## 4 IPTMS Power Management Optimisation  
As presented in Section 2.4, an optimisation strategy for IPTMS can manage thermal and electric energy flows in the aircraft implementing suitable PM strategies. The optimisers aim to (1) reduce operational costs by minimising fuel consumption and battery usage, (2) achieve optimal operations for in-flight components, and (3) satisfy power and thermal demands in different flight conditions. Compared with the conventional optimisers for HEA that optimise power split, the IPTMS optimisers also consider the effect of temperature and auxiliary loads in the TMS. Due to increased electrification, there is a need to manage the power and thermal energies of additional electrical components. Additionally, some TMS components are electrified, such as electrical coolant pumps and cooling fans. These electric components require power supplies and generate heat loads in the aircraft. As a result, it is necessary to assess their performance and optimise their energy flows. Dynamic programming (DP) is a non-causal power management (PM) strategy. It is highly versatile, enabling it to tackle a wide range of problems [73]. It has been applied for HEA [74; 75; 76], with findings indicating that DP can reduce operational costs and improve hydrogen fuel consumption for fuel-cell HEA [75]. Additionally, it also leverages flight scheduling and has nearly no accuracy losses when imitating mechanical operations [74]. However, DP has a disadvantage in computational speed. Pontryagin's Minimum Principle (PMP) is a causal PM strategy derived from DP through variational methods. It can convert the original global optimal problem to a local one [77]. This strategy has been widely studied for HEVs [78]. However, PMP has difficulty solving complex nonlinear problems [79], which presents a challenge for implementing PMP in HEA. Model Predictive Control(MPC) is an online causal PM strategy. In MPC, the algorithm first predicts the future states based on the system model over a predictive horizon of length N at each sampling time. It then selects the optimal control law from the current time k to the time k + N through rolling optimisation. Once the optimising control law is chosen for the current time k, the results are output only for the time k, disregarding the rest, and the system state is updated. As a result, MPC is more sensitive to the aircraft's future performance. Therefore, while uncertainties in the prediction pose challenges to the robustness of MPC, it is less computationally demanding than DP [80]. [PERSON] et al. [81] proposed a hierarchical MPC approach for the IPTMS of a military aircraft. They design the IPTMS to provide power for auxiliary loads, emergency loads, and engine starts, while cooling electric components, such as AC/DC loads, advanced electrical equipment, batteries, generators, and inverters. They also assess the dynamic performance of the components with the graph-based approach developed in [82, 83]. The MPC is three-level, where the upper level is for planning thermal and power demands, state predictions, and power shedding. The lower-level MPC provides optimal solutions for desired battery power and cooling mass flow rates when satisfying state constraints. The tracking-level MPC tracks the mass flow rates of pumps and fans and makes decisions for valves, inverters, batteries, and power loads. Compared with the baseline controller, the MPC has better performance in capability, safety, and efficiency. A hierarchical MPC approach is implemented for the IPTMS of a UAV in [84]. It was found that the fuel burn can be reduced by 16% with the MPC optimisation strategy compared to the baseline strategy, and the MPC is more reliable. [PERSON] et al. [85] implemented PMP to optimise the performance of the rechargeable battery pack and the TMS for a parallel turboelectric aircraft. The battery is cooled by bleed air extracted from the engine, and it can provide power for propulsion.