## 5 Proposed Optimization Problem
### Constraints  
The constraints are critical to guarantee the effective and optimal operation of the HEMS. The equality constraints guarantee that the power balance is maintained, ensuring that the generation and consumption of energy are always matched. The inequality constraints help in maintaining the operational limits of the system, such as the maximum capacity of energy storage and the allowable power drawn from renewable sources. Operational constraints guarantee that the SOC of the battery remains within safe boundaries, and the RESs operate within their specified capacities. Additionally, the constraints on operational and maintenance costs are vital for keeping the system economically viable. These constraints collectively ensure that the HEMS operates reliably, sustainably, and cost-effectively, balancing various objectives such as cost minimization, the maximization of renewable energy usage, and the minimization of environmental impact. The proposed optimization problem includes the following constraints:  
**a. Power balance constraint**  
Ensuring the power balance is a fundamental requirement for the effective functioning of a HEMS. This constraint guarantees that the energy produced by various sources matches the energy consumed by the household at all times. Balancing power generation from photovoltaic panels, WTs, and FCs with the energy demands of household appliancesand devices is crucial to maintain system stability. Any imbalance can result in either energy shortages or surpluses, which can destabilize the system and reduce efficiency. By meticulously coordinating different energy sources and backup systems, such as BESSs and EVBs, the system provides a consistent and dependable power supply. This balance enables optimal resource utilization, minimizes dependence on external power sources, and improves overall system efficiency. It can be expressed as follows:  
\[P_{Lt}(t)=P_{Gent}(t),\forall t\in\tau \tag{25}\]  
\[P_{PV}(t)+P_{WT}(t)+P_{FCs}(t)+k\ \times\ P_{BESS}(t)+k\ \times\ P_{EV}(t)=P_{L}(t) \tag{26}\]  
\[\text{k}=\begin{cases}\ \ \ k=1\ \ \ \text{{discharge}}\\ \ \ \ k=0\ \ \text{{charge stays constant}}\\ \ \ \ k=-1\ \ \ \text{{charge}}\end{cases} \tag{27}\]  
The power balance constraint ensures that the total power demand (\(P_{Lt}(t)\)) is always met by the total generated power (\(P_{Gent}(t)\)) at any given time t, which is crucial for maintaining grid stability. This is achieved by balancing the power contributions from PV systems, WTs, FCs, BESS, and EVs. The term k determines the operation mode of the BESS and EVs. When k = 1, it indicates that the system is discharging, meaning that the BESS and EVs are supplying power to the home. When k = 0, it indicates that there is no change in the charge state, meaning the BESS and EVs are neither charging nor discharging, and their SOC remains constant. When k = \(-1\), it indicates that the system is charging, meaning that the batteries and EVs are absorbing power from the home. This constraint ensures that the sum of power from all sources and storage systems precisely matches the load demand at all times.  
0. 0.  
where  
\[LPSP(t)=\frac{\sum\limits_{t=1}^{T}LPS(t)}{\sum\limits_{t=1}^{T}P \underset{L}{(t)}} \tag{30}\] \[LPS(t)=P_{L}(t)-P_{PV}(t)-P_{WT}(t)-P_{FC}(t)-P_{BESS_{d}}(t)-P_{EV_ {d}}(t) \tag{31}\]  
This reliability constraint ensures that the system maintains a high level of reliability by keeping the LPSP below the acceptable threshold (\(\beta\)). By calculating the LPSP, the system assesses whether the power generated and stored is sufficient to meet the load demand over time, accounting for all available sources of power. The goal is to minimize the risk of power shortages, thereby ensuring that the system consistently meets consumer demand with minimal interruptions.