Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Fully Distributed Economic Dispatch for Cyber-physical Power System with Time Delays and Channel Noises  PDF

  • Yuhang Zhang 1
  • Ming Ni 2,3,4
  • Yonghui Sun 1
1. College of Energy and Electrical Engineering, Hohai University, Nanjing 210098, China; 2. NARI Group Corporation (State Grid Electric Power Research Institute), Nanjing 211106, China,, Nanjing 211106, China; 3. NARI Technology Co., Ltd., Nanjing 211106, China; 4. State Key Laboratory of Smart Grid Protection and Control, Nanjing 211106

Updated:2022-11-19

DOI:10.35833/MPCE.2020.000847

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Abstract

Economic dispatch problem (EDP) is a fundamental optimization problem in power system operation, which aims at minimizing the total generation cost. In fact, the power grid is becoming a cyber-physical power system (CPPS). Therefore, the quality of communication is a key point. In this paper, considering two important factors, i.e., time delays and channel noises, a fully distributed consensus based algorithm is proposed for solving EDP. The critical maximum allowable upper bounds of heterogeneous communication delays and self-delays are obtained. It should be pointed out that the proposed algorithm can be robust against the time-varying delays and channel noises considering generator constraints. In addition, even with time-varying delays and channel noises, the power balance of supply and demand is not broken during the optimization. Several simulation studies are presented to validate the correctness and superiority of the developed results.

I. Introduction

ECONOMIC dispatch strategy for power system has always been a research hotspot. In the past twenty years, a great amount of algorithms such as genetic algorithm [

1], particle swarm optimization (PSO) algorithm [2], and deep reinforcement learning algorithm [3] have been utilized to solve economic dispatch problem (EDP). However, these traditional algorithms for EDP are centralized algorithms which require a center controller to collect information and send control signals. Due to the unsatisfactory performance of centralized scheduling and control modes such as communication congestion, poor flexibility and scalability, single point failure and some privacy issues, distributed economic dispatch (DED) strategy becomes a popular research topic recently, which is more suitable for cyber-physical power system (CPPS) with increasing distributed energy resources [4]. Various distributed methods have been gradually proposed and applied in the real-time economic dispatch [5]-[11]. Especially, consensus-based algorithms have been used widely for their excellent performance. In [12], a distributed leader-follower algorithm for EDP is presented, while a leader agent is required to store the deviation value between power output and load demand. Without a leader node, a two-level strategy is designed in [13]. In [14], with a feasible initialization, the optimal active power output and margin cost can be obtained by the proposed consensus-based algorithm. In [15], a robust distributed power allocation strategy considering cyber attack is proposed. The other relevant results can be found in [16]-[19].

It is worth noting that all the aforementioned distributed consensus-based algorithms are implemented under the perfect communication network. However, in the modern CPPS, the power grid and the cyber network interact with each other closely [

20], [21]. Hence, the communication quality will largely influence the reliability and stability of power systems. In the actual communication network, the information receiving from neighbors is always with time delays induced by the physical distance among them [22]. Therefore, based on the existing distributed consensus-based algorithms, the effects of communication time delays and channel noises are required to be investigated. Meanwhile, it is also necessary to design a novel approach to adapt to the imperfect communication environment. In [23], the effects of delays on the double-level consensus-based algorithm are discussed. According to the case studies presented in [24], it is found that there are some negative results considering time delays, for example, the convergence rate is slower and the results converge to wrong values or even diverge. In [25], a distributed algorithm for EDP on microgrids considering time-varying delays is presented. The distributed continuous-time algorithm proposed in [26] can calculate the optimal economic dispatch result under undirected switching graphs. In [27], the allowable upper bound of time delays is obtained, but only the uniform constant delays are considered and the generation constraints are ignored. Reference [28] analyzes the effects of the heterogeneous time delays on a consensus-based algorithm and obtain the maximum allowable delay theoretically. However, the above-mentioned algorithms only consider the communication delays, while the self-delays are never considered. In fact, a communication unit always has self-delays caused by delayed data measurements and by pre-treating received information. The self-delays can also degrade the performance or even cause the instability of power systems. Only few studies consider EDP with different self-delays and communication delays simultaneously [29], without consideration of the impacts of channel noise, which is also an important factor.

