wide-area measurements at different locations to implement decoupled oscillation damping. The increasingly observable system information (bus voltage magnitudes, angles, and power flows) provides new options for designing POD controllers for specifically targeted oscillation modes. When power oscillation occurs in the power system, it is common that not all the oscillation modes are excited when the oscillation occurs. Those modes that are triggered are usually referred to as dominant modes. Assume that there are q (q < n/2) dominant modes. Let oj(t) = eαjt cos(ωjt + ϕj), j = 1, …, q, denote the time-domain pure oscillation modal signals corresponding to those dominant modes, where αj is the damping ratio, ωj is the angular oscillation frequency and ϕj is the phase shift. Let fi(t), i = 1, …, m, represent the ith wide-area measurement, where we assume that there are m measurements in the system and it is necessary that m ≥ q. Then, the wide-area measurements during oscillation can be approximately represented as the linear combination of those dominant oscillation modal signals, that is, q
fi (t ) ≈
First, we would like to show the results of using PSS with the proposed decoupled control to damp oscillations. In this case, the PSS on Gen 34 (Bus 118) is selected to damp the Alberta mode of 0.292 Hz (pure Alberta modal single is extracted for decoupled control). The conventional control approach has been well tuned following the method in [28]. At time t = 1.0 s, a pulse perturbation is added to the mechanical power of Gen 10 to simulate a small disturbance. The results of bus voltage oscillation and generator rotor speed deviation are shown in Fig. 3. It is noted that the oscillation magnitude is much smaller when the decoupled control is implemented. Therefore, the proposed decoupled control can successfully increase the inter-area oscillation damping effect. To explore which oscillation mode is exactly further damped, a spectrum analysis of the oscillation measurements with the two different control options (conventional and decoupled) is performed using the fast Fourier transform (FFT) [31]. The results are shown in Fig. 4. The Alberta mode and BC mode are the most dominant oscillation modes. When decoupled control is applied to damp the Alberta mode, only that specific mode is damped, and the BC mode is not aggravated. Therefore, the decoupled control method precisely damps the targeted mode without changing the other modes.
q
∑ cij e αj t cos(ωj t + ϕj) = ∑ cij oj (t ), j=1
4.1. Using PSS
j=1
(7)
where cij is the amplitude of the jth dominant oscillation modal signal in the ith PMU measurement. For simplicity, (7) can be rewritten in the following vector form:
f (t ) = Co (t ),
4.2. Using HVDC transmission The proposed decoupled control can be also implemented on HVDC transmission for POD control, and it can work along with the conventional POD schemes to further damp inter-area oscillations. In this paper, a conventional POD method uses the terminal frequency difference as a negative feedback signal to modulate the real power change over the HVDC line [9], as shown in Fig. 5. The frequency difference is obtained from passing signals of electrical angle differences from PMUs through a derivative filter. The feedback signal is defined as
(8)
f(t ) = [f1 (t ) … fm (t )]⊤ ∈ ℝm , C = [cij] ∈ ℝm × q , and where ⊤ q o(t ) = [o1 (t ) … oq (t )] ∈ ℝ . The flowchart of the proposed wide-area measurement-based modal decomposition is shown in Fig. 1. Given the latest-historical PMU smalldisturbance measurements, we can apply Prony's algorithm [29] to determine the coefficient matrix C offline. When m ≥ q and C has a left inverse denoted by C−L, we can save the coefficient matrix C for future real-time application. When real-time disturbance occurs, we can extract the dominant oscillation modal signals o(t) from the PMU measurements by the following:
PΔDC = −K (frec − finv ),
(9)
where frec and finv are the frequencies on the HVDC rectifier and inverter side, K is the modulation gain, and PΔDC is the modulation (change) of real power over HVDC lines. The decoupled control serves as a complement to the conventional POD, by adding the pure oscillation modal signal to conventional feedback signals. In this case, the BC mode of 0.63 Hz is the targeted oscillation mode. Specifically, at time 1.0 s, a small pulse perturbation is added to the mechanical power of Machine 10 at Bus 24. Oscillations are triggered by this small transient event. Fig. 6 shows the extracted
o (t ) = C−Lf (t ). Remark 1. Note that the coefficient matrix C is the key for pure modal signal extraction, and it depends on the system topology and operating point. Therefore, it is recommended that C be determined or updated using the latest-historical PMU small-disturbance measurements. 3
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Fig. 1. Flowchart of the proposed decoupled damping control.
