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represent the entire building. In this paper the heat demand and temperature will be fixed prior to the optimization. As a consequence the building thermal capacity will not be actively used to store thermal energy. The second boundary condition to the heat supply system involves the electricity grid side. The demand side management aspect of the problem will be accounted for by using a time varying energy price p(t). 2.4. Optimal control problem formulation In an optimal control problem the profile of a controlled variable will be varied to obtain an optimal value of a cost function. The controlled variable and all state variables influenced by this control are given appropriate constraints. In the system formulation described above C ̇sup is the controlled variable and E is a dependent state variable. As the building energy demand, and thus it’s thermal comfort are fixed prior to optimization the cost function can be defined as the total energy cost during the optimization period: tf ̇ Csup(Theater,out − Theater,in)p(t)dt (5) 0 This implies the inclusion of the heater efficiency in the energy price signal and the independence of the efficiency of everything but time. An outside temperature dependent efficiency is however still possible as the outside temperature itself is only time dependent. To avoid transient conditions, a periodic operation with relative period tf is assumed and peri- odic boundary conditions are applied: E(0) = E(tf ) (6) C ̇sup(0) = C ̇sup(tf ) The problem is constrained by the maximum amount of energy that can be stored, the maximum power of the heat supply and the maximum heater capacity flow rate: 0 ≤ E ≤ Emax 0 ≤ Q ̇ sup ≤ Q ̇ max (7) 0≤ C ̇sup ≤C ̇max In the perfectly stratified model the second and third constraint are equivalent while in the perfectly mixed model they differ as the heat flow depends on the state of the storage tank. For the perfectly mixed model an additional constraint needs to be added ensuring sufficient energy can be transferred through a finite heat exchanger to the emission system: Q ̇ em ≤ (εC ̇ )max(T − Tem) (8) Where ε is the effectiveness of the heat exchange to the emission system. 2.5. Solution of the optimal control problem To solve the discussed optimization problems the Acado Toolkit [16] and Matlab interface was used. The control profile is discretized in 48 time steps. A sequential quadratic programming approach with exact Hessian calculation is used. To address numerical scale issues and to extend the applicability the model is nondimensionalized using reference quantities derived from the boundary conditions. The period of simulation (tf ) J = 4PDF Image | Energy cost reduction by optimal control of ideal sensible thermal energy storage
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