In our previous work [

30], only the homogeneous time delays were investigated. In this paper, the impacts of heterogeneous time delays and channel noises are taken into account. Besides, the cases on time-varying delays, time-varying power demand, and switching topology are also analyzed. The major contributions of this study can be summarized as follows.

1) An effective fully DED strategy is presented for CPPS under non-ideal communication environment. The sum of load demands is calculated in a distributed fashion.

2) The proposed algorithm is suitable for online operation. The global power balance constraint will always be satisfied even with the effect of uncertain delays.

3) The strict allowable range of heterogeneous time delays is derived. Furthermore, the self-delays are also considered in the theoretical results.

4) The effects of time-varying delays and channel noises are investigated considering generator constraints and time-varying power demand. It verifies that the proposed algorithm can be better applied in the actual system under a non-ideal communication environment.

The rest of this paper is organized as follows. (Section II describes the concepts of CPPS, graph theory, EDP, and distributed consensus-based algorithm.) Then, (a fully distributed consensus-based algorithm for solving EDP in the CPPS considering time delays and channel noises is provided in Section III, where the theoretical analysis is also presented.) (Several case studies are given in Section IV to evaluate the effectiveness of the proposed algorithm.) Finally, Section V draws the conclusions.

II. Preliminary

A. Architecture of CPPS

The architecture of CPPS is demonstrated in Fig. 1. Each generator node or load node corresponds to an agent with communication and computing capabilities. Communication links among agents are flexible, which may not be the same as the physical links. In this paper, we mainly investigate the power communication layer in CPPS.

Fig. 1  Architecture of CPPS.

B. Graph Theory

G=(V,E) describes the network topology of the cyber layer in CPPS. V={V1,V2,,Vn} and EV×V denote the communication nodes and the communication links, respectively. The graph G denotes to an undirected graph with no graph loops.

The adjacency matrix A=(aij)n×n describes the connectivity between communication nodes. The diagonal element aii=0, and aij=1 if nodes i and j are connected, otherwise aij=0. Ni={VjV|(Vj,Vi)E} denotes the neighbour nodes of node i. di=jNiaij denotes the degree of node i. The element of the corresponding Laplacian matrix L is given as:

lii=ijaijlij=-aij (1)

C. EDP

The generation cost function is formulated in a quadratic form [

16] as:

Ci(Pi)=(Pi-αi)22βi+γi (2)

where αi0, βi>0, and γi0 are the cost coefficients of generator unit i; and Pi is the active power of generator unit i.

The classical EDP aims at minimizing the total generation cost subject to the power balance constraint and the generator output constraints. Assume there are n generator units, the EDP is formulated by:

mini=1nCi(Pi) (3)

s.t.

iSGPi=jSLPj=Pload (4)
PiminPiPimax    iSG (5)

where Pj is the local load demand of unit j; Pload is the sum of all local demands in a system; SG is the set of all generators; SL is the set of all loads; and Pimin and Pimax are the lower and upper output power limits, respectively.

D. Distributed Consensus-based Algorithm

The distributed consensus-based algorithm optimizes the global objective by using local and neighbor information. In general, the state updating protocol of agent is briefly represented as:

xi(t+1)=xi(t)+j=1Naij(xj(t)-xi(t))=xi(t)+j=1N(-lijxj(t)) (6)

where xi(t) and xj(t) are the states of agents i and j at iteration t, respectively; and N is the number of agents in the communication network.

III. DED for CPPS with Time Delays and Channel Noises

A. CPPS Unit Framework

In CPPS, the security and stability of system operation rely on the coordination and cooperation of each CPPS unit. The structure of a CPPS unit is illustrated in Fig. 2. Communication units can send and receive the information to their neighbor units. Niin denotes the neighbors which send the information to unit i while Niout denotes the neighbors which receive the information from unit i. The information processing unit is important to achieve the global control. On one hand, these units monitor and sample the continuous operating states of physical devices. On the other hand, they process the information of neighbors, calculate the outgoing information, and make the control instructions. In the following, DED is based on this framework.

Fig. 2  Structure of a CPPS unit.

B. Total Power Demand Discovery Algorithm

In power systems, the power demand of each load unit is time-varying in practical, hence, the total power demand Pload is time-varying as well. Note that most existing results use the centralized methods containing central controllers to figure out Pload. How to obtain Pload in a distributed way for each agent is an interesting issue to be addressed. Herein, a fast distributed finite-step algorithm is designed to find Pload.

