Work on optimisation of IESs initially focussed on ensuring optimal dispatch within the system (Geidl and Andersson, 2007a). Since then, several different algorithms have been proposed, demonstrating the evolving landscape of IES optimisation. Notable among these are nature-inspired metaheuristic algorithms such as particle swarm optimisation and genetic algorithms. Reference (Wang et al., 2010) employed a particle swarm optimisation algorithm to optimise a combined cooling, heating and power system for a building integrated with thermal stores and hybrid cooling systems. The reference shows the energetic, economic, and environmental performance of such a system configuration with redundant connections compared to a conventional system for a building. Similarly, two dispatch-optimisers for centralised energy management systems were presented in (Nemati et al., 2018), including an improved genetic algorithm and an enhanced mixed integer linear programming method. These approaches addressed unit commitment and optimal dispatch while considering network restrictions and unit constraints. The effectiveness of each method was evaluated using a test microgrid model under different operation policies.

Consideration of demand response and integration of renewable energy sources (RES) into IESs complicates the decision-making for power dispatch, urging for the exploration of new methods for the task. For example, an interval optimisation model was presented in (Zhang et al., 2020) for coordinating and scheduling a gas and electricity based IES involving wind power integration and demand response. Modern artificial intelligence-based techniques such as deep reinforcement learning have also been adopted to deal with IESs involving intermittent RES and demand uncertainty. For instance, the ability of deep learning to respond to dynamic changes in energy sources and loads was demonstrated in (Zhou et al., 2023), achieving a performance comparable to traditional programming approaches. The uncertainties associated with volatile wind power input to an IES could also be addressed adopting hydrogen systems as fast power regulators. A relevant example is presented in (Ding et al., 2022), where the optimal power dispatch of an IES is scheduled with a two-stage optimisation approach solved via the column and constraint generation method.

Additional aspects in the analysis of IESs involve optimisation strategies for carbon emissions reduction and introducing operational flexibility. These aspects could be coupled to those stated in previous paragraphs. For example, an integrated demand response (IDR) pricing method and a stepped carbon trading mechanism were introduced in (Ye et al., 2023) to minimise carbon emissions and operating costs in an IES. In another example, an optimal day-ahead dispatch model for an IES with carbon emission trading to tackle environmental challenges was presented in (Zhang et al., 2021). In this reference, the energy hub approach was adopted to establish a non-linear integer programming problem. This, in turn, was used to minimise wind curtailment and improve system efficiency, highlighting the importance of carbon emission costs in guiding energy dispatch decisions. Another relevant example of considering operational flexibility in a low-carbon optimal operation model of an IES was presented in (Wang et al., 2022). In this reference, the IES of an industrial park was optimised with a focus on the response from controllable flexible loads to minimise operating costs.

The optimisation goals for an IES could vary depending on the specific interests of multiple stakeholders. Such scenarios could be analysed by considering an IES governed by multiple agents such as energy service providers, renewable energy owners, and users, and using multi-objective optimisation such as the non-dominated sorting genetic algorithm-III to balance the interests of each agent (Zeng et al., 2019). Besides, a hierarchical framework for trading IDR resources among users in IESs using blockchain and an energy management system could increase user participation, reduce costs, minimise resource loss, and improve system flexibility, as discussed in (Wang et al., 2022).

The emphasis on steady-state methodologies has overlooked the need for dynamic verification of optimal solutions. Recent references in this context have discussed integrating low-carbon energy solutions into an IES to decarbonise the energy sector (de la Cruz Loredo et al., 2022a; 2022b), where emphasis was placed on the need to transition from steady-state to dynamic analysis due to the complex interactions within IESs. A methodology for validating optimised operation strategies in IESs through simulation platforms was presented in (Chen et al., 2022). By comparing the results obtained from optimised simulation models with those from physical simulation models, the reference demonstrated that the optimised strategy, using the interior point optimisation algorithm, produced dynamic simulation results with deviations of less than 10% compared to the physical models.

Implementing a day-ahead operational optimisation module may be helpful to ensure the effective coordination of an IES (Xu et al., 2023). This module would operate every 24 h, optimising system performance to minimise, for instance, operational costs and carbon emissions simultaneously. However, traditional optimisation modules do not account for the slow transients, control loops, and non-linearity of components present in thermal systems (Good et al., 2017; Martinez Cesena et al., 2020), which may impact the daily estimation of optimal power flows. In turn, these may induce significant errors in the calculation of total operational costs in the long-term—hindering, as a result, the decision-making process. By understanding the impact of these effects, a more accurate assessment of operational costs can be achieved, allowing stakeholders and system operators to make more informed decisions on infrastructure management and expansion planning.

