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storage loss when the storage is full minus the empty storage loss rate. Storage loss in full and empty state is established according to Eqs. (11) and (12). Furthermore, the mathematical problem is subject to the bound- ary conditions of maximum production capacity and thermal storage capacities determined by Eq. (8): Pck,t ≤ pck,t Sht ≤ sht (8) Sct ≤ sct As technical considerations suggest that real-world operation should also minimize the number of compressor starts and stops (switches), and that transient characteristics may be ignored for discrete operation (minimum 1 h between switches) the analysis will apply a constraint by which the compressor unit is only allowed to operate at full capacity within each hour. For this purpose, a binary variable U is introduced, which constrains the compres- sor unit’s electricity consumption to either zero or full capacity as expressed in Eq. (9): costs, the optimal operation is associated with an electricity con- sumption profile, which COMPOSE subsequently uses to find the intermittency-friendliness coefficient Rc and marginal CO2 emis- sions in accordance with the methodology described earlier. 3.5. Thermal storage loss Thermal storage losses are calculated on the basis of thermal conduction losses from free standing insulated tanks. Heat losses from radiation and convection are considered to be insignificant and are ignored. The thermal conduction loss for the heat storage in empty state sleh [kWh] is found from Eq. (11). pcHP,t FvHP,t =U× COPc HP,t ,whereU∈{0,1} (9) where [W/(mK)] is the thermal conductivity, A [m2] is the surface area of the storage tank, Tin [K] is the return water temperature, Tambient [K] is the ambient temperature, and r [m] is the thickness of the insulation material. The differential storage loss sldh is the full storage loss rate minus the empty storage loss rate as expressed in Eq. (12). M.B. Blarke et al. / Energy and Buildings 50 (2012) 128–138 133 Fig. 4. Case study: hourly space cooling and water heating demand (kWh/h). sleht = TA(T−T ) in,t ambient,t (11) t=1 r To summarize, the short-term least-cost planning problem for the TB may be expressed as the mixed-integer linear program in Eq. (10): T A(Tout,t − T ) ambient,t − sleht (12) Expressions similar to Eqs. (11) and (12) are used for the cold stor- sldht = r maximize f (x) subject to Eqs. (4)–(9). Notice that the objective function is set to maximize due to the sign convention according to which costs are negative and benefits are positive. COMPOSE solves this mixed-integer linear programming prob- lem according to Eq. (10), i.e. by minimizing the economic cost of operation under given constraints. For any given set of parameter values, COMPOSE will identify the most feasible oper- ational strategy. In addition to the resulting energy balance and (10) age. t=1 4. Proof-of-concept design and case study assumptions The TB solution is designed for a single-family residential build- ing in San Jose, CA (PG&E service area) and compared to the continued use of existing end-use appliances, which use electric- ity for providing hot tap water heating and space cooling services. While viable, the analysis does not consider replacing gas-firedPDF Image | Thermal battery with CO2 compression heat pump
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