In most DED strategy, the communication topology connected by generator agents is adopted. Herein, load agents should also be considered in the cyber network. The new bigger communication topology is G˜ and its Laplacian matrix is L˜. Di(k) is defined as the communication state of agent i at step k, iSGSL, and |SG|=n, |SL|=m. The K-step algorithm [

31], [32] is applied to calculate Pload for each agent. The algorithm is given as follows, where wij(k) is the weight coefficient of the link at iteration k.

Algorithm 1  : discovery algorithm of total power demand

Step 1: initialize the communication state Di(0):

    Di(0)=0      iSG,i=1,2,,nPi    iSL,i=n+1,n+2,,n+m

Step 2: compute K different nonzero eigenvalues of L˜, i.e., μ2,μ3,,μK+1

Step 3: if k<K, update communication weight:

    wij(k)=1-diμk+1    j=i1μk+1           jNi0                 otherwise

   Then, compute the state value for each agent:

    Di(k+1)=wii(k)Di(k)+jNiwij(k)Dj(k)    i=1,2,,n+m

Step 4: else if k=K, figure out Pload:

    Pload=(n+m)Di(K)    i=1,2,,n+m

   end if

Step 5: end if

Remark 1   The initialization process is at Step 1 and Step 2. At Step 3, both wij(k) and Di(k+1) are updated at each iteration. The total number of iteration steps is K, which is determined by L˜. There is no need to preload the Laplacian matrix. When the communication topology changes, each agent can update the Laplacian matrix by itself automatically using a novel graph discovery algorithm [

31]. At Step 4, we obtain D1(K)=D2(K)==Dn+m(K)=1n+mi=1n+mDi(0). Using Algorithm 1, each agent obtains Pload in finite steps in a distributed manner.

C. DED with Heterogeneous Time Delays

In classic DED algorithm, the incremental cost for agent i is:

λi=Ci(Pi)Pi=Pi-αiβi    i=1,2,,n (7)

The incremental cost is utilized as the consensus variable. The distributed algorithm under the ideal communication environment is designed as:

λi(t+1)=Pi(t)-αiβi+εj=1n(-lijβi-1λj(t))Pi(t+1)=Pi(t)+ρj=1n(-lijλj(t)) (8)

where ε and ρ are the positive gains for adjusting the convergence speed. The larger ε and ρ are, the faster the convergence speed of the distributed iteration process is. The feasible positive gains are bounded by 0<ρ4/amax and 0<ε3/amax, where amax is the maximum eigenvalue of β-12Lβ-12, β=diag{β1,β2,,βn} [36].

There are two-layer protocols in algorithm (8). Firstly, the incremental cost information is updated by local power information and neighbor incremental cost information. Then, the power information is updated by the local power information of previous iteration as well as the neighbor incremental cost information. To ensure the privacy, only the incremental cost information is needed to exchange, which is different from the algorithm in [

8], [16].

Remark 2   According to the definition of Laplacian matrix, it is easy to get 1TL=0. Thus, i=1nPi(t+1=i=1nPi(t). So the total output power is invariant at each iteration and only determined by the initialization.

Since we have analyzed the case of homogeneous time delays in our previous work, the heterogeneous communication time delays τij and self-delays Tij are considered in this case. Self-delays are caused by delayed relative measurements and computation in the information processing unit shown in Fig. 2. Considering the two types of delays, the distributed algorithm is described as:

i=1nPi0=Pload=(n+m)Di(K) (9)
λi(t+1)=Pi(t)-αiβi+ρβijNiaij[λj(t-τij)-λi(t-Tij)] (10)
Pi(t+1)=Pi(t)+ρjNiaij[λj(t-τij)-λi(t-Tij)] (11)

Herein, the upper bound of the maximum allowable delay is denoted by τ¯, i.e., τijτ¯ and Tijτ¯. In order to describe the topology graph with time delays, we further modify the adjacency matrix, degree matrix with communication delays, and degree matrix with self-delays as Aτ(s)=aije-τijs/di, Dτ(s)=diag1dij=1naije-τijs, DT(s)=diag1dij=1naije-Tijs, respectively.

Based on the algorithms (9)-(11), the allowable range of heterogeneous time delays can be derived by Theorem 1.

