Cyber-Physical Distributed Systems

2.2.1 CPS Performance Evaluation

2.2.1.1 CPS Performance Index

Having built the hybrid model of the closed‐loop feedback CPS with degraded components, an event‐based MCS method is introduced to quantitatively assess the reliability of such system. The quantitative analysis for system performance considers the domain performance requirements [445].

The qualities of a control process of CPSs consist of the rising/declining time, the percentage overshoot and the settling time, which are all relevant in assessing real‐time operational performances. Descriptions of the qualities are presented in Table 2.1. Rising/declining time is useful while ensuring that the control process responds rapidly to the command signal; percentage overshoot ensures that the control process exhibits sufficiently small oscillations during response; and the settling time ensures that the control process quickly reaches the expected output value and keeps stable.

Table 2.1 Domain Requirements and Descriptions.

If any of the quality parameters exceeds the corresponding maximal operational requirements within the total run time To, it can be concluded that the CPS is unable to satisfy the stipulated requirements. Hence, it has failed at total run time To, because of component degradation.

When considering the quality‐Rising time, it can be represented as

(2.10)

When considering the quality‐Declining time, it can be represented as

(2.11)

where RT, DT, PO and ST are the qualities of the control process at total run time To. x = 0 means that the CPS has failed at total run time To. RTmax, DTmax, POmax and STmax are the maximal operational requirements.

Besides the operational requirements, the nonfunctional requirement‐reliability is especially important. Reliability directly determines the ability of the CPS to maintain the expected qualities despite the presence of degraded components.

2.2.1.2 Reliability Evaluation of CPSs

It is noticed that the degradation process associated with each component is probabilistic and it is difficult to obtain the explicit reliability function of such a system. The event based MCS method is introduced in this book to estimate the ability of the system with a PID control strategy in the design phase [446][447]. The reliability of the CPS is then estimated as a tabulated function of the total run time subject to operational requirements. The detailed procedures for applying MCS method to hybrid model are described below.

INPUT: the PID control strategy Kp, Ti and Td; the mathematical model for the CPS; the degradation paths of components; the total run time To; the operational requirements.

OUTPUT: the estimate R*(To), of system reliability at total run time To.

STEP 1: Set current number of MCS replications h equal to 1.

STEP 2: Define precision interval L, the percentage of simulations belonging to this precision interval p, and the number of simulations NT in each MCS run; set NF, NE and current number of simulations i in this MCS run to 0.

STEP 3: Conduct a simulation using the hybrid model described in section 2.1.1.3. If the simulation fails as judged by (2.10) or (2.11), then set NF = NF + 1, the current reliability

and note the qualities of process RTi/DTi, POi and STi.

STEP 4: If

, NE = NE + 1.

STEP 5: If

or i < NT, let i = i + 1 and go to STEP 3; otherwise,

and go to STEP 6.

STEP 6: If h < H, let h = h + 1 and go to STEP 2; otherwise,

and halt.

The estimated reliability R*(To) has the following several important statistical properties which have been proved in literature [448][449][450]. These properties are rather useful in verifying whether the estimated reliability of system obtained from the event based MCS is a good estimator and defining the relationship among H, ε and α. The unbiased estimator with required precision derived from a small number of simulations is important for reliability assessment. Several properties about the estimated reliability are introduced as follows.

Property 2.1: The expectation of the estimated reliability R*(To) of the CPS with degraded components obtained from MCS method is an unbiased estimator of the exact reliability R(To) at total run time To.