While the theoretical basis of the optimisation algorithms described in the previous section is fundamentally correct, their verification process and applicability in real systems are not straightforward and must be further investigated.

To bridge this gap, this paper presents a novel methodology for dynamically verifying the results of an optimisation algorithm aimed at improving the power dispatch of an integrated electrical-thermal system supported by thermal stores. Verification has been done against results obtained with a dynamic model. By incorporating slow transients of thermal systems, control loops, and non-linearity of components in the dynamic model, often overlooked in traditional steady-state approaches, the presented methodology provides a more accurate assessment of energy consumption and operational costs.

The optimisation algorithm is based on sequential quadratic programming (SQP) (Bonnans et al., 2006; Nocedal and Wright, 2006). It demarcates conventional energy supplies from low-carbon sources by their respective carbon footprints. By incorporating this key environmental factor into the decision-making process, operational costs are minimised while promoting a smart and eco-friendly approach. System optimisation and verification using the dynamic model have been carried out using real data from a civic building in the UK dedicated to health services. Such energy system draws electricity from the local electricity grid and operates power-to-heat units and gas boilers (GBs) to meet energy demand.

The dynamic verification of the optimisation algorithm is conducted through a series of scenarios designed to examine its adaptability and performance under diverse operating conditions of the IES under study. Optimisation has been carried out in MATLAB/Simulink, and a “model-in-the-loop” (MiL) simulation method has been adopted to perform the dynamic verification with Apros—a commercial software for modelling and dynamic simulation of energy systems. The presented methodology offers a practical approach to evaluate optimisation strategies under realistic operating conditions and quantifying discrepancies in power flows and operational costs between the optimisation module and the dynamic model of the IES under study—contributing to the development of more accurate and effective optimisation solutions for IESs.

For this paper, the ‘energy hub’ modelling methodology was adopted (Eladl et al., 2023). The energy hub concept, introduced in (Geidl and Andersson, 2007b), provides a framework for steady-state modelling and simulation of IESs. Essentially, an energy hub links different energy vectors (e.g., heat, gas, electricity, hydrogen) via coupling technologies.

Figure 2 shows an example of an energy hub. The hub receives electricity and natural gas from their respective distribution networks as inputs, converts them to other energy forms, and stores them using different components such as a combined heat and power (CHP) unit, a gas boiler, an electric battery, and a thermal energy storage (TES) unit. These energy conversion and storage components, when operated in a coordinated manner, help fulfil the system’s energy demand—for example, for the system in Figure 2, electricity demand and heat demand. The operation of these elements within the energy hub could be optimised to minimise, for instance, operational costs, among other objective functions (see Section 2.2).

Energy hub’s components establish redundant connections within the IES, offering two key benefits. Firstly, they enhance supply reliability for the loads, reducing dependence on a single network. Secondly, the additional flexibility enables optimal supply management by evaluating energy vector utilisation based on criteria such as cost, carbon emissions, and availability, ensuring the efficient utilisation of resources (Geidl et al., 2007).

The modelling concept of an energy hub enables analysing power flows through converter devices by determining their energy efficiency, which is calculated as the steady-state output and input ratio. When there are multiple energy vectors as inputs and outputs, a conversion matrix may be employed to mathematically establish connections between energy networks (Geidl et al., 2007).

Once a steady-state model of the system under investigation has been developed, as discussed in Section 2.1, an optimisation module is employed to meet operational specifications, such as minimising system operational costs and carbon emissions. The optimisation module requires the definition of an objective function subjected to a number of equality and inequality constraints (Frangopoulos, 2009). These constraints may depend on the characteristics of the system under study.

For an IES linking electricity, heat, and gas energy vectors as in the system shown in Figure 2, electrical and heat demand profiles are used as inputs to the optimisation module. The outputs of the module provide the optimal power flows for the system under study—thereby driving power dispatch and optimising system operation on a 24-h basis.

The optimisation approach here adopted differs from conventional multi-objective optimisation algorithms by employing a single-objective function that minimises operational and carbon emission costs simultaneously (Capone et al., 2021). The implementation is done in MATLAB 2021b, utilising the “fmincon” function (MathWorks, 2024), which employs the SQP algorithm. This algorithm is considered highly effective for non-linear programming and is characterised by a rapid convergence to optimal solutions (Bonnans et al., 2006; Nocedal and Wright, 2006).