Theorem 1   Under the algorithm (9)-(11), λi can converge to the optimal value if τ¯ satisfies:

τ¯<min12ρβi-1di    i=1,2,,n (12)

Proof   Combining (10) and (11), we can obtain:

λi(t+1)=λi(t)+ρβi-1j=1naij[λj(t-τij)-λi(t-Tij)] (14)

Similarly, taking the Laplace transform for (14) yields:

sλi(s)-λi(0)=ρβi-1j=1naije-τijsλj(s)-j=1naije-Tijsλi(s) (15)

Let λ(s)=[λ1(s),λ2(s),,λn(s)]T, we can obtain:

sλ(s)-λ(0)=ρβ-1(Aτ(s)-DT(s))λ(s) (16)

Then, we can obtain:

λ(s)=[sI+diag{ρβi-1di}(DT(s)-Aτ(s))]-1λ(0) (17)

Hence, the characteristic equation can be obtained as:

det{sI+diag{ρβi-1di}(DT(s)-Aτ(s))}=0 (18)

Let Gr(s)=-diag{ρβi-1di/(s+2ρβi-1di)}[2I-(DT(s)-Aτ(s))]. According to (19), we can prove that det{sI+diag{ρβi-1di}(DT(s)-Aτ(s))}=0 and det{I+Gr(s)}=0 have the same roots in the open right-half complex plane.

I+Gr(s)=I-diagρβi-1dis+2ρβi-1di[2I-(DT(s)-Aτ(s))]=diagρβi-1dis+2ρβi-1di(DT(s)-Aτ(s))+diagss+2ρβi-1diI=diag1s+2ρβi-1di[sI+diag{ρβi-1di}(DT(s)-Aτ(s))] (19)

According to [

29], the spectrum of Gr(s) satisfies:

σ(Gr(jω))-Coρβi-1dijω+2ρβi-1diΩ(jωτ¯):i=1,2,,n (20)
Ω(jωτ¯)=Co{2-e-jψ+e-jφ,2-ejψ-e-jφ:ψ,φ[0,ωτ¯] (21)

where Co{} denotes the convex hull; and ω is the angular frequency.

Then, the roots of det{I+Gr(s)}=0 are not in the open right-half complex plane if

-1-Coρβi-1dijω+2ρβi-1diΩ(jωτ¯):i=1,2,,n (22)

i.e.,

Co2+jωρβi-1di-1:i=1,2,,n-1Ω(jωτ¯)= (23)

It is easy to obtain:

Re2+jωρβi-1di=2Im2+jωρβi-1di=ωρβi-1di (24)

Thus, the convex hull of the above formula is a line. Then, we consider the convex hull of Ω(jωτ¯), which contains two cases.

1) Case 1: ωτ¯<π/2.

For convenience, the convex hull of Ω(jωτ¯) in this case and the locus of 2+jω/(ρβi-1di) are illustrated in Fig. 3. In order to ensure that the intersection is an empty set, the imaginary part of 2+jω/(ρβi-1di) must satisfy:

ωρβi-1di>2tanωτ¯2 (25)

Fig. 3  Convex hull of Ω(jωτ¯) in case 1 and locus of 2+jω/(ρβi-1di).

Together with ωτ¯<π/2, we can obtain:

ρβi-1diτ¯<12 (26)

Furthermore, we can obtain:

τ¯<min12ρβi-1di    i=1,2,,n (27)

2) Case 2: π/2<ωτ¯<2π.

Similarly, the convex hull of Ω(jωτ¯) in this case and the locus of 2+jω/(ρβi-1di) are obtained as shown in Fig. 4. Obviously, the intersection is not empty if the imaginary part of 2+jω/(ρβi-1di) is less than 2. Thus, we can obtain:

Fig. 4  Convex hull of Ω(jωτ¯) in case 2 and locus of 2+jω/(ρβi-1di).

