Degrees of Freedom In HVAC Controls Design BY JOHN VARLEY, P.E., HBDP, MEMBER ASHRAE

This article was published in ASHRAE Journal, June 05. Copyright 05 ASHRAE. Posted at www.ashrae.org. This article may not be copied and/or distributed electronically or in paper form without permission of ASHRAE. For more information about ASHRAE Journal, visit www.ashrae.org. Degrees of Freedom In HVAC Controls Design BY JOHN VARLEY, P.E., HBDP, MEMBER ASHRAE Stable control of an HVAC system can only be achieved when the number of controlled variables are less than or equal to the degrees of freedom (DOF), which represents the maximum number of variables in the control system that can be manipulated by it. In complex HVAC applications, the number of points in a system, such as an air handler control sequence, can exceed 00. Therefore, understanding the DOF in such a system is essential in realizing a stable control sequence. This article reviews the process of calculating the DOF and applies it to an air-handling system to inform the HVAC design community of its ease of application and benefits to the design process. Background Evaluating the control DOF is a necessary step when engineers design process control systems in the chemical process industry. However, in the HVAC industry, a DOF analysis is rarely, if ever, performed in the design of control systems. Why is DOF analysis ignored in the HVAC industry? The likely answer is that professionals in the industry do not perceive that it adds a value greater than its cost. Further, it may be ignored due to a lack of knowledge regarding its principles, application, and benefits. Nevertheless, in highly complex systems where dozens of variables are controlled, not performing this analysis increases the potential for instability in the system, which can create system failures and incur sizeable costs to resolve them. If a simple method of calculating the DOF were available and the design community realized it could eliminate uncertainty in their controls designs, DOF analysis would certainly become a standard part of the HVAC system design protocol. The control DOF defines the number of variables that can be influenced in a process: DOF = NV NE () Where NV = number of independent variables NE = number of independent equations relating the variables John Varley, P.E., is an associate at Harley Ellis Devereaux in Chicago. 44 ASHRAE JOURNAL ashrae.org JUNE 05

The DOF of a control system represents the maximum number of variables that can be manipulated by it. Therefore, the total number of independently acting controllers in a system cannot exceed its DOF. However, the accounting of independent variables and equations to determine the DOF is cumbersome and prone to error. As a result, simplified methods for calculating it have been developed. One of these methods, proposed by Rodríguez and Gayoso, is explored in this article with application to HVAC controls. In this approach the DOF for a control system is defined as follows: DOF = S i + S out + H A () Where S i = number of input streams S o = number of output streams H = energy flow in the system (H takes the value of if there is energy transfer to the system or 0 otherwise. In addition, coils add one degree of freedom. If the energy is transferred inside the system boundaries as with a process-to-process heat exchanger, then this variable will not add any DOF.) A = amount of inventories (liquid or gas) that are not controlled. In the case of HVAC systems, this variable includes any volume in the system where the flow rate is not regulated. (For example, each mixing or diverging flow that is not controlled will remove one DOF.) This is illustrated in the examples. For a complete process, the DOF is: DOF = S ip + Σ u (S out + H + A) () Where S ip are the inputs to the process, and Σ u is the sum of all the units in the process. The reader is encouraged to review Reference. Application During the early stages of generating contract documents, the design engineer creates schematic drawings of the HVAC systems to provide a road map for the system design. An example of one of these diagrams is shown in Figure. This diagram shows a typical control system for a laboratory space. Upon completion of FS Supply Air Heating Coil H C HHWR HHWS FIGURE Control schematic for airflow control in a laboratory. General Exhaust TS H C TT FS Temperature Transmitter Temperature Fume Hood Exhaust Air (s) FS FIGURE Simplified schematic for airflow control in a laboratory. the schematic diagrams, a DOF analysis is performed to determine if the controls elements are correctly specified. The system is simplified in Figure. Tagged with red numbers, the VAV valves are represented as throttling valves and are the independent controllers in the system. Therefore, the total number of controllers in the system is two. The fume hood exhaust valve is not included because it is controlled independently of the room control system. Also, the heating coil valve is not included in the count of controllers because it does not regulate the flow of air. Likewise, because the temperature control loops are not evaluated, the inventory in the coil is not included in the calculation of the degrees of freedom. Finally, the streams are labeled by black numbers. The DOF is calculated: Number of streams = (The thermal fluid for the coil is not counted as a stream.) Energy flow to the system = Number of inventories = (the space) DOF = + = (4) JUNE 05 ashrae.org ASHRAE JOURNAL 45

8 9 5 7 5 4 4 7 6 6 0 8 Coil N.O. 7 5 6 4 FIGURE (LEFT) Schematic for air-handling system. FIGURE 4 (RIGHT) Simplified schematic for air-handling system. Advertisement formerly in this space. Therefore, the number of controllers in the system is correctly specified. In fact, another controller could be added to the system without compromising the stability of the system. Next, the DOF analysis is expanded to the air-handling system shown in Figure. It is a variable air volume (VAV) system serving three floors. The system consists of a return fan on each floor, a relief damper at the roof, a return damper near the unit and minimum and maximum outside air dampers. The supply discharge air temperature is controlled by chilled water cooling and glycol heating coils. An array of four fans provides the flow. Not shown in Figure are the multiple VAV boxes that serve each floor. As in the fume hood control in the previous example, these boxes are not controlled by the system and are excluded from the DOF review. A dedicated variable frequency drive (VFD) adjusts the speed of each return fan while a single VFD manages the supply fans. A total of 70 points are provided in the controls scheme. As in the previous example, the first step in calculating the system control DOF is to simplify the system. In this case, the following assumptions are made: Safety control loops act independently of the system control and are not included in the analysis. 46 ASHRAE JOURNAL ashrae.org JUNE 05

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Because the heating and cooling coils are not used concurrently, they will be modeled as a single coil, and, thus, one inventory. Streams: mixing and diverting airflow junctions are each counted as a single inventory. Only the inputs to each inventory are counted if the outputs become inputs to a successive inventory. The floors are represented as space and are inventories. The DOF is calculated: Number of streams = 7 (The thermal fluid for the coil is not counted as a stream.) Energy flow to the system = (Work of the fan motors is neglected.) Number of inventories = 0 (7 mixers and diverters [shown in Figure 4 in the dashed circles], three spaces.) DOF = 7 + 0 = 8 (5) As shown in Figure 4, eight controllers are provided (tagged with red numbers). Therefore, the system control is correctly specified. Despite its complexity, the DOF analysis provides a quick means of verifying the correctness of the controls system design. If the number of controllers exceeds the DOF, controllers must be removed until the number is equal to or less than the DOF. Conclusion Evaluating the control DOF in an HVAC control system is a valuable aid in identifying potential instabilities in a control system. A simple method of calculating DOF is presented in this paper and applied to an air-handling system. References. Luyben, W.L. 996. Design and Control Degrees of Freedom. Industrial & Engineering Chemistry Research 5(7):04 4.. Huesman, A.E.M., O.H. Bosgra, P.M.J. Van den Hof. 007. Degrees of freedom analysis of economic dynamic optimal plantwide operation. 8th International IFAC Symposium on Dynamics and Control of Process Systems ()65-70. Cancún: International Federation of Automatic Control.. Rodríguez, M., J.A. Gayos. 006. Degrees of freedom analysis for process control. 6th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering ()489 494. Advertisement formerly in this space. 48 ASHRAE JOURNAL ashrae.org JUNE 05