The control strategy proposed in this paper has a two-level hierarchical structure shown below in Fig. 6. Low-level feedback controllers ensure setpoint tracking and disturbance rejection for HVAC subsystem. The high-level control subsystem controls the cabin air temperature and allocates references for low-level controllers.
Low-level control system
The evaporator outlet air temperature, i.e. the cabin inlet air temperature Tea,out, is controlled in a feedback loop to provide accurate and high-bandwidth tracking of the reference set by the high-level control system. The superheat temperature ΔTSH is regulated with respect to fixed reference ΔTSH,R = 5 °C, where the main aim of the corresponding feedback controller is to suppress disturbance influence including the one imposed by the action of outlet temperature controller. The linearized input–output HVAC model is characterised by coupled dynamics, which is described in Fig. 4a by four transfer functions linking the control inputs ωcom and av to the controlled outputs Tea,out and ΔTSH:
$$\left[ {\begin{array}{*{20}c} {T_{ea,out} \left( s \right)} \\ {\Delta T_{SH} \left( s \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {G_{11} \left( s \right)} & {G_{12} \left( s \right)} \\ {G_{21} \left( s \right)} & {G_{22} \left( s \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\omega_{com} \left( s \right)} \\ {a_{v} \left( s \right)} \\ \end{array} } \right]$$
(8)
It has been found that reasonably good control performance of superheat temperature regulation and evaporator setpoint tracking can be obtained for the given HVAC model by applying a simplified, decoupled control structure where only two main controllers Gc11(s) and Gc22(s) are used (Fig. 4a). The controllers are of proportional-integral (PI) type, and their parameters are tuned by using a search-algorithm optimisation procedure targeted to single-input single-output (SISO) linear system (Isermann 1981). The cost function to be minimised combines penalisation of closed-loop control error and control effort. Referring to the control structure shown in Fig. 4a, the cost functions for the two control loops are defined as:
$$\begin{aligned} {\min} \,J_{11} & = \frac{1}{1 + M}\sum\limits_{k = 0}^{M} {\left[ {\left( {T_{ea,out,R} - T_{ea,out} } \right)^{2} + r_{11} \left( {\omega_{com,R} - \omega_{com} } \right)^{2} } \right]} \\ {\min} \,J_{22} & = \frac{1}{1 + M}\sum\limits_{k = 0}^{M} {\left[ {\left( {\Delta T_{SH,R} - T_{SH} } \right)^{2} + r_{22} \left( {a_{v,R} - a_{v} } \right)^{2} } \right]} \\ \end{aligned}$$
(9)
where r11 and r22 are weighting coefficients which set the trade-off between control error suppression, i.e. performance, and control effort reduction, i.e. efficiency and relative stability. Since the linearized HVAC dynamics model parameters depend on the operating point, PI controller gain scheduling maps have been obtained by repeating the optimisation procedure for multiple operating points with fixed weighting coefficients r11 and r22. The analysis has showed that the most significant operating point parameters are the evaporator outlet air temperature Tea,out and the blower fan air mass flow \(\dot{m}\)ea, which results in two-dimensional scheduling maps for the controller proportional and integral gains. The final low-level control system structure is shown in Fig. 4b and it consists of two PI controllers with two pairs of gain-scheduling maps.
The low-level control system performance is illustrated in Fig. 5 for the full, 12-th order nonlinear process model, where blue lines denote the response of control system with fixed controller gains (tuned for Tea,out = 15 °C and mea = 0.05 kg/s), while green lines correspond to the control system with gain-scheduling applied. The evaporator air mass flow \(\dot{m}\)ea is kept at 0.075 kg/s, the superheat temperature reference is ΔTSH,R = 5 °C and the evaporator outlet air temperature reference with magnitude of ΔTea,out,R = 5 °C is applied at t = 1000 s. In comparison with the fixed-gain control system, the gain-scheduling control system achieves faster evaporator outlet air temperature response (Fig. 5a) and lower superheat temperature control error (Fig. 5b). The performance improvement is achieved by stronger compressor and expansion valve control efforts (Fig. 5c, d). Figure 5e, f show that optimal controller gains vary significantly, thus making the gain scheduling algorithm necessary to achieve optimal performance over a wide operating range.