ωρβi-1di<2 (28)

Based on ωτ¯>π/2, we can obtain:

ρβi-1diωωτ¯>12π2=π4 (29)

Furthermore, we can obtain:

τ¯>π4ρβi-1di (30)

Thus, it is concluded that the intersection is empty if τ¯ satisfies:

τ¯<minπ4ρβi-1di    i=1,2,,n (31)

According to the analysis of the above two cases, we can finally obtain:

τ¯<minπ4ρβi-1di,12ρβi-1di=min12ρβi-1di    i=1,2,,n (32)

Remark 3   Theorem 1 shows that the delay tolerance of algorithm (9)-(11) is determined by the gain coefficient ρ, the cost coefficient βi, and the degree di. The bigger the degree di is and the stronger-connected the communication topology is, the faster convergence rate is. However, the delay upper bound is smaller, which leads to the poorer robustness against time delays. Therefore, there is a tradeoff between the convergence speed and delay tolerance. In a practical system, since the network topology is usually determined, the gain ρ is designed to adjust the convergence speed and delay tolerance ability according to the actual demand. Besides, we also consider the impact of self-delays, which is more completed compared with the existing results [

27], [28].

D. DED with Time-varying Delays and Channel Noises

Besides time-varying delays and channel noises, the generation output constraints are considered. In algorithms (9)-(11), the gain coefficient ρ is a constant, the results are not convergent due to the impacts of the channel noises. Therefore, we transform the gain coefficient into a time-varying form ρ(t), which satisfies:

t=0ρ(t)=+t=0ρ2(t)<+ (33)

Then, the improved distributed algorithm considering time-varying delays and channel noises can be obtained as:

i=1nPi(0)=Pload=(n+m)Di(K)λi(t+1)=Pi(t)-αiφi(t)+ρ(t)φi(t)j=1n[-lij(t)(λj(t-τ(t))+ηij(t))]Pi(t+1)=Pi(t)+ρ(t)j=1n[-lij(t)(λj(t-τ(t))+ηij(t))] (34)

where ηij(t) is the channel noise in the communication link from unit i to unit j at instant t; and ρ(t)=θ1ln(θ2t+1)/(θ2t+1), in which θ1>0 and θ2>0 are the convergence factors. The expression of φi(t) is omitted here and can be found in [

33].

Considering that the total load demand is time-varying, we have to update Pload at time intervals, which are assumed as 15 min [

34], [35]. The fully DED with time-varying delays and channel noises is presented by Algorithm 2.

Algorithm 2  : fully DED with time-varying delays and channel noises

Step 1: calculate Pload by Algorithm 1, i.e., i=1nPi0=Pload=(n+m)Di(K), and set the initial output power Pi(0) of each generator according to its capacity and operation state

Step 2: calculate the optimal incremental cost λ* and the corresponding output power Pi* by iterations:

     λi(t+1)=Pi(t)-αiφi(t)+ρ(t)φi(t)j=1n[-lij(t)(λj(t-τ(t))+ηij(t))]

     Pi(t+1)=Pi(t)+ρ(t)j=1n[-lij(t)(λj(t-τ(t))+ηij(t))]

Step 3: every other 15 min, update the new total power demand Pload' by Algorithm 1

Step 4: if ΔP=Pload'-Pload0, allocate the mismatch power ΔP by the virtual command node

   Then, go to Step 2

   else if ΔP=0, go to Step 3

   end if

Step 5: end if

Remark 4   This algorithm is a fully DED implementation. All the state data are calculated by information exchange among their neighbors. Firstly, Pload is figured out using Algorithm 1 under the communication graph G˜ which contains generator units and load units. Then, the optimal dispatch results are calculated under the communication graph G which contains generator units only. That is to say, load units merely participate in network communication when updating the Pload. Thus, the communication burden is alleviated effectively by this algorithm. The result is recalculated every 15 min, so it means the cyber network just needs to switch every 15 min. The switch frequency is acceptable in the future CPPS. Besides, Algorithm 2 is robust to channel noises, which are ignored in the existing results [

27], [28].

IV. Case Studies

The case studies are simulated in the MATLAB R2018a environment on a laptop with Intel Core i5-7300U CPU @ 2.60 GHz and 8 GB RAM. The IEEE 14-bus system containing 5 generator units and 9 load units is utilized. The communication topology of the 14-bus CPPS is presented in Fig. 5.

Fig. 5  Communication topology of 14-bus CPPS.

G contains five schedulable generator units only, while G˜ includes all units. For graph G, the sets of neighbours of each unit are denoted as NG1={G2,G5}, NG2={G1,G3,G4}, NG3={G2,G4}, NG4={G2,G3}, NG5={G1} and the adjacency matrix A and the Laplacian matrix L are given as:

A=0100110110010100110010000 (35)
L=2-100-1-13-1-100-12-100-1-120-10001 (36)

In the testing power grid, the corresponding parameters of cost functions and capacity constraints of each generator are provided in Table I [

33].

