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Min Xie Cyber-Physical Distributed Systems
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As the response is oscillatory, the filter Fd(s) given in (3.47) is used to cancel the effect of the oscillatory poles and reject the disturbance. Based on (3.48), with n = 2 and the designed γ = 1.2, the a2 and a1 in Fd(s) are derived as 0.154 and 0.403. Therefore, the Fd(s) is expressed as
(3.57)
To quantitatively evaluate the influence of communication delays on the LFC of the single‐area WAPS, this chapter employs an aggregated indicator, which measures frequency fluctuations and control efforts:
(3.58)
where ω = 2 × 103 is a normalized factor for the deviation of system frequency.
3.3.3.2 Analysis of Case 1
For the time‐varying delays, a discrete PI controller (Kp = 1 and Ti = 0.6) can tolerate a maximal allowable delay of 0.281s, based on Table III in [665]. As the upper bound of the cumulative time‐varying delays in the C‐A channel and the S‐C channel is 0.280s, this PI controller can preserve the stability of the LFC.
Figure 3.13 shows the LFC performance of the WAPS under the delay‐margin‐based PI controller with and without the proposed Smith predictor and the designed disturbance filter. The PI controller takes more than 200s to stabilize the system frequency and the designed disturbance filter cannot reduce frequency oscillations as illustrated by Figure 3.13 (a). The Smith predictor can largely reduce the frequency deviations and settling time as shown by Figure 3.13 (b). Figure 3.13 (c) shows that the use of the Smith predictor and the designed disturbance filter can provide enough disturbance rejection, and, therefore, further reduce the active frequency oscillations, which could damage generators, trip tie lines, and increase wear and tear on power system components.
Figure 3.13 LFC performance of WAPS (a) without Smith predictor, (b) with Smith predictor and (c) with Smith predictor and disturbance filter.
Figure 3.13 (b) and Figure 3.13 (c) show that the Smith predictor based on the DHMM reduces the frequency drop and the setting time as compared to the EWMA. Additionally, the accuracy of the delay prediction by K‐means‐based DHMM (Table 3.5) enhances the system LFC performance against the positive load disturbance as compared to the uniform‐based DHMM.
Table 3.6 shows the statistical properties of the indicator (3.58) for different scenarios under case 1, computed using 1000 samples. The Smith predictor greatly reduces frequency fluctuations as compared to the original WAPS.
Table 3.6 Statistical properties of aggregated indicator of the case 1.
The comparison of Scenario 3 and 4, and of Scenario 3 and 5, shows that the uniform‐based DHMM and K‐means‐based DHMM enhances the performance of the LFC by 4.6% and by 6.9%, respectively, as compared to the EWMA, according to the indicator in (3.58). The comparison of Scenario 3 and 6, of Scenario 4 and 7, and of Scenario 5 and 8, shows that use of the disturbance filter further improves the performance of the LFC by 33.3%, by 36.8% and by 36.5%, respectively, as compared to scenarios without using the filter.
3.3.3.3 Analysis of Case 2
For the robust PID controller, Kp, Ti and Td are set to 0.20, 0.20 and 1.01, via the GA‐based robust LFC strategy [666][667].
Figure 3.14 shows the LFC performance of the WAPS under the robust PID controller with and without the proposed Smith predictor and the designed disturbance filter. The comparison of Figure 3.13 (a) and Figure 3.14 (a) shows that the robust PID controller is better than the PI controller in stabilizing system frequency. Consistently, the introduction of Smith predictor is effective in eliminating the system frequency oscillation and reducing the settling time. The usage of disturbance filter can further improve the LFC performance.
Figure 3.14 Dynamic performance of single‐area LFC (PID controller) of WAPS (a) without Smith predictor, (b) with Smith predictor and (c) with Smith predictor and disturbance filter.
Even though the robust LFC strategy is well designed by [668], the Smith predictor can still improve LFC performance, based on the comparison of Figure 3.14 (a) and Figure 3.14 (b). The use of disturbance filter also works for the robust PID controller and further reduces frequency oscillations, based on the comparison of Figure 3.14 (b) and Figure 3.14 (c).
Table 3.7 shows the statistical properties of the indicator (3.58) for different scenarios and for Case 2, computed using 1000 samples. The comparison of Scenario 3, 4 and 5, and of Scenario 6, 7 and 8 presented in Table 3.7, shows that the K‐means‐based DHMM has the smallest indicator value and the best performance in damping the WAPS frequency oscillations, because it can provide the best time delay predictions as compared to the EWMA and the uniform‐based DHMM.
