We then proposed an algorithm which finds a feasible point by starting from the dual-based optimum solution and searching among the dual variables of the rate constraints.
The calculation method of electric nodal prices can be found in [32], while the natural gas nodal prices are determined by the dual variables of the equality constraint (36) at the optimal solution point of the optimization model.
The pricing problem's objective function is the reduced cost function of the RMP with respect to server type t and ( {u}_{r,{n}_h}^t ) coefficients denote the values of the dual variables of the RMP for type t server.
After the commitment decisions are made, the model calculates the optimal dual variable of demand and generation balance constraint as the energy or imbalance price.
Using the optimal dual variables of energy constraints, the sub problem is brought up to check whether current solution is optimal, if not, solving the sub problem will generate a new pattern that will further improve current solution.
Gutman, Robust redundancy determination and evaluation of the dual variables of linear programming problems in the presence of uncertainty, 1, in, V. Kucera, M. Sebek (Eds)., Proceedings of 3rd IFAC Symposium on Robust Control Design (ROCOND 2000), IFAC, Prague, Czech Republic, Elsevier Science, Amsterdam, 2000, paper 115] is essentially used.
It is in particular shown that the time-varying optimal routing problem can be solved by applying an internal model controller to the dual variables of a certain convex network optimization problem.
Where Ρ S represents the projection operator onto the constraint set S. Finally, update the scaled dual variables of ADMM {left{begin{array}{l}{mathbf{u}}_1^{(t)}={mathbf{u}}_1^{left t-1right)}+gamma left(mathbf{F}{mathbf{x}}^{(t)}-{mathbf{z}}^{(t)}right) {mathbf{u}}_2^{(t)}={mathbf{u}}_2^{left t-1right)}+gamma left({mathbf{y}}^{(t)}-{mathbf{x}}^{(t)}right)end{array}right.}_ (23).
where λ=[λ1,…2,…,λ m ] T is the vector of dual variables for the constraints.
The corresponding dual variables for the constraints are shown in the rightmost column of Problem (44).
