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Min Xie Cyber-Physical Distributed Systems
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In this sense, both the PSO and GA can be regarded as generating new solutions in the neighborhood of two parents, i.e., using crossover in the GA and using attractions to two pbest positions in the PSO. This multi‐parent effect is a key advantage over single‐point techniques such as the simulated annealing and tabu search.
As compared to GA, the PSO utilizes three available information sets to facilitate the search process. These are the local best‐known positions (pbest), the global best‐known position (gbest), and the current positions [480][481][482]. This allows greater diversity and exploration over a single population (which with elitism would only be a population of pbests).
In addition, the momentum effects on particle movement allows fast convergence (e.g., when a particle is moving in the direction of a gradient) and increased variety/diversity in search trajectories. The performance of PSO has been shown superior to GA in parameters optimization of PID controller [483][484]. The MOPSO is preferred to the multi‐objective GA because no evolution operators, i.e., the crossover and mutation, are required, and the information of the current optimum particles effectively identifies candidate solutions throughout the problem space.
The detailed introduction to PSO and MOPSO are provided in [485][486][487] and [488][489][490][491][492][493], respectively. The core part of the MOPSO algorithm is detailed in the following.
Notation to be used in MOPSO Algorithm:
A: solution set which provides optimal values of the PID controller parameters KP, KI and KD
NP: number of particles
xi and vi: current position and velocity of particle i
MP: number of dimensions (MP = 3)
I and NI: current and maximum number of iterations
ω: inertia coefficient (ω = 0.5 × 0.99I which tunes the impact of the previous history velocities on the current velocity)
c1 and c2: cognitive and social factors (which can accelerate the search towards the local and global best directions, and equal to c1 = 1 and c2 = 2)
r1 and r2: random numbers drawn from the uniform distribution [0, 1].
Procedures of MOPSO Algorithm:
STEP 1: Randomly initialize the position xi of the particle i (i = 1, 2, …, NP) sampling from a uniform distribution within the solution space, i.e., G = [0, 10]3. The coordinates of the particle position represent the PID controller parameters, i.e., KP, KI and KD. Initialize vi, pbesti and gbesti to zero.
STEP 2: Compute the fitness value for each particle based on (2.20).
STEP 3: Update the values of pbest and gbest based on calculated fitness values; determine the velocity and position values for each particle in the next iteration using following equations:
(2.22)
(2.23)
where Pi = [pbesti1, pbesti2, …, pbestiMP]T, Gi = [gbesti1, gbesti2, …, gbestiMP]T and d = 1, 2, …, MP. The current best solution is put into the set A.
STEP 4: If the maximum number of iterations NI is reached, go to STEP 5; otherwise, go to STEP 2.
STEP 5: Decode the particle with the minimum fitness value and choose the result as the optimally designed PID controller.
As indicated by [494][495][496][497][498][499], the multi‐objective optimization generates the Pareto optimal set of the non‐dominated solutions from which the global optimum solution is selected based on the specific application. In this study, fuzzy set theory is applied to model the trade‐off between the two objectives [500][501][502][503][504][505]. First, a linear membership function is defined for each objective function
(2.24)
According to (2.24), the minimum of the objective function Ji is associated to the minimum of the membership function, therefore it has maximum degree of achievement of the fuzzy objective. Then, for every non‐dominated solution k, the aggregate membership function for the PID controller optimization is computed as:
(2.25)
The solution which minimizes (2.25) is regarded as the best compromise between the two objectives and named “Best Tradeoff I”. For comparison, the distance of each non‐dominated solution k from the origin in the fuzzified coordinate is also computed [506][507][508][509][510][511]:
(2.26)
And the solution with minimum MF2k is chosen as the best compromise between the two objectives and named “Best Tradeoff II”. The convergence of optimization problem solved by the single PSO has been investigated in detail in works [512][513]. These works investigated similar optimization problem of PID controller, where both the objective value and the controller parameters become almost constant toward the end of 50 iterations (the number of populations is 30). As for the convergence of optimization problem of PID controller solved by the MOPSO, works [514][515][516][517][518][519] showed fast convergence to the Pareto front in 50 iterations (the number of populations is 50).
Note
Part of the contents are from [520][521][522][523][524] and permission has been obtained to use the contents.
3
Stability Enhancement of CPSs
In this chapter, we preform the stability analysis and enhancement method on one of the most widely used Cyber‐Physical Systems (CPSs) ‐ microgrids from two perspectives as illustrated example: 1) the operations of the integrated Distributed Energy Resource (DER) systems and open communication network, and 2) the design of optimal Automatic Generation Control (AGC) strategies in the face of communication degradation.
This chapter focuses on microgrids in islanded operations with real power generation and demand from a systemic perspective. The total generation of DERs supplies the demand. Therefore, the initial conditions of DERs are determined to balance the stochastic Renewable Energy Resources (RERs) and the demand. As a result, voltage profiles and reactive power are neglected. This simplified model is adopted by several works [525][526][527], and is also integrated with microgrid power management systems [528][529][530][531][532][533].
