The thesis is structured as follows.
The first part of the thesis is an introduction to multiparametric program-ming. In this thesis, parametric programming is the main technique used to characterize the impact of system uncertainty on the real-time operation in the electric grid. Specifically, a parametric framework is proposed for economic dispatch in which the system random variables are formulated as parameters.
The optimal solution structures are explicitly exploited as algebraic functions of random variables. This parametric framework and optimal solution structures constitute the basic tools for operations under uncertainty in the electric grid.
In the second part of the thesis we focus on probabilistic forecasting of real-time LMP and network congestion. A new probabilistic forecasting technique is
proposed based on a multiparametric programming formulation that partitions the uncertainty parameter space into critical regions from which the conditional probability distribution of the real-time LMP/congestion is obtained. The pro-posed method incorporates load/generation forecast, time varying operation constraints, and contingency models. By shifting the computation associated with multiparametric programs offline, the online computational cost is signif-icantly reduced. An online forecasting technique by generating critical regions dynamically is also proposed, which results in several orders of magnitude im-provement in the computational cost over standard Monte Carlo methods.
In the third part of the thesis we focus on the multi-area interchange schedul-ing under uncertainty. An interchange schedulschedul-ing technique based on a two-stage stochastic minimization of expected operating cost is proposed. The mul-tiparametric framework is used in the second stage by formulating interchange and random load and generation as parameters. To solve the stochastic opti-mization for optimal interchange, an equivalent problem that maximizes the expected social welfare is formulated for each separate scheduling interface.
The proposed technique leverages the operator’s capability of forecasting lo-cational marginal prices and obtains the optimal interchange schedule without iterations among operators.
For the scheduling problem in a multi-area multi-interface system, an itera-tive algorithm is also proposed based on the coordinate descent method. In par-ticular, the proposed algorithm iteratively optimizes the interface flows using a multidimensional demand and supply functions. Optimality and convergence are guaranteed for both synchronous and asynchronous scheduling under
nom-inal assumptions.
We conclude this thesis in the last chapter.
CHAPTER 2
MULTIPARAMETRIC PROGRAMMING
Parametric programming is a type of mathematical optimization, where the op-timization problem is solved as a function of one or multiple parameters. In contrary to sensitivity analysis, which characterizes the change of the solution with respect to small perturbations of the parameters, parametric programming systematical analyzes the effect of uncertainty and variability in mathematical programming problems.
In general, parameters may reside in the objective function or (and) the con-straints of an optimization. In this thesis, we focus on a special type of para-metric programs — the right-hand side multiparapara-metric programs with linear constraints.
In this chapter, we introduce the key concepts and main results of mul-tiparametric programming which provide the theoretical foundations for the proposed parametric framework. The definitions and theoretical results follow chapter “Multiparametric Programming: a Geometric Approach” in [16].
2.1 Problem Formulation and Critical Region
Consider a general right-hand side multiparametric program as follows:
minz(x)subject to Ax ≤ b+ Eθ (y) (2.1)
where x is the decision vector, θ the parameter vector, z(·) the cost function, y the Lagrangian multiplier vector, and A, E, b are coefficient matrix/vector with compatible dimensions.
We denote Θ the region of parameters such that the mathematical program (2.1) is feasible. For any given θ ∈ Θ, let z∗(θ), X∗(θ) and Y∗(θ) be the optimal objective value, the set of optimal primal solutions and the set of optimal dual solutions in problem (2.1) for θ.
In general, there may exist multiple optimal solutions to (2.1). In this thesis, we focus on the special case where the optimization (2.1) has a unique optimal primal solution and a unique optimal dual solution for each parameter value θ. The uniqueness can be guaranteed by the assumption that problem (2.1) is neither primal degenerate nor dual degenerate for all θ ∈ Θ. The definition of primal and dual degeneracy is given below.
Definition 1. ([16], p. 25, 32) For any given θ ∈Θ, the mathematical program (2.1) is said to be primal degenerate if there exists a x∗(θ) ∈X∗(θ)such that the number of active constraints at the optimizer is greater than the dimension of parameter θ.
Definition 2. ([16], p. 26, 32) For any given θ ∈Θ, the mathematical program (2.1) is said to be dual degenerate if its dual problem is primal degenerate.
The multiparametric programming analysis builds on the concept of critical region. A critical region is a set of the parameters space where the local condi-tions for optimality of a multiparametric program remain unchanged.
Definition 3. ([16], p. 112) LetJ denotes the set of constraint indices in (2.1). For any subsetI ⊆ J, let AI and EI be the corresponding submatrices of A and E, respectively,
consisting of rows indexed byI. An optimal partition of the index set J associated with parameter θ ∈Θ is the partition (I(θ), Ic(θ))where
I(θ) , {i ∈ J|Aix∗(θ)= b + Eiθ, for all x∗(θ) ∈X∗(θ)}, Ic(θ), {i ∈ J|Aix∗(θ) < b+ Eiθ, for some x∗(θ) ∈X∗(θ)}.
Definition 4. ([16], p. 112) For a given θ0∈Θ, let (I0, Ic0), (I(θ0),Ic(θ0)). The critical region related to the index setI0is defined as
ΘI0 , {θ ∈ Θ|I(θ) = I0},
which is the set of all parameters θ ∈ Θ with the same active constraint set I0 at the optimum of problem (2.1).
By Definition 3, the optimal partition specifies two sets of constraints: one set is a combination of active constraints at the optimum of problem (2.1) and the other inactive. By Definition 4, a critical region is essentially a subset of the parameter spaceΘ on which a certain set of constraints is active at the optimum of problem (2.1).
The multiparametric programming determines the optimal solution struc-ture of problem (2.1) consisting of
1. the feasible parameter spaceΘ and its critical region partition {Θi},
2. the optimal objective function z∗(θ), the optimal primal solution1 x∗(θ), and the optimal dual solution y∗(θ)as functions of the parameter θ ∈Θ.
In this thesis, we concentrate on multiparametric linear programs (MPLPs) and multiparametric quadratic programs (MPQPs).
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