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Distributed fixed step-size algorithm for dynamic economic dispatch with power flow limits

Abstract

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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Abbreviations

ai, bi, ci :

Distributed generator cost factor.

P i,h :

Power out of the distributed generator i at time slot h.

D j,h :

Load demand of the user i at time slot h.

D j,h :

Renewable energy output of the user in node i at time slot h.

\(P_i^M\) :

Maximum output of the distributed generator i.

\(P_i^m\) :

Minimum output of the distributed generator i.

\(D_i^M\) :

Maximum value of the load demand i.

\(D_i^m\) :

Minimum value of the load demand i.

\(P_i^R\) :

Ramp-rate constraint parameter of the generator i.

T l :

Transmission line power flow limits of the line l.

\(T_l^N\) :

Vector comprising the Tl, and \(T_l^N = \left({1/N} \right){T_l}\).

B :

Admittance matrix of the power grid.

W :

Reduced incidence matrix, and W M × (N1).

E :

Matrix of distribution factors.

E i :

Vector comprising the ith column elements of the distribution matrix E.

L :

Laplacian matrix of the communication network, and L ∈ ℝN × N.

L M :

Expanded Laplacian matrix, with LM = LIM ∈ ℝMN × MN.

⨂:

Kronecker product.

I M :

M × M unit matrix.

H :

Total number of time slots involved in scheduling.

M :

Number of transmission lines.

N :

Number of the node, and N = NP + ND.

N P :

Number of distributed generations.

N R :

Number of renewable generations, and NRND.

N D :

Number of users.

\({{\cal N}_P}\) :

Set of the generator and \({{\cal N}_P} = \left[{1, \ldots ,{N_P}} \right]\).

\({{\cal N}_R}\) :

Set of the generator and \({{\cal N}_R} = \left[{1, \ldots ,{N_R}} \right]\).

\({{\cal N}_D}\) :

Set of the user and \({{\cal N}_D} = \left[{1, \ldots ,{N_D}} \right]\).

\({\cal H}\) :

Index set of h, and \({\cal H} = \left\{{1, \ldots, H} \right\}\).

μ i, h :

Lagrangian multiplier of the upper ramp-rate constraint.

ν i, h :

Lagrangian multiplier of the lower ramp-rate constraint.

θ i, h :

Lagrangian multiplier of the lower transmission line loss.

γ i, h :

Lagrangian multiplier of the upper transmission line loss.

\({\cal L}\) :

Lagrangian function of the converted primal problem.

λ h :

Vector form of Lagrangian multiplier λi, h, and λ = [λ1, h, …, λN, h] ∈ ℝN.

θ h :

Vector form of Lagrangian multiplier θi, h, and \({\theta _h} = \left[{\theta _{1,h}^T, \cdots \theta _{N,h}^T} \right] \in \mathbb{R}{^{M\,N}}\).

ξ h :

Vector form of Lagrangian multiplier ξi, h, and \({\xi _h} = \left[{\xi _{1,h}^T, \cdots \xi _{N,h}^T} \right] \in \mathbb{R}{^{M\,N}}\).

γ h :

Vector form of Lagrangian multiplier γi, h, and \(\gamma = \left[{\gamma _{1,}^T, \cdots \gamma _{N,h}^T} \right] \in \mathbb{R}{^{M\,N}}\).

ζ h :

Vector form of Lagrangian multiplier ζi, h, and \({\zeta _h} = \left[{\zeta _{1,}^{\rm{T}}, \cdots \zeta _{N,h}^{\rm{T}}} \right] \in \mathbb{R}{^{M\,N}}\).

β i :

Transmission line loss coefficient.

NΩ(Pi, h):

Normal cone of the constraint set of the generator node i. \({N_{\rm{\Omega}}}\left({{P_{i,h}}} \right) = \left\{{{z_i}\left| {{z_i}\left({{P_i} - P_i^\prime} \right) \leqslant 0,\forall {P_i},P_i^\prime \in {{\rm{\Omega}}_{{P_{i,h}}}}} \right.} \right\}\).

NΩ(Di, h):

Normal cone of the constraint set of the user i. \({N_{\rm{\Omega}}}\left({{D_{i,h}}} \right) = \left\{{{z_i}\left| {{z_i}} \right.\left({{D_i} - D_i^\prime} \right) \leqslant 0,\forall {D_i},D_i^\prime \in \,{{\rm{\Omega}}_{{D_{i,h}}}}} \right\}\).

ρ i, h :

Node injection active power of node i at time slot h. \(\forall i \in {{\cal N}_D},{\rho _{i,h}} = {P_{i,h}} - {\beta _i}P_{i,h}^2;\,\forall i \in {{\cal N}_D},{\rho _{i,h}} = {R_{i,h}} - {D_{i,h}}\).

ρ h :

The vector form of node injection active power, and ρh = [ρ1 h, …, ρN, h].

ο:

Hadamard product.

υi, ωi :

Users’ utility function parameters.

λ i, h :

Lagrangian multiplier of the supply and demand balance constraints.

φ i, h :

Lagrangian multiplier of the consensus constraint h = 0.

φ h :

The vector form of Lagrangian multiplier φi, h and φh = [φ1, h,…, φN, h].

ξ i, h :

Lagrangian multiplier of the consensus constraint LMθh = 0.

ζ i, h :

Lagrangian multiplier of the consensus constraint LMγh = 0.

Π :

Projection operator, and ΠΩ (x) = min‖x − y‖, ∀y ∈ Ω.

\({{\rm{\Omega}}_{{P_{i,h}}}}\) :

Feasible set of \({P_{i,h}},\,{{\rm{\Omega}}_{{P_{i,h}}}} = \left\{{{P_{i,h}}\left| {P_i^m} \right. \leqslant {P_{i,h}} \le P_i^M} \right\}\).

\({{\rm{\Omega}}_{{D_{i,h}}}}\) :

Feasible set of \({D_{i,h}},\,{{\rm{\Omega}}_{{D_{i,h}}}} = \left\{{{D_h}\left| {D_i^m} \right. \leqslant {D_{i,h}} \le D_i^M} \right\}\).

ε P :

Pre-set algorithm convergence tolerance threshold.

α :

Algorithm iteration step size.

\(\mathbb{R}_ + ^N\) :

N-dimension positive space.

\({{\cal N}_i}\) :

Neighbor set of the agent i.

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Correspondence to Wenwu Yu.

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Wang, K., Fu, Z., Xu, Q. et al. Distributed fixed step-size algorithm for dynamic economic dispatch with power flow limits. Sci. China Inf. Sci. 64, 112202 (2021). https://doi.org/10.1007/s11432-019-2638-2

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  • DOI: https://doi.org/10.1007/s11432-019-2638-2

Keywords

  • dynamic economic dispatch
  • distributed optimization
  • discrete-time algorithm
  • active power flow limit
  • transmission line loss