Fig. 2. Modified MinniWECC system.
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Table 1 Inter-area oscillation modes of the modified MinniWECC system. Mode
Frequency (Hz)
Damping (%)
N-S Alberta E-W south 1 Montana BC E-W south 2 Middle
0.134 0.292 0.529 0.548 0.632 0.694 0.714
35.8 2.8 8.6 5.4 1.0 5.3 6.1
oscillation signal using the proposed method, it is found the decoupled method can accurately separate the concerned BC mode, as the extracted signal is the very close to the pure mode. Three control options (no control, conventional control, and conventional plus decoupled control) using the PDCI are used to damp the inter-area oscillations. Here the parameters for conventional controller are fine-tuned and the feedback gain is K = 1500. The results of HVDC damping control are shown in Fig. 7. Fig. 7(a) shows the oscillation observed on the power flow in the AC transmission line from Bus 89 to Bus 38, which is the tie-line that parallels the PDCI; Fig. 7(b) shows the oscillation observed in voltage magnitude at Bus 24. As seen in the figure, the conventional POD controller increases the damping of the system compared to the base case where no POD controller exists. With the supplementary decoupled control approach, it provides even better damping than the conventional controller only. The oscillation magnitudes are dramatically decreased and the system returns to steady state much faster. A spectrum analysis is also performed using the FFT method on the oscillation measurements for the different control options. The results are shown in Fig. 8. Clearly, the Alberta mode and BC mode are the most dominant ones among the oscillation modes. Compared with the base case, the conventional POD controller increases the overall damping of all oscillation modes. When the decoupled controller is also applied, only the BC mode is further precisely damped and the other modes remain almost unchanged (compared to the conventional method). The results validate that only the corresponding decoupled eigenvalue is affected by the decoupled control approach, which complements the conventional POD method by further improving the
Fig. 4. Spectrum analysis of oscillation measurements associated with PSS case.
damping performance. 5. Discussion The case studies in the previous section demonstrate the advantage of using the proposed decoupled control to improve power system small signal stability, where pure modal signals can be extracted using the C matrix. However, the C matrix depends on the system topology and operating point. If large transient events occur we cannot linearize the system to get a stationary C matrix. In this section, we propose to test the feasibility of applying the proposed decoupled approach on large transients with the saved C matrix calculated using small-disturbance measurements. To test the performance of proposed approach, we have designed four typical large transient events that might happen in the system, and the results are presented as follows. 5.1. Large transient case one: three-phase fault In this case, a three-phase fault occurs on Line 86-87 (in parallel with the PDCI) at time 1.0 s. The faulted line is tripped two and a half cycles later, causing large transients in the system as well as a topology change. The PDCI and the three control options are used to damp the inter-area oscillations. The results of machine speed deviation, Bus 24
Fig. 3. Oscillations from small-signal stability studies using PSS. 5
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Fig. 5. Decoupled control work along with conventional HVDC POD control.
Fig. 6. Extracted BC mode using the proposed method.
Fig. 7. Oscillations from small-signal stability studies using HVDC.
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5.3. Large transient case three: switching, line loss In this case, one major transmission path Line 90-44, which carries 1970 MW real power, is tripped at time 1.0 s because of an open switching event. This loss of line changes the system topology and causes large transients in the system. The PDCI and the three control options are used to damp the inter-area oscillations. The results of machine speed deviation, Bus 24 voltage oscillation, and frequency difference of north-south areas are shown in Fig. 11. The results are again similar to small-signal stability studies: the conventional POD controller increases the damping of the system compared to the baseline, while with the supplementary decoupled control approach it provides even better damping performance. 5.4. Large transient case four: loss of large load Fig. 8. Spectrum analysis of oscillation measurements associated with HVDC case.
