Abstract—Thermostatic Radiator Valves (TRV) had proved their significant contribution in energy savings for several years. However, at low heat demands, an unstable oscillatory behavior is usually observed and well known for these devices. This instability is due to the nonlinear dynamics of the radiator itself which result in a large time constant and high gain for radiator at low flows. A remedy to this problem is to make the controller of TRVs adaptable with the operating point instead of widely used fixed PI controllers. To this end, we have derived a linear parameter varying model of radiator, formulated based on the operating flow rate, room temperature and the radiator specifications. In order to derive such formulation, the partial differential equation of the radiator heat transfer dynamics is solved analytically. Using the model, a gain schedule controller among various possible control strategies is designed for the TRV. It is shown via simulations that the designed controller based on the proposed LPV model performs excellent and stable in the whole operating conditions.21672
I. INTRODUCTION
Efficient control of heating, ventilation and air conditioning (HVAC) systems has a great influence on the thermal comfort of residents. The other important objective is energy savings, mainly because of the growth of energy consumption, costs and also correlated environmental impacts. Hydronic radiators controlled by thermostatic radiator valves (TRV) provide good comfort under normal operating conditions. Thermal analysis of the experimental results of a renovated villa in Denmark, built before 1950, has demonstrated that energy savings near 50% were achieved by mounting TRVs on all radiators and fortifying thermal envelope insulation [1].
A.    System Description
The case study is composed of a room and a radiator with thermostatic valve. Disturbances which excite the system are ambient temperature and heat dissipated by radiator. It is assumed that heat transfer to the ground is negligible having thick layers of insulation beneath the concrete floor. Block diagram of the system is shown in Fig. 1. All of the symbols, subscripts and the parameters value are listed in table I and table II. The chosen values for all parameters are in accordance with the typical experimental and standard values [2]. As mentioned before, the case study is adopted to the one previously studied in [3]. The TRV is driven by a batterized stepper motor. Pressure drop across the radiator valve is maintained constant unlike what is taken as the control strategy in [4]. Instead, flow control is assumed to be feasible by the accurate adjustment of the valve opening. The valve opening is regulated by the stepper motor which allows the concrete adjustment.
 
Fig.1. Closed loop control system of room and radiator
B. Problem Definition
To maintain the temperature set point in a high load situation, TRVs are usually tuned with a high controller gain. The inefficiency appears in the seasons with low heat demand especially when the water pump or radiator is over dimensioned [5]. In this situation, due to a low flow rate, loop gain increases; and as a result oscillations in room temperature may occur. Besides discomfort, oscillations decrease the life time of the actuators. This problem is addressed in [4] for a central heating system with gas-expansion based TRVs. It is proposed to control the differential pressure across the TRV to keep it in a suitable operating area using an estimate of the valve position.
The dilemma between stability and performance arises when TRV is controlled by a fixed linear controller. Designing TRV controller for high demand seasonal condition, usually leads to instability in low demand weather condition. A high loop gain and long time constant are the main reasons of this phenomenon. In contrast, selecting a smaller controller gain to handle the instability situation, will result in a poor radiator reaction while the heat demand is high.
Figures 2 and 3 show the results of a simulation where oscillations and low performance occur. In the shown simulation results, the forward water temperature is at 50℃. The proportional integral (PI) controller of TRV is tuned based on Ziegler-Nichols step response method [6]. 散热器恒温阀英文文献和中文翻译:http://www.751com.cn/fanyi/lunwen_13992.html