Proof: Let

be the expected reliability of the CPS with degraded components at total run time To, where h = 1, 2, …, H. Then,

(2.12)

Thus,

(2.13)

Property 2.2: R*(To) is an unbiased, consistent estimator of R(To):

(2.14)

Property 2.3: If the absolute error ε and the confidence level (1 − α)% of the MCS method are required, the total number of MCS replication H satisfies below approximate relation

(2.15)

Proof: Let R*(To) be the estimated reliability and R(To) be the actual reliability. Consider the absolute error ε and the confidence level (1 − α)%, it exists

Using Zα/2 as the z‐value of the (1 − α/2) percentile of the standard normal distribution, set the initial estimate of the replications required for MCS as the smallest integer H such that

where S0 is the sample standard deviation of R*(To) obtained from a sample with H0 MCS replications. In this book, H0 is selected as 1000. Setting the number of MCS replications at 1500 as well as the sample size of simulations in each MCS run at 5000 is adequate for an absolute error 0.005 at the confidence level 95%. It should be noted that if H0 is small, it would be more appropriate to use t‐distribution

instead of Zα/2.

A high reliability indicates that the designed CPS should be able to satisfy the operational requirements in most cases while considering the degradation of the components. Otherwise, the system should be redesigned by optimizing the PID control strategy, in order to improve the ability of the feedback system compensating the loss effectiveness caused by degraded components, and finally provide a required reliability.

2.2.2 CPS Model Verification

The validation of data‐driven degradation model of the component in the CPS is usually via comparison and this chapter takes the model of generator degradation as the example to show how to conduct the cross‐verification with existing estimation model.

Example 2.2: The power system with a natural gas power plant consisting of four gas turbine units is shown in Figure 2.1. The values of the power system parameters are given in [451][452], i.e., b = 1, c = 0.05, X = 0.6, Y = 1, Tc = 0.3, Tf = 0.23, Tt = 0.2, Kp = 120, Tp = 20, B = − 0.425 and R = 2.2. The discrete PID controller for the LFC is optimally tuned via the two‐degree‐of‐freedom internal model control design method and the PID approximation procedure [453].

This tuning method provides the system with disturbance rejection ability, and results in KP = 0.13, KI = 0.42 and KD = 1.32, which ensure optimum LFC performance (POO =  − 0.015 pu and STO = 2.5 s), against the sudden load increase and the requirements in [454], i.e., failure thresholds LPO =  − 0.016 pu and LST = 30 s. The steady output of the gas turbines is SOO = 2.5 · 10−3 pu.

The degradation of the gas turbines and of the PMU reduces the LFC performance and ultimately results in the failure of the power system [455][456][457]. For simplicity, time T, λ and σB are expressed in units of 10000 h in this example, unless other units are specified. The degradation rates of the compressors flow capacity and efficiency are estimated by applying the EM algorithm to the dataset in [458]:

• the flow capacity degradation of gas turbine is described by: λ∼N(1.8568, 0.04732), σB = 0.1, a = 1;

• the efficiency degradation of gas turbine is described by: λ∼N(1.0252, 0.04272), σB = 0.055, a = 1.

The model parameters for the measurement drift and the measurement error of the PMU (Ts = 0.1 s) are given in [459]:

• the measurement drift degradation of sensor is described by: λ∼N(1.1575, 0.06052), σB = 0.04, and a = 1 [460];

• the measurement error is described by: w[mHz]∼N(0, 22).

In addition, Table 2.2 presents the parameters of the distribution of the aging rate, i.e., the drift parameter λ, obtained by the EM algorithm and the 95% confidence interval.

Table 2.2 The parameters and the 95% confidence interval of the distribution of the drift parameter λ.

Figure 2.4 (a) and Figure 2.4 (b) present the comparisons among the degradation models used in [461][462], and the degradation model of (2.3) for two gas turbines.

Figure 2.4 Comparisons among the Wiener degradation model of (2.3) and linear and quadratic degradation models used in [463][464] for two gas turbines.

The Wiener degradation model accounting for unit‐to‐unit variability better represents the degradation path of the turbine health index. The mean squared errors (MSE) of the Wiener process are 0.7498 and 0.7145; the MSE of the linear and quadratic regression‐based methods are, respectively, 0.8512 and 0.9665, and 0.8198 and 0.8757. Additionally, the Wiener degradation path can track the variable trend of the flow capacity drop. As such, the drift parameter λ introduces a stochastic component capturing the unit‐to‐unit variability.