In this case, one very large load on Bus 29 is suddenly tripped at time 1.0 s, losing 1570 MW load on that bus. This loss of large load changes the system topology and causes large transients in the system. The PDCI and the three control options are used to damp the inter-area oscillations. The results of machine speed deviation, Bus 24 voltage oscillation, and frequency difference between north and south areas are shown in Fig. 12. The results are again similar to small-signal stability studies: the conventional POD controller increases the damping of the system compared to the baseline, while with the supplementary decoupled control approach it provides even better damping performance.
voltage oscillation, and frequency difference of north-south areas are shown in Fig. 9. The results are similar to small-signal stability studies: the conventional POD controller increases the damping of the system compared to the baseline, while with the supplementary decoupled control approach it provides even better damping performance.
5.2. Large transient case two: generator trip In this case, Gen 11 on Bus 25 is suddenly tripped at time 1.0 s, losing 2150 MW generation. This sudden generator trip changes the system topology and causes large transients in the system. The PDCI and the three control options are used to damp the inter-area oscillations. The results of machine speed deviation, Bus 24 voltage oscillation, and frequency difference of north-south areas are shown in Fig. 10. The results are again similar to small-signal stability studies: the conventional POD controller increases the damping of the system compared to the baseline, while with the supplementary decoupled control approach it provides even better damping performance.
5.5. Discussion summary The four large transient events have something in common: the conventional POD controller can damp the inter-area oscillations quite well, while the best performance is achieved when it works along with the proposed decoupled control approach. To have a better view of how much the decoupled control contributes to oscillation damping, we used the Oscillation Baselining and Analysis Tool (OBAT) [32] developed at Pacific Northwest National Laboratory to examine the damping ratios of dominant modes in each large transient event. The results are summarized in Table 2. The dominant modes for baseline oscillations are still the Alberta and BC mode. The conventional POD control
Fig. 9. Results of large transient case one: three-phase fault. 7
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Fig. 10. Results of large transient case two: generator trip.
Fig. 11. Results of large transient case three: switching, line loss.
6. Conclusion and future work
sufficiently damps the Alberta modes, and slightly improves the damping ratio of the BC mode. After decoupled control is also implemented, the BC mode is much further damped. Therefore, the proposed decoupled control enhances the transient stability (precisely on the BC mode) in the typical four large transient events, where the C matrix might be changed. We could not guarantee the performance of decoupled control on all other large transient events, but at least we are sure that it works well for small-signal transients as well as the above four typical large transient events in the system.
A decoupled control strategy has been proposed to damp inter-area oscillations in power systems. The extracted pure oscillation mode is leveraged in the feedback control of PSS, HVDC, or other actuators that help damp a specific mode without affecting other modes. This decoupled approach can be implemented independently or along with other POD controllers. The decoupled control strategy is powerful for providing precise damping of targeted inter-area oscillation modes without deteriorating other modes unnecessarily. Extensive results of studies of both small-signal transients and four typical large transients have demonstrated the excellent performance of the proposed 8
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Fig. 12. Results of large transient case four: loss of large load. Table 2 Oscillation analysis of large-transient events. Event
Alberta damping ratio (%)
BC damping ratio (%)
Base
Conv.
Conv. + Decouple
Base
Conv.
Conv. + Decouple
2.8 2.7 2.7 3.0
> 10.0 > 10.0 > 10.0 > 10.0
> 10.0 > 10.0 > 10.0 > 10.0
0.9 0.9 1.0 1.1
2.1 1.8 2.2 1.9
3.9 4.2 4.3 3.1
[9]
[10] One Two Three Four
[11]
[12]
decoupled approach. In future work, we plan to study how to efficiently update the modal decomposition matrix. We also plan to investigate the impact of communication delay and measurement noises on the proposed decoupled POD controller and design compensations for these network effects.
[13]
[14] [15]
Conflict of interest [16]
None declared.
[17]
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