The output variable is the controlled variable, also known as the control objective. When HVAC system energy saving is studied in this paper, chiller profile should be incorporated into system modeling if electricity consumption is chosen as the control variable. This would increase the model complexity and therefore electricity consumption is not adopted as the control output variable in this research. The flow rate of supply chilled water is approximately constant due to the constant speed pumps. Therefore, energy saving can be realized by increasing the supply water temperature while maintaining the zone comfort. There is no need to obtain the quantitative relationship between the supply chilled water and chiller energy consumption because such relationship is not needed in the optimization calculation.

In Eq. (13), the supply water temperature is not used as a controlled variable. However, in this study, the bypass valve opening level is selected as the controlled variable as shown in Fig. 14. Here the HVAC system with the existing control system is demonstrated in the dash box in Fig. 8 and is illustrated in Fig. 9. Although the bypass valve opening level is a manipulate variable in the original system as shown in Fig. 8, it is used as a controlled variable as shown in Fig. 14 for the following reason. If the supply water temperature is selected as the controlled variable, the control structure in Fig. 14 has to be changed in Eq. (14) as a constraint condition. Because in order to ensure the indoor comfort, the supply water temperature cannot increase without limit, in other words, the bypass valve opening level must not be closed. Solving this constraint is same as the handling the opening level when it is used as the controlled variable, therefore, can also be solved through Eqs. (13)–(16).

The prediction horizon, N is normally selected as large as possible in the premise of ensuring the computation ability. In this study, the HVAC system is turned off in every night. Thus, the prediction horizon at each sample time is variable. In each sampling time, the optimal sequence  is obtained by using Eqs. (13)–(16). The first one of the sequence y*(k) is implemented on the process. As new information becomes available at the next sampling time, the above process will be repeated. Then a new sequence is generated, and the first value of the new sequence is used to control.

The cost function in Eq. (13) minimizes a sum of the deviations of the bypass valve opening level from the set point, which means the control target will be achieved. And the control structure of the system is shown in Fig. 15. In the figure, random variables are outdoor temperature and solar radiation, and they directly act on the prediction model and play the role of the feedforward. The prediction model, which is composed of the three transfer functions described above, predicts the future value of bypass valve opening level based on the two random variables and the supply chilled water temperature. The predicted supply chilled water temperature enters the controller as a feedback value, and then the controller, according to the optimization control algorithm, adjusts the supply chilled water temperature to control the bypass valve opening level to the setpoint.

4. Results and discussions

4.1. Model validation

The performance of the model under random variables is evaluated by the normalized root mean squared error (NRMSE) fitness value:

where y is the real output value and yˆ is the calculated output value. If the matching between the calculated value and the real value is too low, the parameters of the PSD method should be changed, and then the PSD method is re-applied to modeling until a satisfactory matching degree is obtained. The parameters include samples of overlap from section to section and the selected window function. After sensitivity test, parameters for PSD were set as follows: the window is 10 that means a Hamming window of equal length is used, and the overlap is the default value 50%.