Table I  Parameters of Cost Functions and Capacity Constraints of Each Generator
Generatorαi (MW)βi (MW2h/$)γi ($/h)Pimin (MW)Pimax (MW)
G1 -2535.2 352.1 -8616.8 150 500
G2 -2535.2 352.1 -8616.8 150 500
G3 -2023.2 257.7 -7631.0 100 400
G4 -826.8 103.7 -3216.7 50 200
G5 -2023.2 257.7 -7631.0 100 400

A. Total Power Demand Calculation

The power demand of each load is shown in Table II. By using Algorithm 1, we can obtain the average power demand in 13 steps. The iteration process is shown in Fig. 6. The final average value is 107.1429 MW.

Table II  Power Demand of Each Load
LoadDemand (MW)LoadDemand (MW)
L1 100 L6 80
L2 240 L7 330
L3 60 L8 50
L4 120 L9 120
L5 400

Fig. 6  Power demand calculation results using Algorithm 1.

Consequently, the total power demand Pload=14×107.1429=1500 MW. Assume that the initial output power of each generator is set as P1(0)=400 MW, P2(0)=300 MW, P3(0)=300 MW, P4(0)=150 MW, P5(0)=350 MW. For convenience, the initial values are used in all the following case studies.

B. Case Study 1

The heterogeneous time delays including self-delays are considered. Let ρ=5, we obtain the allowable delay upper bound τ¯=5.1850 s by Theorem 2. The elements in the communication delay matrix τ and self-delay matrix T are determined randomly within the allowable range, as shown in (37) and (38), respectively.

τ=0500440350020500350040000 (37)
T=0400450230030500550040000 (38)

As shown in Fig. 7(a) and (b), under the graph G, the optimal output power results of each generator are obtained as P1opt=509.7 MW, P2opt=509.7 MW, P3opt=205.3 MW, P4opt=70.0 MW, P5opt=205.3 MW, respectively. Meanwhile, the incremental costs converge at about 200 s and the optimal incremental cost λopt=8.6478. Then, the graph G is changed into a more strongly connected topology graph G', which is illustrated in Fig. 8. By Theorem 1, the allowable delay upper bound is reduced to τ¯'=2.5925 s. As shown in Fig. 7(c) and (d), under the graph G', the incremental cost does not achieve consensus so that the output power cannot converge. The algebraic connectivities of the graph G and the graph G' are λ2(G)=0.5188 and λ2(G')=5, respectively. Since the graph G' is more stronger-connected, the system may have a better convergence rate. However, the delay tolerance decreases. Therefore, there exists a trade-off mentioned in Section III.

Fig. 7  Simulation results under topology graphs G and G'. (a) Output power under graph G. (b) Incremental costs under graph G. (c) Output power under graph G'. (d) Incremental costs under graph G'.

Fig. 8  Strongly connected topology graph G'.

C. Case Study 2

In this case study, the generator constraints, the time-varying power demand, and the time-varying delays are considered. These delays are all assumed to be bounded by τ=6. The tested probability distribution of these delays is given in Table III.

Table III  Probability Distributions of Time Delays
DelayProbability distribution
Scenario 1Scenario 2
0 0.05 0.05
1 0.35 0.05
2 0.35 0.15
3 0.10 0.15
4 0.05 0.30
5 0.05 0.25
6 0.05 0.05

The noises are assumed randomly within the range [-0.2,0.2], and in this case ρ(t)=10ln(0.1t+1)/(0.1t+1). The total power demand is 1500 MW at first, then G1 and G2 increase the local demand 50 MW, respectively. The total power demand is updated to 1600 MW at 900 s. Similarly, the total power demand is updated to 1700 MW at 1800 s and 1400 MW at 2700 s, respectively. From Fig. 9, it can be observed that the proposed algorithm can keep fast convergence during the four processes.

Fig. 9  Simulation results with generation constraints considering time-varying delays of scenario 1 and channel noises. (a) Output power. (b) Incremental costs. (c) Total power supply and demand.

