A remedy to this dilemma is choosing an adaptive controller instead of the current fixed PI controller.
It is, also, worth mentioning that the same problem was investigated via simulation based studies in [3]. The LPV control oriented model of radiator was, however, developed based on simulations.
In order to validate the controller performance, we utilized simulation models of the HVAC components. Two approaches for HVAC systems modeling are the forward, [7], [8], [9] and the data-driven methods [10], [11], [12] indicated by [2].
Fig.2. Undamped oscillations in room temperature and radiator flow which occur in low demand situation while the controller is designed for high demand condition.
Fig.3.Poor performance in the cold weather condition, applying the controller designed for the low demand situation
In this paper, we adopted heat balance equations of the room model in accordance to the analogous electric circuit, described formerly in [13]. Radiator dynamics are formulated as a distributed system in order to analyze the radiator transferred heat.
Rest of the paper is organized as follows: In section II, the radiator transferred heat is derived analytically. Based on the result, control oriented models are developed in section III. Utilizing the models, the control structure based on flow adaptation is proposed in the same section. A simulation-based test is conducted in section IV. Discussion and conclusions are given finally in Sections V.
II. SYSTEM MODELING
A. Heat Balance Equations
Radiator is modeled as a lumped system with N elements in series. The nth section temperature is given by, [14]:
in which Crad is the heat capacity of the water and radiator material, Tn is the temperature of the radiator’s nth element and n = 1,2,…,N. The temperature of the radiator ending points are inlet temperature: T0 = Tin, and return temperature: TN = Tout. In this formulation, we assumed the same temperature of the radiator surface as the water inside radiator. Besides, heat transfer only via convection is considered. Kr represents the radiator equivalent heat transfer coefficient which is defined based on one exponent formula, [14] in the following:
in which θ0 is the radiator nominal power in nominal condition which is Tin,0 = 90℃, Tout,0 = 70℃ and Ta = 20℃. △Tm, 0 expresses the mean temperature difference which is defined as △Tm, 0 = -Ta in nominal condition. n1 is the radiator exponent which varies between 1.2 and 1.4, but 1.3 is the value of the exponent for most radiators. In such case, we can approximate the non fixed, nonlinear term in Kr with a constant between 2.5 and 3.2 for a wide enough range of temperature values. Picking 2.8 as the approximation value, Kr = 2.8× .
The power transferred to the room can be described as:
Heat balance equations of the room are governed by the following lumped model [9]:
in which Te represents the envelop temperature, T f the temperature of the concrete floor and Ta the room air temperature. Q r is the heat power transferred to the room by radiator. Each of the envelope, floor and room air is considered as a single lump with uniform temperature distribution.
Assuming a constant pressure drop across the valve, the thermostatic valve is modeled with a static polynomial function mapping the valve opening δ to the flow rate q:
The above presented radiator model is highly nonlinear and not suitable for design of controller; thus a simplified control oriented LPV model is developed in the next section.
B. Control Oriented Models
The relationship between room air temperature and radiator output heat can be well approximated by a 1st order transfer function. 散热器恒温阀英文文献和中文翻译(2):http://www.751com.cn/fanyi/lunwen_13992.html