The optimisation approach has been borrowed from (Morales Sandoval et al., 2023). To prevent duplication of published work, interested readers are referred to the reference for full details.

The dynamic models used for components within a thermal-hydraulic system are based on differential equations that capture the transport phenomena of a heat transfer fluid (HTF), which is essentially a fluid employed to move thermal energy across two locations (Incropera et al., 2011). These equations include both the hydraulic characteristics, which detail the fluid flow, and the thermal attributes, which describe the temperature propagation. Valves, pipes, heat exchangers, hydraulic pumps, hydraulic separators, and energy storage tanks are typical elements considered in these models (De la Cruz-Loredo et al., 2022b).

In general, for an IES integrating heat as an energy vector, a dynamic model of the heating system is utilised to capture the slow thermal transients, which are often overlooked in traditional optimisation modules (Martinez Cesena et al., 2020).

By comparing the outcomes of the optimisation module (described in Section 2.2) with those of the dynamic model (Section 2.3), potential deviations in power flows and operational costs are quantified. This method allows assessing the impact of slow thermal dynamics on system performance, thereby providing stakeholders and system operators with valuable insights for decision-making. This step is described in further detail in Section 3.5 for the system under study, where the operational costs for two seasons of a calendar year are investigated.

The system under study operates based on the electricity and heat consumption of QEH. The facility is connected to both an electricity network and a gas supply network and is structured into heating zones (HZs). Each heating zone is treated as an independent heat consumption unit within the primary heating system.

The architecture of the primary heating system involves 1,200 m of pipelines connecting seven HZs through main and boiler pipeline loops, as shown in Figure 3A. The boiler loop incorporates two CHP units with an output power of 1,400 kW and four GBs with an output power of 5,200 kW located in HZs 1, 2, 4, and 5 to meet electricity and heat demand (see Figure 3B).

The original system configuration shown in Figure 3 was modified to incorporate sustainable energy technologies, in line with UK’s National Health Service and national targets to reduce greenhouse gas emissions (NHS Foundation Trust, 2015).

The upgraded IES is shown in Figure 4. This system configuration includes the two CHP units, 3 GBs located in HZs 1, 2, and 4, and an electric boiler (EB) in HZ5 within the boiler loop. In addition, a 100 m³ TES system consisting of four interconnected hot water tanks was considered. This sensible heat TES system is connected at the bottom to the return pipeline of the main loop and at the top to the boiler loop, after the GB in HZ1, for charging purposes. To avoid any disturbance to the supply temperature in the boiler loop, the TES system discharges immediately before the GB in HZ1 (De la Cruz-Loredo et al., 2022b).

In both the optimisation module and the dynamic model utilised for this study, the two individual CHP units within the physical system are considered as a single CHP unit with an equivalent output power for simplicity. Therefore, reference to the CHP units will be made using the term “CHP plant” to denote the collective operation of the units as a single entity.

Table 1 shows the considered capacities of the retrofitted technologies. These have been selected based on practical specifications and availability in the UK market (London Engineers Company, 2023; Refrigeration Technology Co. Ltd, 2023). For the different system configurations studied in this paper, the schematics and essential mathematical formulations for each are outlined in Supplementary Appendix SA.

Real electricity and heat demand profiles from QEH were used as inputs to the optimisation module. These profiles correspond to different seasons of the year and are provided in Supplementary Appendix SB. However, to maintain control over supply and return temperatures and prevent fluctuations in the dynamic model, an overestimation of electrical demand and underestimation of heat demand were considered. A 1-h granularity was employed.

Figure 5 shows the daily price profiles for gas and electricity adopted in this paper based on energy prices available at the hospital site in 2020 and average market prices (Guelpa, 2024).

The following assumptions were considered for system modelling within the optimisation module (based on the data provided by QEH and information available in the literature) (Morales Sandoval et al., 2023):

i. The system is considered to be in steady-state between any two time periods.

A detailed dynamic model of the heating system was implemented. However, representation of the electricity system, for simplicity, was limited to reproduce the energy balance between supply and demand based on real historical data. The dynamic model was implemented in Apros using available library models (de la Cruz Loredo et al., 2022a; 2022b).