During the first 15 min (process 1), the optimal output power results of each generator are P1opt=500.0 MW, P2opt=500.0 MW, P3opt=213.4 MW, P4opt=73.2 MW, P5opt=213.4 MW. The optimal incremental cost λopt=8.68. During the second 15 min (process 2), the optimal output power results of each generator are P1opt=500.0 MW, P2opt=500.0 MW, P3opt=255.0 MW, P4opt=90.0 MW, P5opt=255.0 MW, and λopt=8.84. During the third 15 min (process 3), the optimal output power results of each generator are P1opt=500.0 MW, P2opt=500.0 MW, P3opt=296.6 MW, P4opt=106.7 MW, P5opt=296.6 MW, and λopt=9.00. During the fourth 15 min (process 4), the optimal output power results of each generator are P1opt=483.1 MW, P2opt=483.1 MW, P3opt=185.9 MW, P4opt=62.1 MW, P5opt=185.9 MW, and λopt=8.57.

In scenario 2, the probabilities of τ=4 and τ=5 are set to be higher, which affect the stability of the system more seriously. Figure 10 shows that the optimal results are the same as those in scenario 1. Therefore, if the time-varying delays are within the allowable range, Algorithm 2 can maintain good convergence. It verifies that Algorithm 2 has a good robustness to the time delays and it can adapt to the non-ideal communication environment. In addition, the total power demand always equals to the total output power, while the algorithm proposed in [

28] cannot ensure the real-time power balance.

Fig. 10  Simulation results with generation constraints considering time-varying delays of scenario 2 and channel noises. (a) Output power. (b) Incremental costs. (c) Total power supply and demand.

D. Case Study 3

We further test the effectiveness of the proposed algorithm under a more fragile communication condition, i.e., the switching topology. The time delay values are adopted from Table III. During the first 200 s, the disconnected topology A shown in Fig. 11 is used for communication.

Fig. 11  Switching topology graph.

Then, the topology switches between topology A and topology B every 10 s. The simulation results with switching topology and time-varying delays are shown in Fig. 12. During the first 200 s, the communication topologies are two separate parts so that the results converge to different optimal values. After 200 s, under the switching topology, there exists slight fluctuation during the convergence process because of the effects of time delays. Nevertheless, the whole system finally converges. In addition, no matter under what kind of communication topology, the supply-demand balance of the whole system is not broken. Hence, the proposed algorithm has a good robustness against time delays and adapts to the switching topology.

Fig. 12  Simulation results with switching topology and time-varying delays. (a) Output power. (b) Incremental costs. (c) Total power supply and demand.

E. Case Study 4

We further apply the proposed algorithm to a larger system, i.e., IEEE 118-bus system, which is composed by 54 dispatchable generators. Without loss of generality, the cost coefficients of all generators are chosen by the following ranges [

33]:

αi[-65,-25]i=1,2,,54βi[12,16]i=1,2,,54 (34)

The circle topology is used in this case study and several additional edges are added to increase the convergence rate appropriately. According to the determined communication topology, the feasible gain coefficient is calculated as 0<ρ<3.58. In this case, ρ(t)=0.3ln(0.1t+1)/(0.1t+1). The time-varying delays and channel noises are bounded like case study 3. The initial output power of each generator is set to be 50 MW. The total load demand is 2700 MW. From Fig. 13, it is verified that the proposed algorithm has a good robustness to the time delays and can well adapt to the non-ideal communication environment in a large system.

Fig. 13  Simulation results with time-varying delays and channel noises in IEEE 118-bus system. (a) Output power. (b) Incremental costs. (c) Total power supply and demand.

V. Conclusion

Considering the non-ideal communication environment, a fully DED algorithm is proposed for CPPS. The heterogeneous communication delays and self-delays are investigated. The allowable delay upper bound can be figured out under a specified communication topology. Furthermore, the algorithm performance under the time-varying delays and channel noises are discussed simultaneously considering the generator constraints, time-varying power demand, and switching topology by several case studies.

In a practical system, if delays and channel noises have been approximately estimated in advance, based on the theoretical results of Theorem 1, we can appropriately adjust the gain coefficient and the communication topology to satisfy the need for convergence speed and delay tolerance. Therefore, our work is valuable for engineering practice. In our future work, the cyber attack in CPPS will be investigated. And we will further focus on the CPPS modeling and the application of distributed techniques.

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