Augmented Lagrange Hopfield Network for Combined Economic and Emission Dispatch with Fuel Constraint

Nomenclature

• a_i, b_i, c_i emission coefficients for thermal unit i

• d_i, e_f, f_i fuel cost coefficients for thermal unit i

• B_ij, B_0i, B₀₀ transmission loss formula coefficients

• F_ik fuel delivery for thermal unit i during subinterval k, in tons

• F_i^min, F_i^max lower and upper fuel delivery limits for thermal unit i, in tons

• M number of subintervals of scheduled period

• N total number of thermal units;

• P_Dk load demand of the system during subinterval k, in MW

• P_Lk transmission loss of the system during subinterval k, in MW

• P_ik output power of thermal unit i during subinterval k, in MW

• P_i^min, P_i^max lower and upper generation limits of thermal unit i, in MW

• Q_ik fuel consumption function of thermal unit i in subinterval k, in tons/h

• t_k duration of subinterval k, in hours

• X_i0 initial fuel storage for unit i, in tons

• X_ik fuel storage for unit i during subinterval k, in tons

• X_i^min, X_i^max lower and upper fuel storage limits for thermal unit i, in tons

• V_p,ik output of continuous neuron representing for output power P_ik

• V_f,ik output of continuous neuron representing for fuel delivery F_ik

• V_x,ik output of continuous neuron representing for fuel storage X_ik

• V_λ,k output of multiplier neuron associated with power balance constraint

• V_γ,k output of multiplier neuron associated with fuel delivery constraint

• V_η,ik output of multiplier neuron associated with fuel storage constraint

• U_p,ik, U_f,ik, U_x,ik inputs of continuous neurons corresponding to the outputs V_p,ik, V_f,ik and V_x,ik, respectively

• U_λ,k, U_γ,k, U_η,ik inputs of multiplier neurons corresponding to the outputs V_λ,k, V_γ,k and V_η,ik, respectively

• ΔP_k power balance constraint error in sub-interval k, in MW

• ΔF_k fuel delivery constraint error in subinterval k, in tons

• ΔX_ik fuel storage constraint error for unit i during subinterval k, in tons

• ΔV_p,ik, ΔV_f,ik, ΔV_x,ik iterative errors of continuous neurons

• σ slope of sigmoid function of continuous neurons

• α_p, α_f, α_x updating step sizes for continuous neurons

• α_λ, α_γ, α_η updating step sizes for multiplier neurons

• φ_0i, φ_1i, φ_2i fuel consumption coefficients for thermal unit i

• λ_k, γ_k, η_ik Lagrangian multipliers associated with power balance, fuel delivery and fuel storage constraints, respectively

• β_λ,k, β_γ,k, β_η,sk penalty factors associated with power balance, fuel delivery and fuel storage constraints, respectively