In this study, a discrete-time distributed algorithm is proposed for solving the dynamic economic dispatch problem with active power flow limits and transmission line loss. To avoid the communication burden and implement the algorithm in a favorably distributed manner, the splitting method is used to bypass the centralized updating of the algorithm's parameters, which is unavoidable when implementing conventional Lagrangian methods. The use of a fixed step-size and distributed update enhances the applicability of the algorithm. The performance and effectiveness of the proposed distributed algorithm are verified via numerical studies on the IEEE 14-bus system.
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Figure 1
(Color online) Optimal dispatch of each generator.
Figure 2
(Color online) (a) Structure chart of IEEE 14-bus; (b) topology of IEEE 14-bus.
Figure 3
(Color online) (a) Power output of the distributed generators; (b) load demand of the users.
Figure 4
(Color online) (a) Mismatch of entire grid and (b) the incremental cost of each generation in five time slots.
Figure 5
(Color online) Values of (a) $\mu_{i,h}$ and (b) $\nu_{i,h}$ in five time slots.
Figure 6
(Color online) Consensus values of the Lagrangian multipliers $\theta_i$ (a) and $\gamma_i$ (b) of the power flow constraints.
Figure 7
(Color online) Active power flow of the transmission lines.
| Parameter | Generator number | ||||
| 1 | 3 | 4 | 13 | 14 | |
| $a_i$ | 0.08 | 0.062 | 0.075 | 0.072 | 0.066 |
| $b_i$ | 2.25 | 4.2 | 3.25 | 6.25 | 3.2 |
| $c_i$ | 23 | 12 | 23 | 14 | 19 |
| $P_i^m$ | 20 | 10 | 20 | 9 | 5 |
| $P_i^M$ | 80 | 80 | 95 | 70 | 80 |
| $P_i^R$ | 10 | 10 | 10 | 8 | 7 |
| $\beta_i$ | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |
| User number | Time 1 | Time 2 | Time 3 | Time 4 | Time 5 |
| 7 | 10 | 15 | 6 | 13 | 6 |
| 8 | 5 | 12 | 5 | 15 | 10 |
| 10 | 4 | 17 | 3 | 12 | 18 |
| 11 | 15 | 32 | 10 | 20 | 13 |
| User number | $\omega_i$ | $\upsilon_i$ | $d_i^m$ | $d_i^M$ |
| 2 | 0.06 | 15.12 | 30 | 60 |
| 5 | 0.082 | 11.984 | 15 | 39 |
| 6 | 0.065 | 12.992 | 15 | 35 |
| 7 | 0.062 | 13.216 | 15 | 37 |
| 8 | 0.066 | 12.04 | 15 | 35 |
| 9 | 0.071 | 15.904 | 20 | 58 |
| 10 | 0.062 | 12.544 | 10 | 25 |
| 11 | 0.075 | 13.3 | 15 | 30 |
| 12 | 0.076 | 12.572 | 18 | 32 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
| From | 1 | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 7 | 7 | 9 | 9 | 10 | 12 | 13 |
| To | 2 | 5 | 3 | 4 | 5 | 4 | 5 | 7 | 9 | 6 | 11 | 12 | 13 | 8 | 9 | 10 | 14 | 11 | 13 | 14 |
| $T_l$ | 54 | 66 | 36 | 42 | 69 | 96 | 28 | 48 | 36 | 36 | 46.8 | 36 | 36 | 36 | 54 | 42 | 54 | 25.2 | 46.8 | 72 |
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