Because wind and solar sources have intermittent characteristic, their power generations have stochastic behaviors. The probabilistic power patterns can be well justified by probability distribution functions, such as Weibull and normal distribution for wind speed and solar irradiation [534][535]. Alternatively, in this chapter we consider actual wind speed and solar irradiation as in [536][537]. The historical wind speed and solar radiation dataset are from the Elia Grid, Belgium.
The ability of the integrated system to maintain system frequency deviation within tolerance margins quantifies system reliability and is evaluated by Monte Carlo simulation (MCS) as detailed in Section 2.2.1.2. Stochastic time delays are modelled by generating random congestion based on Media Access Control (MAC) protocols. Congestion of network channels depends on the activity level of the interfering traffic, which is the root cause of network‐induced delays and packet dropouts [538][539][540][541]. The open communication network model is implemented via TrueTime simulator testing different MAC protocols [542][543] as detailed in Section 2.1.2.
To stabilize system frequency against RER, demand variability and communication degradation, a discrete Proportional–Integral–Derivative (PID) controller is used [544][545]. The heuristic algorithms such as PSO [546][547][548][549][550], Genetic Algorithms (GA) [551], flower pollination algorithm [552], quasi‐oppositional harmony search algorithm [553], Cuckoo Search algorithm [554], and artificial bee colony algorithm [555], have been introduced to provide easy implementation, conceptual simplicity, more flexibility, and independency to the initialization of the controller optimization procedure.
PSO is adopted to minimize the stochastic objective function and achieve the optimal PID controller for various architectures and conditions of the open communication network, because the movement of particles is influenced by the local best‐known positions as well as by the global best‐known positions in the search space, PSO can avoid being trapped into local minimum [556][557][558][559][560]. Finally, the robustness and effectiveness of the proposed MOPSO‐based‐PID‐controlled AGC against communication degradation is assessed.
In addition, we conduct various simulations to test the performance of the integrated system with the MOPSO‐optimized control strategy for three communication network architectures, under different uncertainty conditions and operating scenarios of the DERs and load. The performance of the integrated system with the perfect communication is used as a benchmark.
The other two network architectures are the Ethernet and the hybrid network (a mix of Ethernet and 802.11b/g). Different network configurations, i.e., traffic conditions of the open communication networks, are assessed. The performance of the integrated system using the three network architectures are compared under the uncertainty conditions and the effectiveness and robustness of the proposed control strategy is shown.
3.1 Integration of Physical and Cyber Models
3.1.1 Basics of WAPS
3.1.1.1 Physical Layer
Figure 3.1 shows the control block diagram of a single‐area Wide‐Area Power System (WAPS) in which one generator supplies power to the service area and the transmissions of frequency measurements and control signals occur via the open communication network [561][562]. It is extended from the diagram presented in Figure 2.1.
Figure 3.1 Schematic of the single‐area WAPS with communication delays.
The WAPS physical configuration is provided in Figure 3.1. ΔPv, ΔPm, ΔPd, uk and Δfk are the valve position, the mechanical output of the turbine, the load, the control signal computed by the controller in the period k and the deviations of frequency measured by the Phasor Measurement Units (PMU) in the period k, respectively. The droop characteristic is a feedback gain to improve the damping characteristics of the power system and is usually set to 1/R.
The dynamics of the governor, the non‐reheat turbine, and the load and machine are described by the first order transfer functions with specified gain and time constant as:
(3.1)
(3.2)
(3.3)
where TG, R, TT, M and D denote the time constant of the governor, speed drop, time constant of the turbine, moment of inertia of the generator and generator damping coefficient.
As there is no power exchange to other areas, the area control error (ACE) in period k is defined as:
(3.4)
where β > 0 denote the frequency bias factor.
The PID controller is tuned to improve the dynamic performance of Load Frequency Control (LFC) strategy by balancing the generation production and the demand load in real time. It receives real‐time system frequency measurements from the PMU via the open communication network and then computes the control signals based on (3.5).
If Δfk is negative, the PID controller will send out a command to increase the generation production; and if Δfk is positive, the PID controller is going to reduce the generation production. However, the PMU takes measurements at every sampling time, therefore the PID controller is time‐discrete rather than time‐continuous as in [563]. Its transfer function C(z) is:
(3.5)
where Kp, Ti and Td are proportional, integral and derivative gain of the PID controller. a(z) and b(z) are determined by the integrator method and filter method. N is the filter constant. The forward Euler method is most appropriate for the case with a small sampling interval, where the Nyquist limit is large compared to the bandwidth of the controller.
As the sampling interval of PMU is rather small, the integrator method and filter method in this chapter apply the forward Euler method, where a(z) and b(z) are all equal to
3.1.1.2 Cyber Layer
The cyber layer in WAPS is mainly based on the open communication network, where the total time delay can be expressed as [564]:
(3.6)
where Tpre denotes the preprocessing time at the source which is the sum of the computation time Tscomp and the encoding time Tscode. Twait denotes the waiting time at the source which is the sum of the queue time Tqueue and blocking time Tblock. Ttx denotes the network time delay which is the sum of the time required to send the data to the channel Tframe and the propagation time Tprop. Tpost denotes the postprocessing time at the destination, which is the sum of the decoding time Tdcode and the computation time Tdcomp.
