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.
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 [
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 [
In our previous work [
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.
The architecture of CPPS is demonstrated in

Fig. 1 Architecture of CPPS.
describes the network topology of the cyber layer in CPPS. and denote the communication nodes and the communication links, respectively. The graph denotes to an undirected graph with no graph loops.
The adjacency matrix describes the connectivity between communication nodes. The diagonal element , and if nodes and are connected, otherwise . denotes the neighbour nodes of node . denotes the degree of node . The element of the corresponding Laplacian matrix is given as:
(1) |
The generation cost function is formulated in a quadratic form [
(2) |
where , , and are the cost coefficients of generator unit ; and is the active power of generator unit .
The classical EDP aims at minimizing the total generation cost subject to the power balance constraint and the generator output constraints. Assume there are generator units, the EDP is formulated by:
(3) |
s.t.
(4) |
(5) |
where is the local load demand of unit ; is the sum of all local demands in a system; is the set of all generators; is the set of all loads; and and are the lower and upper output power limits, respectively.
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:
(6) |
where and are the states of agents and j at iteration , respectively; and is the number of agents in the communication network.
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 Structure of a CPPS unit.
In power systems, the power demand of each load unit is time-varying in practical, hence, the total power demand is time-varying as well. Note that most existing results use the centralized methods containing central controllers to figure out . How to obtain 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 .
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 and its Laplacian matrix is . is defined as the communication state of agent at step , , and , . The -step algorithm [
Remark 1 The initialization process is at Step 1 and Step 2. At Step 3, both and are updated at each iteration. The total number of iteration steps is , which is determined by . 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 [
In classic DED algorithm, the incremental cost for agent is:
(7) |
The incremental cost is utilized as the consensus variable. The distributed algorithm under the ideal communication environment is designed as:
(8) |
where and are the positive gains for adjusting the convergence speed. The larger and are, the faster the convergence speed of the distributed iteration process is. The feasible positive gains are bounded by and , where is the maximum eigenvalue of , [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 [
Remark 2 According to the definition of Laplacian matrix, it is easy to get . Thus, . 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 and self-delays are considered in this case. Self-delays are caused by delayed relative measurements and computation in the information processing unit shown in
(9) |
(10) |
(11) |
Herein, the upper bound of the maximum allowable delay is denoted by , i.e., and . 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 , , , 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), can converge to the optimal value if satisfies:
(12) |
Proof Combining (10) and (11), we can obtain:
(14) |
Similarly, taking the Laplace transform for (14) yields:
(15) |
Let , we can obtain:
(16) |
Then, we can obtain:
(17) |
Hence, the characteristic equation can be obtained as:
(18) |
Let . According to (19), we can prove that and have the same roots in the open right-half complex plane.
(19) |
According to [
(20) |
(21) |
where denotes the convex hull; and is the angular frequency.
Then, the roots of are not in the open right-half complex plane if
(22) |
i.e.,
(23) |
It is easy to obtain:
(24) |
Thus, the convex hull of the above formula is a line. Then, we consider the convex hull of , which contains two cases.
1) Case 1: .
For convenience, the convex hull of in this case and the locus of are illustrated in
(25) |

Fig. 3 Convex hull of in case 1 and locus of .
Together with , we can obtain:
(26) |
Furthermore, we can obtain:
(27) |
2) Case 2: .
Similarly, the convex hull of in this case and the locus of are obtained as shown in

Fig. 4 Convex hull of in case 2 and locus of .
(28) |
Based on , we can obtain:
(29) |
Furthermore, we can obtain:
(30) |
Thus, it is concluded that the intersection is empty if satisfies:
(31) |
According to the analysis of the above two cases, we can finally obtain:
(32) |
Remark 3 Theorem 1 shows that the delay tolerance of algorithm (9)-(11) is determined by the gain coefficient , the cost coefficient , and the degree . The bigger the degree 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 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 [
Besides time-varying delays and channel noises, the generation output constraints are considered. In algorithms (9)-(11), the gain coefficient 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 , which satisfies:
(33) |
Then, the improved distributed algorithm considering time-varying delays and channel noises can be obtained as:
(34) |
where is the channel noise in the communication link from unit to unit at instant ; and , in which and are the convergence factors. The expression of is omitted here and can be found in [
Considering that the total load demand is time-varying, we have to update at time intervals, which are assumed as 15 min [
Remark 4 This algorithm is a fully DED implementation. All the state data are calculated by information exchange among their neighbors. Firstly, is figured out using
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 Communication topology of 14-bus CPPS.
contains five schedulable generator units only, while includes all units. For graph , the sets of neighbours of each unit are denoted as , , , , and the adjacency matrix and the Laplacian matrix are given as:
(35) |
(36) |
In the testing power grid, the corresponding parameters of cost functions and capacity constraints of each generator are provided in
The power demand of each load is shown in

Fig. 6 Power demand calculation results using Algorithm 1.
Consequently, the total power demand MW. Assume that the initial output power of each generator is set as MW, MW, MW, MW, MW. For convenience, the initial values are used in all the following case studies.
The heterogeneous time delays including self-delays are considered. Let , we obtain the allowable delay upper bound s by Theorem 2. The elements in the communication delay matrix and self-delay matrix are determined randomly within the allowable range, as shown in (37) and (38), respectively.
(37) |
(38) |
As shown in

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

Fig. 8 Strongly connected topology graph .
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 . The tested probability distribution of these delays is given in
The noises are assumed randomly within the range , and in this case . The total power demand is MW at first, then G1 and G2 increase the local demand MW, respectively. The total power demand is updated to MW at 900 s. Similarly, the total power demand is updated to MW at 1800 s and MW at 2700 s, respectively. From

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 MW, MW, MW, MW, MW. The optimal incremental cost . During the second 15 min (process 2), the optimal output power results of each generator are MW, MW, MW, MW, MW, and . During the third 15 min (process 3), the optimal output power results of each generator are MW, MW, MW, MW, MW, and . During the fourth 15 min (process 4), the optimal output power results of each generator are MW, MW, MW, MW, MW, and .
In scenario 2, the probabilities of and are set to be higher, which affect the stability of the system more seriously.

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.
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

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 Simulation results with switching topology and time-varying delays. (a) Output power. (b) Incremental costs. (c) Total power supply and demand.
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 [
(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 . In this case, . The time-varying delays and channel noises are bounded like case study 3. The initial output power of each generator is set to be MW. The total load demand is MW. From

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.
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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