SCIENCE CHINA Information Sciences, Volume 61, Issue 1: 012204(2018) https://doi.org/10.1007/s11432-016-9114-y

## Economic power dispatch in smart grids: a framework for distributed optimization and consensus dynamics

Wenwu YU1,5,*, Jinhu LV3,4
• AcceptedApr 20, 2017
• PublishedAug 29, 2017
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### Abstract

By using the distributed consensus theory in multi-agentsystems, the strategy of economic power dispatch is studied in a smartgrid, where many generation units work cooperatively to achieve an optimal solution in a local area. The relationship between thedistributed optimization solution and consensus in multi-agentsystems is first revealed in this paper, which can serve as a general framework forfuture studies of this topic. First, without the constraints ofcapacity limitations, it is found that the total cost forall the generators in a smart grid can achieve the minimal value if theconsensus can be reached for the incremental cost of all the generation unitsand the balance between the supply and demand of powers iskept. Then, by designing a distributed consensus control protocol inmulti-agent systems with appropriateinitial conditions, incremental cost consensus can be realized and the balance for the powers can also besatisfied. Furthermore, the difficult problem for distributedoptimization of the total cost function with bounded capacitylimitations is also discussed. A reformulated barrier function isproposed to simplify the analysis, under which the total cost canreach the minimal value if consensus can be achieved for themodified incremental cost with some appropriate initial values.Thus, the distributed optimization problems for the costfunction of all generation units with and without bounded capacitylimitations can both be solved by using the idea of consensus inmulti-agent systems, whose theoretical analysis is still lackingnowadays. Finally, some simulation examples are given toverify the effectiveness of the results in this paper.

### Acknowledgment

This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB0800401), National Natural Science Foundation of China (Grant Nos. 61673107, 61673104, 61621003, 61532020), National Ten Thousand Talent Program for Young Top-notch Talents, Cheung Kong Scholars Programme of China for Young Scholars, Six Talent Peaks of Jiangsu Province of China (Grant No. 2014-DZXX-004), and Fundamental Research Funds for the Central Universities of China (Grant No. 2242016K41058).

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• Figure 1

Illustration of networks with pinning control.

• Figure 2

(Color online) (a) The incremental cost of all three generation units as well as (b) their corresponding output powers in Table tab1.

• Figure 3

(Color online) (a) The incremental cost of all three generation units as well as (b) their corresponding output powers at power demands 650 MWh and 850 MWh.

• Figure 4

(Color online) (a) The incremental cost of all generation units as well as (b) their corresponding output powers in a scale-free network structure.

• Figure 5

(Color online) (a) The modified incremental cost of all three generation units with capacity limitations as well as (b) their corresponding output powers in Table 3.

• Figure 6

The diagram of IEEE 30-bus.

• Figure 7

The communication topology of IEEE 30-bus.

• Figure 8

(Color online) (a) The states for the modified incremental cost of all generation units with capacity limitations and (b) their corresponding output powers in Table 4for IEEE 30-bus test system.

• Table 1   Parameters of three-unit system
 Unit $\alpha_i$ $\beta_i$ $\gamma_i$ $P_{Gi}(0)$ 1 561 7.92 0.001562 300 2 310 7.85 0.00194 250 3 78 7.8 0.00482 100
• Table 2   Parameters of 100-unit system in a scale-free network
 Unit $\alpha_i$ $\beta_i$ $\gamma_i$ $P_{Gi}(0)$ $i$ [100,~500] [7.5,~8] [0.001,~0.004] [175,~225]
• Table 3   Parameters of three-unit system
 Unit $P_i^m$ $P_i^M$ $P_{Gi}(0)$ 1 250 300 260 2 200 300 220 3 150 200 170
• Table 4   Parameters of generations in IEEE 30-bus
 Unit (Generator No.) $\alpha_i$ $\beta_i$ $\gamma_i$ $P_{\min}$ $P_{\max}$ 1 (1) 0 2.00 0.00375 50 200 2 (2) 0 1.75 0.01750 20 80 3 (5) 0 1.00 0.06250 15 50 4 (8) 0 3.25 0.00834 10 35 5 (11) 0 3.00 0.02500 10 30 6 (13) 0 3.00 0.02500 12 40

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