This section introduces the reliability enhancement of CPSs via using heuristic algorithms, such as GA and PSO, and the control performance improvement of CPSs via solving a multi‐objective optimization problem. This section provides the basics of maintaining the performance indexes of CPSs.

2.3.1 PSO‐Based Reliability Enhancement

In the present section, the PSO method is selected as an example, where a solution consisting of three parameters of the PID control strategy ‐ Kp, Ti and Td ‐ is encoded as “a particle” in PSO. The fitness value of each particle is determined by the same fitness function. It should be aware of the fact that the reliability of the system may stay at 1 if the total run time To is in an early stage of the system's life time, which indicates that the degradation of component is not so severe.

In such a case, the PID control strategy to be adopted not only needs to make the system reliability stay at 1, but also maintain the qualities of the control process at high levels. Therefore, the fitness function should consider two cases: system reliability at 1 and not. If system reliability is 1, the fitness function is a function of qualities of the control process; otherwise, the fitness function is a function of system reliability plus a penalty for system reliability not being at 1.

Thus, the fitness value of a particle can be evaluated by the following equation:

(2.16)

where

and

During each PSO iteration, information about the best position of each particle and the best particle in the entire swarm can be obtained after computing the fitness values of all particles based on (2.16). And then in the next iteration, the movements of the particles are guided by their previous best‐known positions within the search space, called pbest, as well as the best known position of the entire swarm, called gbest.

For a particle j with D dimensions, its local best known position can be represented by pbestj = (pj1, pj2, …, pjD) and the global best known position of the entire swarm is gbest = pg. Therefore, the velocity

(2.17)

(2.18)

where m ∈ {1, …., M}; j ∈ {1, …., J}; d ∈ {1, …., D}; w is the inertia weight; c1 and c2 are the cognition learning factor and the social learning factor; c1 + c2 generally equals to 4 and ρ1 and ρ2 are random factors restricted to [0, 1].

The followings are some important details of the proposed MCS‐PSO method:

STEP 1: Randomly initialize the particles representing the PID control strategies, including the velocity and the position.

STEP 2: Apply the MCS proposed in section 2.2.2.2 to obtain the R*(To); then compute the fitness value for each particle based on (2.16).

STEP 3: Update the values of pbest and gbest and determine the velocity and position values for each particle in the next iteration on the basis of (2.17) and (2.18).

STEP 4: If the stopping condition ‐ the maximum number of iterations M is reached, go to STEP 5; otherwise, go to STEP 2.

STEP 5: Decode the particle with the minimum fitness value and take the result as the optimally designed PID control strategy which can ensure the maximum possible reliability or the best quality parameters associated with the control process with reliability at 1 at total run time To.

2.3.2 Optimal PID‐AGC

The AGC performance of the integrated system depends on the discrete‐time PID controller. Therefore, the PID controller is optimized to mitigate the communication network disturbances and offer optimum AGC performance by reducing system frequency fluctuations. As a result, system reliability is also optimized. In previous works, the objective function for the optimization of the PID controller is an integral performance index over the total operating time T, which quantifies frequency and control signal deviations [465][466]:

(2.19)

where η1 indicates the relative importance of the two terms and η2 is the normalizing constant to scale both terms in a uniform range and is set to 0.002. The first term is directly related to the reliability of the integrated system and the second term measures the disturbance rejection ability of the controller, which is the total control effort to be minimized. The total process time T is set to 30 s [467].

Motivated by [468][469][470][471][472][473], the objective function in (2.19) is separated into two objectives and multi‐objective optimization is applied. In fact, the weighting factors in the objective function in (2.19) can be changed dynamically using multi‐objective PSO algorithm:

(2.20)

where

The fitness value of the stochastic population‐based algorithms is the expectation of the stochastic objective function obtained from MCS:

(2.21)

where Ji is defined in (2.20) and N = 200 samples.

In this section, we use the PSO rather than other heuristic algorithms, i.e., GA [475], to solve the optimization problem for