The following modelling assumptions were considered:

i. A constant pressure loss coefficient K L = 0.1 was adopted for all components.

The operation of the hospital’s heating system was regulated by a process control system consisting of four control elements: the marginal differential pressure control, the supply temperature control, the return temperature control, and the TES system control. These elements regulate pressure, flow, and temperature within different sections of the IES by adjusting parameters such as pump speed, valve opening range, and heat flow from the heat generation units. For this purpose, proportional-integral (PI) controllers were employed. Table 2 shows relevant specifications of these controllers.

Similar to the optimisation module, real electricity and heat demand profiles from QEH were used. As mentioned before, these are provided in Supplementary Appendix SB. However, the dynamic model adopted a 30-min granularity for the energy demand data.

The energy system needs to select the optimal proportion of electricity and gas intake to reduce energy costs and the production of CO₂. To minimise the daily operational cost of the system while also minimising carbon emissions, the objective function is expressed as (Morales Sandoval et al., 2023) C = ∑ i = 1 24 C g r i d , i E × P g r i d , i E + C g r i d , i G × P g r i d , i G + C C O 2 E + C C O 2 G (4) where C is the daily operational cost, C g r i d , i E and C g r i d , i G are the electricity and gas unit costs at hour i, P g r i d , i E and P g r i d , i G are the power inputs to the energy system from the external electricity and gas networks at hour i, and C C O 2 E and C C O 2 G are the costs of CO₂ production associated with electricity and gas consumption.

Objective function in Eq. 4 is subject to the following equality constraints (representing the electricity and heat balance equations) P g r i d , i E + η C H P g / e P C H P , i G − η E B P E B , i = P d , i E   i = 1 , 2 , 3 , … 24 (5) η C H P g / h P C H P , i G + η G B P G B , i G + η E B P E B , i − P c h , i T E S + P d i s , i T E S = P d , i H   i = 1 , 2 , 3 , … 24 (6) where P C H P , i G and P G B , i G are the power inputs to the CHP plant and the GBs from the gas grid; P E B , i is the electrical power input to the EB either from the electric grid or from the surplus produced by the CHP plant; and P c h , i T E S and P d i s , i T E S represent the charging and discharging powers of the TES system.

Based on the dimensions of hot water tanks available in the market and considering the supply and return temperatures of the IES under study (de la Cruz Loredo et al., 2022b; Refrigeration Technology Co. Ltd, 2023), the capacity of the TES system used in this paper was selected as 1,000 kWh (Majić et al., 2013) (250 kWh per individual hot water tank, see Table 1). Its charging and discharging powers P c h , i T E S and P d i s , i T E S per time period (i.e. 1 h) are constrained as 0 ≤ P c h , i T E S ≤ 250 kW (7) 0 ≤ P d i s , i T E S ≤ 250 kW (8)

To ensure the energy levels in the thermal store remain within the predefined maximum and minimum limits, the following constraints were imposed: ∑ i = 1 j η T E S P c h , i T E S − 1 / η T E S P d i s , i T E S × ∆ t ≤ L max T E S   j = 1 , 2 , 3 , … 24 (9) ∑ i = 1 j η T E S P c h , i T E S − 1 / η T E S P d i s , i T E S × ∆ t ≥ L min   T E S   j = 1 , 2 , 3 , … 24 (10) where ∆ t is the hourly time period considered in the optimisation algorithm, and L max T E S and L min T E S are the maximum and minimum capacities of the TES system.

In the previous formulation, Eqs 5, 6 denote the equality constraints, while Eqs 7-10 denote inequality constraints to meet the objective function in Eq. 4.

To ensure the energy stored in the TES system at the end of the diurnal cycle remains the same as during the start of the cycle, the following equality constraint was included: ∑ i = 1 24 P c h , i T E S − P d i s , i T E S = 0 (11)

Constraint in Eq. 11 enables the cyclic operation of the energy storage unit.

To assess operational costs, two contrasting seasons, summer and winter, were selected to represent periods of high and low heat demand across the year. For simplicity and to provide a basis for discussion, weekly costs were extrapolated to account for 52 weeks in a year. This way, the annual operating cost for each scenario under investigation was determined using C a = C w S × 26 + C w W × 26 (12) where C a is the estimated annual cost of the IES and C w S and C w W represent the weekly operational cost for summer and winter.