Tpre, Ttx and Tpost depend heavily on the processing speed of software or firmware of devices, the packet size and the length of the network cable, which are typically constant and neglectable.
For example, to send one bit of data to a 10 Mb/s Ethernet requires Tframe = 67.2 μs and Tprop = 10 μs for a 2500 m Ethernet with typical transmission rate 2 × 108 m/s. Due to the uncertain amount of data to be sent and the stochastic traffic on the network channel, Twait is the major source of jitter and usually determined by the MAC protocol [565].
This section considers the Ethernet with Carrier Sense Multiple Access with Collision Detection (CSMA/CD) based MAC protocol and then Twait can be analyzed. Tqueue is the time that a packet waits in the buffer of the source node while previous packets in the queue are sent, depending on the blocking of previous packet in queue, the periodicity of packets and the processing load. As Tblock plays a critical role in Tqueue, the work mainly investigates Tblock.
Tblock in the Ethernet is consisted of the time taken by collisions with other packets and the subsequent retransmitted time. As Tblock is probabilistic indicated by the BEB algorithm, it is very difficult to derive an exact expression for Tblock and thereby most current studies focus more on its expectation. When the k‐th collision happens, the sender at source node would back off for a time Tbo defined by
(3.7)
where RN follows a discrete uniform distribution taking value from (0, 2k − 1). For the Ethernet with CSMA/CD MAC protocol, the maximum number of k is 10 [566].
Therefore, the expected Tblock can be expressed by the following equation as
(3.8)
where
3.1.1.3 WAPS Realized in TrueTime
The integrated model of the single‐area WAPS and the CSMA/CD MAC protocol‐based Ethernet is implemented via the TrueTime [568], and the data rate is 8 · 104 bits/s. The architecture of the integrated system and its process overview during the control period k, as shown in Figure 3.1, are listed as follows. It is more detailed than the general framework presented in Section 2.1.2.2.
The PMU node (periodic): based on the NREL report [569] and the standard C37.118.1 [570], the synchronized PMU is assumed to sample AC waveforms with a sampling interval Ts of 0.3 second. The packet (average size = 400 bits [571][572]), containing the time delay between the control center to the plant
The control center node (event‐driven): when the control center receives a frequency measurement Δfk from the PMU, the delivering packet is re‐labelled with a GPS time stamp t2; thus, the time delay between the PMU signal and the control center is
The plant node (event‐driven): the plant acts based on the control signal received, which is re‐labelled with a GPS time stamp t4. The time delay in the communication from the control center to the plant is
The interfering node (periodic): it sends out the disturbing traffic over the network to mimic congestion and generate time delays. This node sends out packets (size ∈ [80, 400] bits [578]) with a probability indicating the fraction of the network bandwidth, i.e., BWshare, occupied by other users in the open communication network. The period of interfering trigger TIT denotes the number of active end‐users.
The time delays can be obtained from the GPS‐based time stamp technique and the network‐induced time delays in C‐A channel are displayed in Figure 3.2. These delays depend heavily on the network state, which reflects the severity of network load and congestion levels [579]. The network state is categorized into ‘good’, ‘normal’ and ‘bad’.
Figure 3.2 An illustrative example for the time delays in C‐A channel.
In period k, the controller cannot have any knowledge of the future
3.1.2 An Illustrative WAPS
The schematic of the illustrative WAPS is detailed in this section, which is made of a physical layer as shown by Figure 3.3, a cyber layer as shown by Figure 3.5 and resulting integrated system as shown by Figure 3.6.
Figure 3.3 Schematic of illustrative physical.
3.1.2.1 Illustrative Physical Layer
The structure of physical layer in Figure 3.3 is general, representative of the Distributed Generation Systems (DGS), and widely adopted in the literature [580][581][582][583]. It models a hybrid microgrid, which consists of conventional generators (Diesel Engine Generator (DEG)), RERs (wind turbine generator, WTG, and photovoltaic power, PV) and ESSs (Battery Energy Storage System (BESS) and Flywheel Energy Storage System (FESS)).
The small signal stability analysis of the hybrid microgrid in Figure 3.5, is based on time‐domain simulations, i.e., transfer function models. In the AGC, the WTG, PV, DEG, FESS and BESS are described by first order transfer functions with specified gain and time constant. A centralized controller is used [584], as opposed to multiple decentralized controllers for each controllable component [585][586][587]. It enables easier maintenance and reduces wiring cost and makes the AGC design problem traceable by reducing the number of controller parameters [588].
On the other hand, the centralized controller impacts the AGC performance, because a unique control signal is used by all the components. Nevertheless, current studies show that the centralized controller can ensure acceptable time‐domain AGC performance [589][590][591].
The transfer functions GWTG(s), GPV(s), and GDEG(s) of the WTG, PV and DEG, respectively, are expressed as
(3.9)
(3.10)
(3.11)
where TWTG, TPV and TDEG are the time constant of the WTG, PV and DEG. PWTG and PPV are the electrical power produced from the RERs.
