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Min Xie Cyber-Physical Distributed Systems
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Wind power PW and solar power Psol, which are generated from the data sets of wind speed and solar radiation of the Elia Grid, Belgium, and their curves are provided in the Figure 3.4 (a), Figure 3.4 (b) and Figure 3.4 (c), respectively.
Figure 3.4 Real‐time curves of (a) wind speed, (b) solar irradiance and (b) comparison between the total output power of RERs and the load.
The DEG is controlled by the control signal
The ESSs are critical in eliminating frequency fluctuations due to their fast response to the control signal. Based on [592][593][594][595], the transfer functions GFESS(s) and GBESS(s) of the FESS and BESS are given as
(3.12)
(3.13)
where TFESS and TBESS are the time constant, PFESS and PBESS are the output power,
Remark 3.1
The DEG, FESS and BESS have rate constraint, i.e.,
The transfer function GHPS(s) of the hybrid power system (HPS) models the relationship between the power imbalance, i.e., ΔPS − ΔPL, and the system frequency Δf(t)
(3.14)
where M and D are the inertia constant and damping constant of the HPS [596], and PS is the total power generated, denoted by PWTG + PPV + PDEG + PFESS + PBESS. PL is the power demand and its curve is provided in Figure 3.4 (c).
3.1.2.2 Illustrative Cyber Layer
The cyber layer Figure 3.5 is responsible for the real‐time communication between the control center and subsystems, i.e., providing data exchanges between the control center and controllable BESS, FESS and DEG in the physical layer. In addition, PMU measurements and control signals are transmitted via the shared and open communication network.
Figure 3.5 Schematic of illustrative cyber layer.
The time delays τsc and τca are determined by (3.6), and their impact on the transfer function from the control signal U(s) to the system output Y(s) is described as
(3.15)
where G(s) denotes the transfer functions of the DERs.
A second source of disturbance in the source‐destination communication is packet dropout, which occurs for three major reasons, i.e., the network disconnection, time‐out transmission and time‐out re‐transmission [597]. The PMU sensor is time triggered, i.e., it takes measurement at every sampling interval Ts. The relationship between the frequency measured by the PMU and the frequency received by the control center is
(3.16)
where γk = 1 indicates the transmission of Δf(kTs) is successful and tn is the time instant when the n‐th data packet is received by the control center. If γk = 1,
In real implementations of the CSMA/CD MAC protocol, the source node discards the packet and reports an error if ten consecutive data collisions occur [598]. Therefore, if congestion is severe, time delays increase and so does the number of packet dropouts. However, network congestion and packet dropout are challenging to link, and, therefore, packet dropout is usually modelled as a stochastic process [599][600][601]. Binary switching sequences are usually applied [602], which specify the expected packet dropout probability.
The stochastic parameter γk of the binary switching sequence is Bernoulli distributed [603][604], taking value of 0 or 1 with
(3.17)
where 0 < Ld < 1 is the expected packet loss probability.
The frequency measurement
(3.18)
On the other hand, the controller is event‐driven, i.e., it updates the control signal and sends it to the DERs as soon as the control center receives an updated frequency measurement. Therefore, if the n‐th packet is received, the control signal
(3.19)
(3.20)
where γn = 1 if the control signal u(tn) computed based on
3.1.2.3 Illustrative Integrated System
The cyber layer and the physical layers have the same topology. Power imbalance is the root cause of system frequency fluctuations. The control center remotely monitors the Energy Storage System (ESS) and the diesel generator to reduce the imbalance and to ensure good AGC performance. Following [605][606], the PID controller in the control center is responsible for the system frequency stabilization, and therefore we use it to represent the control center in the control block diagram shown in Figure 3.6.
Figure 3.6 Control block diagram of illustrative integrated system.
According to work [607], the linearized state‐space realization of the microgrid in the physical layer of Figure 3.6 can be expressed as
(3.21)
where
(3.22)
(3.23)
(3.24)
An elaborated expression of the microgrid state‐space model is given as follows.
(3.25)
(3.26)
3.2 Settings of Stability Analysis
In this chapter, case studies are carried out on the LFC strategy of the illustrative single‐area WAPS with communication delays and are compared with results of the delay margin‐based method [608] and the evolution algorithm‐based method [609]. This section introduces the simulation settings for time delay predictions and different cases used in the LFC of DERs.
3.2.1 Settings for Delay Predictions
The parameters of the integrated system are in Table 3.1 [610][611][612], where Ts is the sampling interval of the PMU, BWshare is the threshold used in the interfering node and TIT is the period of the interference trigger. This table is specifically used in the time delay predictions in the Section 3.3.
Table 3.1 Parameters of the single‐area WAPS with communication network.