Different software packages may be adopted for simulating the complex dynamics of real IESs, facilitating the adoption of a model-based design approach. This approach enables the development of dynamic models, control, and communication systems to conduct virtual tests—eliminating the need for costly verification in real systems. Following this approach, the effectiveness of the optimisation algorithm was demonstrated through an MiL configuration.

MiL simulation was carried out as a co-simulation utilising two software platforms, as shown in Figure 6. The optimisation algorithm and dynamic models of the CHP plant and the EB were implemented in MATLAB/Simulink, while the control schemes and dynamic models of the TES system and the heating network were developed in Apros. The connection between the two software platforms was achieved through the utilisation of an open platform communications (OPC) protocol, which is available in both MATLAB and Apros (OPC foundation, 2024). Such a co-simulation architecture has been successfully adopted in the literature (Morales Sandoval et al., 2021; de la Cruz Loredo et al., 2022a, 2022b; Bastida et al., 2023).

Three different system configurations—namely, Scenario 1 (base case, see Supplementary Figure SA1), Scenario 2 (IES with a TES system, Supplementary Figure SA2), and Scenario 3 (IES with EB and a TES system Supplementary Figure SA3)—were chosen to quantify discrepancies in power flows and operational cost by comparing the outcomes of the optimisation module and the dynamic model.

The base case, similar to the current system configuration at QEH, is examined in more detail in this section. In this case, the electricity and heat demand are met through the external electricity grid, CHP plant and GBs. Figure 7 compares the optimal heat power output Q ˙ C H P , o p t produced by the CHP plant during a day in summer (see Figure 7A, green trace) and winter (Figure 7B, blue trace) with the heat power output Q ˙ C H P , d y n obtained with the dynamic model (orange and purple traces in the figures, with markers). Given that operation of the CHP plant is dictated by the optimisation module and the electricity demand fluctuations being balanced with the electricity grid, a good agreement is observed in heat power flows. In response to the higher energy demand during winter, an increase in the use of the CHP plant can be noticed in both cases.

Figure 8 shows a comparison between the heat power flows Q ˙ G B s , o p t of GBs for a summer day (Figure 8A, green trace) and a winter day (Figure 8B, blue trace) obtained with the optimisation module and via dynamic simulation ( Q ˙ G B s , d y n , orange and purple traces). Here, the differences in heat power flows become evident with higher energy demand in winter, leading to a significant increase in the utilisation of GBs. While the dynamic model aims to replicate the optimal heat power flows, the influence of system dynamics is evident in the model output. Therefore, in scenarios involving large quantities of heat flows (e.g., in winter months with higher heat demand), the discrepancy between the static optimisation module and the dynamic system model becomes more pronounced.

The heat power flow variations exhibited by GBs have an impact on the estimation of daily and weekly operational costs in summer and winter. The weekly operational costs are shown in Table 4. During the week in summer, the operational cost resulting from the dynamic model deviates by £836.28 (equivalent to 4.87%) from the estimated optimal operational cost. In the winter week, with the increased GBs usage, a notable deviation in operational cost is observed, where the output of the dynamic model differs from that of the optimal model by £2289.34, or 5.75%.

In Scenario 2, the electricity and heat demand are met through the external electricity grid, CHP plant, GBs, and TES system. Figure 9 compares the optimal heat power output Q ˙ C H P , o p t produced by the CHP plant during a day in summer (Figure 9A, green trace) and winter (Figure 9B, blue trace) with that obtained with the dynamic model ( Q ˙ C H P , d y n , orange and purple traces with markers).

Comparing Figure 9A with Figure 7A from Scenario 1, it can be observed that both the optimal power output produced by the CHP plant ( Q ˙ C H P , o p t , green traces in both figures) and the heat power output obtained with the dynamic model ( Q ˙ C H P , d y n , orange traces with markers in both figures) vary during the summer day when a TES system is present in the system. This variation is influenced by energy demand and cost. For instance, during hours 15 to 19, electricity demand is high while heat demand is low (refer to Supplementary Figure SB1). Consequently, surplus heat from the CHP plant is directed to the TES system. This stored heat can be utilised during periods of increased energy cost, resulting in operational cost savings. The heat power output produced by the CHP plant during the winter day is similar as in the base case considered as Scenario 1 (Figures 7B, 9B, see all traces in the figures). This is because the incorporation of a TES system into the IES does not have any effect in the winter season as it is not operative.

