Theories and Mathematical Backgrounds

How btso works?

The mathematical formulation for the Building Technology Size Optimization and also how the objective formulation looks like are described here. The building technologies for electricity production under consideration are photovoltaics, wind turbine, fuel cell, electrolysis with the technology for heat production such as boiler, geothermal heat pump, air-water heat pump, solar panel and geothermal energy. Combined heat and power are also considered that produced both electricity and heat. For the storage systems are technologies such as battery storage, hydrogen storage, heat and cold storage taken into account. Detailed description of the storage system could be seen here.

The objective function is the minimization of the combination of network interaction costs and component costs and also the emissions emitted by the technology. The mathematical formulation our problem are:

Objective Function

\[min \; (C_{Component}+C_{Grid})\]

with

Component costs/emissions

\[\begin{split}\begin{align*} C_{Component} = \; &C_{Air_{HP}} + C_{HT} + C_{PV} + C_{Batt} + C_{STC} + C_{STA} + C_{HP} + C_{CT} + \\ &C_{P_{el}}+C_{Boil}+ C_{Wind} + C_{FC} + C_{ES} + C_{H_2S} + C_{Q_h}+ C_{Q_c} \end{align*}\end{split}\]

Network interaction costs/emissions

\[\begin{split}C_{Grid} = \; \sum_{i=1}^{l_t} \delta {t_h}^t \cdot (&{P_{buy}}^t\cdot {c_{buy}}^t - {P_{sell_{PV}}}^t\cdot {c_{sell_{PV}}}^t - \\ &{P_{sell_{Wind}}}^t\cdot {c_{sell_{Wind}}}^t - {P_{sell_{CHP}}}^t\cdot {c_{sell_{CHP}}}^t + \\ &{M_{Boil}}^t\cdot {c_{Fos}}^t + {M_{CHP}}^t\cdot {c_{Fos}}^t)\end{split}\]

Energy balance constraints

\[\begin{split}&{P_{buy}}^t - {P_{sell_{PV}}}^t - {P_{sell_{Wind}}}^t - {P_{sell_{CHP}}}^t - {P_{Air-HP}}^t - {P_{el}}^t + \\ &{P_{PV}}^t - {P_{ch}}^t + {P_{dis}}^t - {P_{HP}}^t + {P_{CHP}}^t +{P_{Wind}}^t + {P_{FC}}^t - {P_{ES}}^t = {P_{el,d}}^t\end{split}\]

The detailed mathematical formulation for each building technologies are described in the corresponding page. The costs and/or emissions equations applies for all building technologies, so it can be represented once in this page.