## 3 Results and Discussion
### Effect of Cut-Off Temperature  
The effect of cut-off temperature on the seasonal energy performance of HHP systems is more important than in monovalent systems since it directly affects the participation of the backup system. Based on weather conditions, two additional cut-off temperatures apart from 0 \({}^{\circ}\)C (T\({}_{\text{cut-off}}=\pm 3\)\({}^{\circ}\)C) were introduced, and their results were compared to the baseline simulation. The influence of T\({}_{\text{cut-off}}\) and the heat pump capacity on the seasonal energy performance and the on-off cycles of the HHP system is illustrated in Figure 8. According to Figure 8a, the general trend shows that in almost all configurations, the \(\eta_{\text{S}}\) improves by decreasing the T\({}_{\text{cut-off}}\), with the highest value of the heating seasonal efficiency belonging to configuration 2. In fact, for each heat pump capacity representing the bivalent temperature, \(\eta_{\text{S}}\) increases as T\({}_{\text{cut-off}}\) decreases, with the maximum value occurring at T\({}_{\text{cut-off}}=-3\)\({}^{\circ}\)C. The energy performance results can be fully explained by observing Figure 8b, in which the values of SCOP\({}_{\text{net}}\) are shown as functions of the heat pump capacity and the cut-off temperature. It is evident that for each heat pump size, the seasonal energy performance increases as the cut-off temperature increases [40]. There are two reasons for the improvement in SCOP\({}_{\text{net}}\) when the heat pump size is reduced. First, as T\({}_{\text{cut-off}}\) increases, the number of defrost cycles performed by the heat pump decreases. Second, because of low external temperatures and high compressor frequencies, the periods during which the heat pump operates with low COP values are significantly reduced. It is important to mention that when the cut-off temperature increases, the peak load ratio decreases, and the share of heating produced by the boiler increases, as shown in Figure 8c. For this reason, the energy performance of the whole HHP system is more influenced by the boiler's seasonal efficiency. Another interesting point is that the net seasonal coefficient of performance of the series operation (configurations 1 and 2) is much better than that of the parallel ones (configurations 3 and 4). In series operation, the HHP system with configuration 2 shows the best performance, and cut-off temperature changes have a greater impact on it. Also, the best SCOP\({}_{\text{net}}\) results are for the 8 kW heat pump, when \(\beta=0.66\). In parallel arrangements, T\({}_{\text{cut-off}}\) does not have a significant effect on SCOP\({}_{\text{net}}\). In contrast to the series configuration, the best performance was achieved at \(\beta=0.33\) with a 4 kW heat pump. Additionally, using two storage tanks and control valves to regulate the mass flow based on the building's demand improves the HHP's performance. In Figure 9a, the yearly frequency of defrost and on-off cycles for air-to-water heat pumps is shown as a function of both the cut-off temperature and the heat pump's capacity. It is evident that as the cut-off temperature increases, the number of defrost cycles performed by the heat pump decreases. For example, the number of defrost cycles decreases by 21% when the cut-off temperature is increased from \(-3\)\({}^{\circ}\)C to 0\({}^{\circ}\)C, and by 73% when the cut-off temperature is increased to 3 \({}^{\circ}\)C. Moreover, the data shown in Figure 9a indicate that in configuration 2, the number of heat pump on-off cycles increases when the HHP system uses a larger heat pump. This trend is more pronounced at lower cut-off temperatures because the heat pump provides a larger portion of the heating demand and operates more frequently. For a constant heat pump capacity, there is a slight decrease in the number of on-off cycles performed by the heat pump during the heating season as the cut-off temperature rises [41].