Figure 10 shows a comparison between the heat power flows Q ˙ G B s , o p t of GBs for a day in summer (Figure 10A, green trace) and winter (Figure 10B, blue trace) obtained with the optimisation module and dynamic simulation ( Q ˙ G B s , d y n , orange and purple traces). A decrease in the heat power output produced by the GBs during summer is noticed when compared to Scenario 1 (comparing Figure 10A with Figure 8A, green traces in both figures). For instance, here the maximum peak of the optimal heat power output produced by the GBs is around 700 kW, against the peak of approximately 850 kW observed in Scenario 1 (Figure 8A). However, it is important to note that the heat power output produced by GBs in the winter day for Scenario 2 remains unchanged compared to that of Scenario 1 (see all traces in Figures 8B, 10B).

Figure 11 compares the performance of the TES system during both summer (Figure 11A) and winter (Figure 11B) days. In Figure 11A, negative values indicate the TES system is charging, while positive values mean the TES system is discharging. A clear discrepancy is observed between the optimal performance indicated by the optimisation module ( Q ˙ T E S , o p t , green trace) and the dynamic performance achieved by the dynamic simulation ( Q ˙ T E S , d y n , orange trace with markers) during the day in summer. Figure 11B confirms that the TES system is not operational during the winter day, with no charging or discharging occurring. This is consistent with the discussion around Figure 9B.

The variations in heat power flow exhibited for Scenario 2 affect the estimation of daily and weekly operational costs in both summer and winter, as detailed in Table 5. During the summer week, the operational cost resulting from the dynamic model deviates by £736.7 (equivalent to 4.38%) from the estimated optimal operational cost. In the winter week, a deviation of 5.75% exists between the output of the dynamic model and that of the optimisation module, as the TES system remains idle—consistent with results obtained in Scenario 1.

For Scenario 3, electricity and heat demand are met through the external electricity grid, CHP plant, GBs, EB, and TES system. Figure 12 compares the optimal heat power output Q ˙ C H P , o p t produced by the CHP plant during the day in summer (Figure 12A, green trace) and winter (Figure 12B, blue trace) with those obtained with the dynamic model ( Q ˙ C H P , d y n , orange and purple traces with markers). An increase in the heat output power provided by the CHP plant during the summer day (Figure 12A) compared to both Scenario 1 (see Figure 7A) and Scenario 2 (Figure 9A) is observed in the initial hours of the day by both the optimisation module and the dynamic simulation. This increase is attributed to the surplus electricity generated by the CHP plant being utilised to power the EB, which entirely converts the electricity consumed to heat supply due to its high efficiency. The same behaviour occurs in the winter day (Figure 12B), where the CHP plant operates at maximum capacity to generate surplus electricity while meeting the heat demand.

Figure 13 shows a comparison between the heat power flows Q ˙ G B s , o p t of GBs for the summer (Figure 13A) and winter (Figure 13B) days obtained with the optimisation module (green and blue traces) and dynamic simulation ( Q ˙ G B s , d y n , orange and purple traces). During the summer day (Figure 13A), the optimisation module does not operate the GBs (see green trace, with values of zero throughout) as the heat demand is met with the EB. However, the dynamic model requires GBs to regulate the supply temperature within the system and prevent disturbances in temperature, as illustrated by the orange trace. During the winter day (Figure 13B), a reduction in the utilisation of the GBs is observed compared to Scenario 1 (Figure 8B) and Scenario 2 (Figure 10A). This decrease is due to the support provided by the EB in meeting the heat demand.

Similar to the management of the CHP plant, the operation of the EB is governed by the optimisation module for power dispatch. As a result, the discrepancies between the optimal heat power output ( Q ˙ E B , d y n ) and that obtained with the dynamic model ( Q ˙ E B , d y n ) are minimal, as shown in Figure 14.

Figure 15 compares the optimal and dynamic performance of the TES system during both summer (Figure 15A) and winter (Figure 15B) days. As in Scenario 2, a discrepancy is observed between the optimal behaviour suggested by the optimisation module ( Q ˙ T E S , o p t ) and the dynamic performance achieved via simulations ( Q ˙ T E S , d y n ) during the summer day. As before, the TES system remains idle during the winter day (see Figure 15B).

Table 6 compares the weekly operational costs for Scenario 3. During the summer week, the operational cost resulting from the dynamic simulations deviates by £731.58 (equivalent to 4.36%) from the estimated optimal operational cost. In the winter week, the operational cost obtained with dynamic simulations differs from that of the optimal model by £2290.37, or 5.83%.