The standard C37.118.1 is applied for the synchronized PMU in power systems, in a single‐area WAPS with one PMU and two phasors [613]. The data frame of PMU measurement is of the order of few hundreds of bytes and the transmitted data rate is 80000 bits/s. The time‐tagged phasors are sent to the control center at rates up to 60 samples per second. Due to the small packet size of PMU measurement, single‐packet transmission is considered, i.e., the data are lumped into one packet. Single‐packet transmission is adopted by the MAC protocols with large packet size, e.g., the maximum transmission unit in the Ethernet is 1500 bytes and the maximum transmission unit in the Wireless Local Area Networks (WLAN) is 7981 bytes. In this chapter, the open communication network employs the Ethernet.
The results demonstrate the accuracy of the proposed Discrete Hidden Markov Model (DHMM) method in predicting time delays (Section 3.3.2). Simulation studies investigate the performance of LFC of the single‐area WAPS, which is equipped with a Proportional–Integral (PI) controller tuned via the delay margin method (Case 1 in Section 3.3.3.2) and a PID controller tuned via evolutionary algorithms (Case 2 in Section 3.3.3.3), with and without the Smith predictor. This comparison demonstrates the effectiveness of the DHMM‐based Smith predictor in improving the LFC performance and reducing frequency oscillations.
3.2.2 Settings for Illustrative WAPS
In the illustrated WAPS to be used for the LFC of RERs, the physical system operates in nominal conditions, i.e., the stochastic wind speed, the variable sun irradiance and uncertain load, respectively, PWTG, PPV and PL, given in [614][615][616]. The coupled algebraic and ordinary differential equations for the DGS and the open communication network in Figure 3.3, are numerically integrated using the Dormand‐Prince method implemented in Matlab ode45 function with a fixed step size of 0.005 s. The parameters of the transfer functions for the DERs in Figure 3.3 are provided in [617][618][619]. Additionally, the physical layer has the following specifications:
The base value for the apparent power is 150 kW.
The rated apparent power of the WTG (Nordtank NTK 150) is 150 kW. The cut‐in wind speed, the rated wind speed and the cut‐out wind speed are 4.0 m/s, 13.5 m/s and 25 m/s, respectively. The wind farm consists of two WTGs.
The structure of a 750 kW PV generation has 465 parallel strings and consists of 7 series‐connected modules (SunPower SPR‐230E‐WHT‐D). The rated power of each cell is 230 W under the nominal operation temperature 45 °C. The nominal efficiency is 18.5%. The length and width of each cell are 1.599 m and 0.798 m. The maximum power temperature coefficient is 3.37 × 10−5.
The rated power of the DEG (MQP300IV) is 300 kW.
The rated power of the BESS (one Tesla Powerpacks) is 50 kW, whose energy capacity is 210 kWh.
The rated power of the FESS (Amber Kinetics) is 25 kW, whose energy capacity is 40 kWh.
Based on Remark 3.1, 0 ≤ ΔPDEG ≤ 2 pu, |ΔPBESS| ≤ 0.33 pu and |ΔPFESS| ≤ 0.17 pu, provide the largest rated output of the DEG, FESS and BESS [620]. The data sets of wind speed and solar radiation for computing real‐time output of RERs are provided by the Elia Grid, Belgium, with a data frequency 1 Hz. The details can be referred to Figure 3.4. The relevant system parameters [621][622] for a typical microgrid in the physical layer of Figure 3.3 are presented in Table 3.2.
Table 3.2 Parameters of frequency response model in the Figure 3.3.
The open communication network is implemented in TrueTime simulator [623]. The detailed procedures are provided in the Section 2.1.2. The data rate of the communication architectures is 800 Kbits/s, and the 802.11b/g is further characterized by transmission power 20 dbm, receiver signal threshold ‐ 48 dbm, ACK timeout 0.04 ms and the retry limit 3. The descriptions of simulation parameters Ti, BWShare and Ld is discussed in Remark 2.1, Remark 2.2 and (3.17) and presented in Table 3.3.
Table 3.3 Parameters of communication network in Figure 3.5.
3.2.3 Cases for Illustrative WAPS
In order to simulate diverse operating conditions of the islanded microgrid, four exemplary cases of the DERs management are considered. The power generated by the RERs and the microgrid load is categorized into low and high, and the excess generation is used to charge ESS. In particular, these four different cases are:
• Case 1: The power generated by RERs is low. The load is low and consumes most of the power generated by RERs. As a result, the DEG is OFF. Stochastic RERs caused by the dynamic weather conditions lead to unbalance between supply and demand, reflected by system frequency fluctuations. These can be mitigated by the ESS via the controller whose input is the system frequency derivation, and therefore system frequency is stabilized.
• Case 2: The power generated by RERs is low and the load is high. It indicates that RERs cannot supply sufficient power, and therefore complementary power is provided by the DEG and the discharging ESS.
• Case 3: The power generated by RERs is high and the load is low. RERs can supply sufficient power, and therefore the excess power is used to charge ESS and the DEG is OFF.
