A qualitative study of the optimal control model for an electric power generating system

Abstract

The economic independence of any nation depends largely on the supply of abundant and reliable electric power and the extension of electricity services to all towns and villages in the country. In this work, the mathematical study of an electric power generating system model was presented via optimal control theory, in an attempt to maximize the power generating output and minimize the cost of generation. The factors affecting power generation at minimum cost are operating efficiencies of generators, fuel cost and transmission losses, but the most efficient generator in the system may not guarantee minimum cost as it may be located in an area where fuel cost is high. We choose the generator capacity as our control ui(t), since we cannot neglect the operation limitation on the equipment because of its lifespan, the upper bound for ui(t) is choosing to be 1 to represent the total capability of the machine and 0 to be the lower bound. The model is analyzed, generation loss free equilibrium and stability is established, and finally applications using real life data is presented using one generator and three generator systems respectively.

Boundary Optimal Control for Triple Nonlinear Hyperbolic Boundary Value Problem with State Constraints

“…We symbolize by (v_1,v_2 )_Ω and ‖v‖_0 the inner product (IP) and the norm (NR) in L^2 (Ω), by (u,u)_Γ and ‖u‖_Γ IP and the NR in L^2 (Σ), by (v_1,v_2 )_1 and ‖v‖_1, the IP and the NR in H^1 (Ω), by (v ⃗,v ⃗ )_Ω and ‖v ⃗ ‖_0 the IP and the NR in L^2 (Ω), by (v ⃗,v ⃗ )_Γ and ‖v‖_Γ the IP and the NR in i L^2 (Σ), by (v ⃗,v ⃗ )_1=∑_(i=1)^3▒(v_i,v_i )_1 and ‖v ⃗ ‖_1^2=∑_(i=1)^3▒‖v_i ‖_1^2 the IP and the NR in V ⃗, and finally V ⃗^* is the dual of V ⃗. The weak form (WKF) of problem (1)(2)(3)(4)(5)(6)(7)(8)(9)…”

“…Then w_i C ] L^∞ (I,U) L^2 ( ) et h _i1 (y_1 ) h_i1 (y_i ) w_i, then h _i1:Q×R→R is of CTHDT. Now, utilizing proposition 2.1 to give that the integral ∫_Q^ ▒ h_i1 (y_ik ) w_i dxdt is continuous w r t y_i But y_i □( )y_i strongly in L^2 (Q), therefore (1)(2)(3)(4)(5)(6)(7)(8)(9), then finally adding each resulting pair of equations together, we obtain:…”

“…The problems of optimal control (OCPs) have a major significant and vital role in numerous fields, such as biology [1], electric power [2], robotics [3], economic [4], and many other different fields. This significance has motivated many investigators to be concerned with studding the OCPs for mathematical modules dominated by the three types of nonlinear PDEs; elliptic [5], hyperbolic [6] and parabolic [7], whilst many others [8][9][10] are concerned with studying the boundary OCPs (BOCPs).…”

The Continuous Classical Boundary Optimal Control of Couple Nonlinear Hyperbolic Boundary Value Problem with Equality and Inequality Constraints

“…1 , 2 the definition of the "Fréchet derivative", the result of Lemma 1, and then using "Minkowiski inequality", we have Let ⃗⃗ = ( 1 , 2 ) & ̅ ⃗⃗ = ( ̅ 1 , ̅ 2 ) are two given controls vectors, then = ( 1 , 2 ) = ( 1 , 2 ) & ̅ = ( ̅ ̅ 1 , ̅ ̅ 2 ) = ( ̅ 1 , ̅ 2 ) are their corresponding stats solutions. Substituting the pair (⃗ , ) in equations (1)(2)(3)(4)(5)(6) and multiplying all the obtained equations by ∈ [0,1] once and then substituting the pair ( ̅ ⃗⃗ , ̅ ) in (1-6) and multiplying all the obtained equations by 1 = (1 − ) once again, finally adding each pair from the corresponding equations together one gets:…”

“…Open Access ∀ = 1,2, and each of , is a unit vector normal outer to the boundary Σ. The set of admissible controls is ⃗⃗⃗ = {⃗⃗ ∈ ⃗⃗⃗ = 2 (Σ) × 2 (Σ)|⃗⃗ ∈ ⃗⃗⃗ a. e. in Σ, 1 (⃗⃗ ) = 0, 2 (⃗⃗⃗ ) ≤ 0} , ⃗⃗⃗ ⊂ ℝ 2 The cost function is The continuous optimal control problem is to find ⃗⃗ ∈ ⃗⃗⃗ such that 0 (⃗ ⃗ ) = 0 (⃗⃗ )…”