Table 3.7 Statistical properties of aggregated indicator of the case 2.
The comparison between Scenario 1 in Table 3.6 and Scenario 1 in Table 3.7 shows that the robust PID is more effective in reducing frequency fluctuations than the delay‐margin based PI controller and improves the value of aggregated indicator by 14.1%. Indeed, the robust PID controller carries out an additional step to search for its optimal parameters, to achieve compromise between the LFC performance and the delay margin [669], besides just computing the delay margin.
As a result, the line‐by‐line cross comparison between Table 3.6 and Table 3.7 shows that the robust PID controller improves the value of aggregated indicator of real systems by 6.8% to 14.5%, as compared to the delay‐margin based PI controller. Since the delay‐margin based PI controller and the robust PID controller are widely implemented in practice [670][671][672], it is significant to compare the results of our method against the real case studies, at which above two controllers are used.
The comparison of Scenario 1 and 8 in Table 3.6, and of Scenario 1 and 8 in Table 3.7, shows that our method, i.e., the Smith predictor with disturbance filter, reduces the value of aggregated indicator of real systems by 66.9% and by 64.1%, respectively, as compared to the real case studies.
3.4 Stability Enhancement of Illustrative WAPS
3.4.1 Eigenvalue Analysis and Delay Impact
According to the state‐space model of the microgrid given in Section 3.1.2.3, the eigenvalue analysis is conducted following the procedure detailed in [673][674][675]. Figure 3.15 compares the effect of the time delay in the proposed MOPSO‐based PID controller with the case of a PI controller [676], when packet dropout is neglected, and the period is 0.01 s.
Figure 3.15 Eigenvalue spectrum of the microgrid in the presence of time delays with (a) the proposed MOPSO‐based PID controller when the time delay increases from 0 to 390 ms, and (b) the PI controller when the time delay increases from 0 to 24 ms.
Figure 3.15 (a) shows the MOPSO‐based PID controller eigenvalue spectrum when the time delay is as large as 390 ms (maximum allowable time delay). As demonstrated, the system with MOPSO‐PID controller remains stable in the presence of large time delays. The advantage of MOPSO‐PID controller over the PI controller [677] is better represented by Figure 3.15 (b); for the latter, the instability can occur when the time delay is as small as 20 ms.
As indicated by [678][679], typical time delays in reality can be in the order of 100‐300 ms. Therefore, the MOPSO‐PID controller is effective and robust in the reliable management of RERs in the microgrid to ensure stable system frequency.
Figure 3.16 shows the system frequency curves of the PI controller [680] and the proposed MOPSO‐based PID controller in the presence of 20 ms and 390 ms, respectively. Consistently with the eigenvalue analysis shown in Figure 3.15 (b), the time delay causes instability in the PI controller. By using the MOPSO‐based PID controller, the system stability greatly improves against delays of the feedback frequency signal and of the control signal.
Figure 3.16 System frequency responses of (a) the PI controller with 20 ms time delay, and (b) the MOPSO‐based PID controller with 390 ms time delay.
3.4.2 Sensitivity Analysis of Network Parameters
For illustrating purpose, the maximum permissible instantaneous frequency deviation is set to ±0.8 Hz according to [681]. The parameters of the discrete PID controller are tuned as KP = 0.52, KI = 2.21 and KD = 1.00 [682][683]. Five combinations of network configurations, i.e., BWShare, Ti and Ld, are taken into account to investigate multiple communication scenarios in the Ethernet and in the hybrid network.
For comparison purpose, the reliability and objective values of the integrated system with perfect communication, i.e., no time delays and no packet dropouts, are R = 94.48% and
Figure 3.17 presents the impact of the length of data traffic of the interference node on the reliability of the integrated system with the Ethernet ([0.2, 0.005, 5%]). The length of data traffic is from 20 to 200 bytes. As expected, the system reliability decreases as the length increases. It is in line with our common knowledge that a larger data packet consumes more bandwidth of the communication channel, which deteriorates network conditions and results in larger time delays and more packet dropouts.
Figure 3.17 System reliability as a function of the length of data traffic.
Figure 3.18 illustrates the relationship between the reliability of the integrated system with Ethernet, and the communication configurations [L
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