• Case 4: The power generated by RERs is high and the load is high. The combination of RERs and ESS is sufficient to supply the total load and the DEG is disconnected.
For the above cases, the initial conditions of the DERs are listed in Table 3.4. These conditions match the realistic situations in Figure 3.4. The reference system frequency is set to 50 Hz.
Table 3.4 State of DERs under different microgrid operating conditions.
3.3 HMM‐Based Stability Improvement
3.3.1 On‐line Smith Predictor
The Ethernet‐based open communication network is mapped into five levels of congestion, i.e., very low, low, medium, high and very high, which are characterized by the DHMM as shown in Figure 3.2. Feasible scalar quantization techniques map the measured random delays into the discrete observation space of the DHMM [624][625][626].
The Missing Data Expectation Maximization (MDEM)‐based Baum‐Welch algorithm and the Viterbi algorithm are introduced to estimate the parameters of the DHMM and predict the time delay, respectively. The predictions
3.3.1.1 Initialization of DHMM
As shown by Figure 3.2, the communication network has 5 states and the state space is denoted by Q = {1, 2, ⋯, 5}. The state in period k is qk, qk ∈ Q. The discrete delay observation space O has M elements and O = {1, 2, ⋯, M}. According to the uniform quantization technique, we construct M complete subintervals:
(3.27)
where h0 equals to hL and hM equals to hU. The delay τk is assumed to be comprised within the interval (hL, hU] and a new observation ok should be defined as ok = l, ok ∈ O.
In real open communication network, time delays are not always uniformly distributed within the interval (hL, hU]. In fact, when the network has light traffic, the time delays are usually concentrated around the subintervals in (3.27) with small statistical mean. Conversely, the time delays are usually concentrated in the subintervals in (3.27) with large statistical mean in case of communication congestion.
Therefore, the distribution of time delays cannot be properly modelled by the uniform quantization technique, and a better quantization technique is needed to provide more accurate time delays predictions, and better LFC performance.
To this aim, the clustering quantization technique is more suitable for non‐uniform distributed time delays. The K‐means algorithm is one of the popular unsupervised clustering algorithms and partitions all observations into different clusters in which each observation belongs to the cluster with the nearest mean. This algorithm has been proved to be very effective in handling large amounts of observations.
In the initialization step, the number of clusters is pre‐assigned as M. Given a set of time delay measurements = {τ1, τ2, ⋯, τK}, where K is the number of measurements, the K‐means clustering algorithm aims to partition the measurements into M (N < M < K ) sets, i.e., S = {S1, S2, ⋯, SM}, to minimize the within‐cluster sum of square (WCSS), i.e., the sum of distance function of each point in the cluster to the center:
(3.28)
where ci is the centroid of elements in Si. As the sum of squares is the squared Euclidean distance, the obtained result is intuitively the “nearest” centroid.
Thus, the quantization process based on the K‐means clustering algorithm is given as follows:
• Input: number of clusters M and the time delay measurements τ = {τ1, τ2, ⋯, τK}
• Step 1: randomly generate an initial set of M centroids
• Step 2: (assignment step) assign each time delay measurement to the cluster whose means has the least WCSS in v‐th iteration. Since time delay measurements and all cluster centroids are in one‐dimensional Euclidean space, this chapter only uses the one‐dimensional Euclidean distance:
(3.29) where each τk can only be assigned to exactly one
• Step 3: (update step) calculate the new means to be the centroids for the observations in the new clusters:
(3.30) where the centroid is a least‐squares estimator and also minimizes the WCSS objective.
• Step 4: Repeat Step 3 and Step 4 until the maximum number of iterations is reached.
Output: All clusters S = {S1, S2, ⋯, SM} and their centroids c1, c2, ⋯, cM.
Thus, based on the K‐means clustering quantization, when τk ∈ Si, a new observation ok is defined as ok = i, i ∈ O holds.
After K periods, a set of underlying network states q = {q1, q2, ⋯, qK} and a set of time delays τ = {τ1, τ2, ⋯, τK} is obtained. Then, a set of corresponding observations o = {o1, o2, ⋯, oK} is created by applying the scalar quantization techniques to τ. The sets q, τ and o are the fundamental elements of the DHMM for the channels of the open communication network [627].
During each signal transmission in the channel, the network transitions among states or stays in the current state [628][629]. These state transitions define a Markov chain across the congestion states, as shown in Figure 3.7. The Markov chain takes values from Q = {1, 2, ⋯, N} and the state transition probability matrix is P = [pij]. The transition probability from the state i in period k to the state j in period k + 1 is
(3.31)
Figure 3.7 Schematic of DHMM: Observed time delays are generated by the underlying and unobserved Markov chain representing the network states.
where pij ≥ 0, i, j ∈ Q, and