It has been found that the closed-loop system performance can be further improved by taking into account the coupled dynamics of HVAC model, which are described in Fig. 4a by the cross-coupling transfer functions G12(s) and G21(s). In this case, the parameters of both PI controller were optimised simultaneously, with an option to include the cross-coupling controller terms/gains Gc12(s) and Gc21(s), as well (Fig. 4a). A multi-objective genetic algorithm was used as optimisation algorithm, because it allowed for overcoming the appearance of local optima and presenting the results in the form of Pareto frontier that enables the designer to select optimal solution based on his/her preference (Cvok et al. 2020). However, such procedure is more time consuming, especially when gain-scheduling is concerned.
High-level control system
In order to achieve favourable cabin thermal comfort while maintaining the maximum HVAC system efficiency, a supervisory high-level control system has been developed. According to the block diagram shown in Fig. 6, the high-level control system regulates the cabin air temperature Tc by commanding the cooling capacity \(\dot{Q}\)d. The cooling capacity \(\dot{Q}\)d is then transformed within a control allocation map to low-level controller inputs/references, which in this case include the evaporator outlet air temperature and air mass flow references Tea,out,R and \(\dot{m}\)ea,R, respectively, while in a more general case more inputs are possible, such as the condenser air mass flow \(\dot{m}\)ca,R. Using the cabin air temperature Tc and the cooling capacity demand \(\dot{Q}\)d as inputs to the control allocation map allows for omitting the cabin dynamics model when designing the control allocation map. This significantly facilitates allocation map generation, and, more importantly, makes the allocation map independent of cabin model and related disturbances (see Fig. 1).
To achieve optimal system performance, it is crucial to base the design of control allocation map on optimisation (Johansen and Fossen 2013). For the specific HVAC system and design case, control allocation is based on instantaneous, on-line optimisation. A linear search-based method is applied starting from the minimum blower fan air mass flow setpoint and corresponding evaporator outlet air temperature as initial guesses. The on-line optimisation relies on PMV and COP maps, both of which are prepared off-line as functions of two inputs (Fig. 2). However, in more general case when using multiple control inputs (e.g. \(\dot{m}\)ca), the dimension of COP map grows, which can lead to poor computational efficiency when using a linear search or may result in local optima when a more advanced, directional search approach is applied. To overcome these weaknesses, an alternative, off-line optimisation approach based on a multi-objective genetic algorithm can be applied, as presented in Cvok et al. (2020). That approach results in control input maps as functions of cabin temperature and cooling capacity demand (Fig. 6), which could be fitted by analytical models/functions, to facilitate the control strategy implementation and calibration.
At the superimposed level, a fixed-gain PI-type cabin air temperature controller Gc,CAB(s) is used with an option to add a gain scheduling algorithm in more general case (Fig. 6). Since the cabin air temperature dynamics are slow, the cabin air temperature controller and the control allocation strategy can have higher sampling time than the low-level controllers (10 s vs. 0.1 s).
The optimal control allocation map is obtained by minimising the following cost function for a wide range of operating points (\(\dot{Q}\)d, Tc):
$$J_{c} = K_{PMV} \left| {PMV\left( {\dot{m}_{ea,R} ,T_{c} } \right)} \right| + K_{COP} \frac{1}{{COP\left( {\dot{m}_{ea,R} ,T_{ea,out,R} } \right)}}$$
(10)
where KPMV and KCOP are weighting coefficients that set the trade-off between the two conflicting criteria: thermal comfort (PMV) and efficiency (COP). The control variables Tea,out,R and \(\dot{m}\)ea,R, are subject to the following constraints:
$$\begin{aligned} & \dot{Q}_{d} = \dot{m}_{ea,R} c_{p,ea} \left( {T_{ea,out,R} - T_{c} } \right) \\ & \dot{m}_{ea,R,{\min} } \le \dot{m}_{ea,R} \le \dot{m}_{ea,R,{\max} } \\ & T_{ea,out,R,{\min} } \left( {\dot{m}_{ea,R} } \right) \le T_{ea,out,R} \le T_{ea,out,R,{\max} } \left( {\dot{m}_{ea,R} } \right) \\ \end{aligned}$$
(11)