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# Transient Thermal Systems: Dynamics and ControlOPEN ACCESS

[+] Author Notes

Andrew Alleyne received his B.S.E. from Princeton University in 1989 in Mechanical and Aerospace Engineering and the M.S. and Ph.D. degrees in Mechanical Engineering in 1992 and 1994, respectively, from The University of California at Berkeley. He joined the University of Illinois, Urbana- Champaign in 1994 where he currently holds the Ralph M. and Catherine V. Fisher Professorship in the College of Engineering. He is the recipient of the 2008 ASME Gustus L. Larson Memorial Award and is also a Fellow of ASME. His research interests are a mix of theory and implementation with a broad application focus.

Neera Jain received her S.B. from the Massachusetts Institute of Technology in 2006 in Mechanical Engineering and her M.S. and Ph.D. degrees in Mechanical Engineering in 2009 and 2013, respectively, from the University of Illinois at Urbana-Champaign. She is a recipient of the Department of Energy Office of Science Graduate Fellowship (2010) and the ASME Graduate Teaching Fellowship (2011). She is currently a visiting member of the research staff in the Mechatronics Group at Mitsubishi Electric Research Laboratories. Her research interests include dynamic modeling and control design for multi-domain and large-scale energy systems.

Mechanical Engineering 136(03), S4-S12 (Mar 01, 2014) (9 pages) Paper No: ME-14-MAR4; doi: 10.1115/1.2014-Mar-4

## Abstract

This is a study focused on the criticality of thermal systems in almost every domain of energy conversion. Thermal systems are critically important to nearly all domains of energy conversion, and controls are vital to extracting maximal efficiency from the overall system. Understanding the dynamics of transient thermal systems is the first step towards effective control design. While a great deal of understanding of steady-state performance of an overall system already exists, the combined performance of coupled and interconnected systems during transients is still not well described or understood. This becomes more important with increased system complexity or increased transient relevance. Continued improvement in control-oriented modeling will be very valuable in terms of accuracy, speed, etc. With energy as a crucial theme for a sustainable future, it is clear that the Mechanical Engineering community must play a key leadership role in achieving this potential, since the thermal energy domain is one with which we are most familiar.

## Article

Energy exists in multiple domains, such as mechanical energy, electrical energy, and chemical energy. Thermal energy, the subject of this article, is similarly ubiquitous. The transfer of energy from one domain to another usually involves a thermal component in some portion of the process. The understanding of thermal energy transfer and conversion is critical to maintaining and increasing efficiency measures of all aspects of the energy supply chain, consisting of energy conversion, transmission, conservation, and scavenging, because it allows for better control of available energy resources.

The research communities in the fields of thermodynamics or heat and mass transport number in the thousands, and a great deal of excellent work is being performed across the world. However, it should be noted that the majority of the attention has focused on the spatial dependence of thermal system behavior as opposed to the temporal behavior 1,2. Most thermodynamics that is taught and researched assumes a static equilibrium approach, illustrated in the pressure-enthalpy (P-h) diagram depicted in Figure 12. Similarly, heat transfer is most often analyzed from a steady state perspective1. It is certainly possible to analyze temporally varying thermal systems, but the analytical solutions are usually available for only the simplest physical geometries or systems- an example of which involves heat conduction in a cylinder. For more complex geometries, the most common approach uses correlations for good spatial properties and numerical methods for determining time variations3. These numerical models may be good for system design, but they are less suited for analysis and potential control design since they do not present the types of functionality control engineers typically need, such as inputs, dynamic states and outputs.

FIGURE 1 Pressure-enthalpy (P-h) diagram of a standard vapor-compression thermodynamic cycle.

To maximize both the performance and efficiency of transient thermal systems, it is imperative that their dynamics be well understood and their control systems well designed. Many of these systems operate within constraints; pushing them closer to their constraints increases their performance/efficiency but may also increase their susceptibility to harm. One example is the tradeoff of evaporator efficiency versus compressor protection in a Vapor Cycle System (VCS) that manifests itself as a safe minimization of evaporator superheat. This is shown as location “1” in Figure 1 depicting the increase in enthalpy beyond the vapor dome occurring at the compressor inlet. Too little superheat risks crossing under the saturation dome and exposing the compressor to undesirable liquid ingestion that increases compressor failure rates. Too large a superheat, while safe, greatly reduces the efficiency of the evaporator by not maximizing heat transfer through the heat exchanger.

Being able to control these systems close to the edges of acceptable safety thus enables us to extract greater system efficiencies, but guaranteed safety, often through control, is imperative for any practical acceptance.

In this article we outline several approaches to control-oriented dynamic modeling of transient thermal systems. We also illustrate the advantage of model-based control design to improve system performance.

## Modeling and System Dynamics

In order to understand transient thermal systems, it is imperative that we are able to construct relatively accurate models for them. Here we review three standard approaches, in varying levels of complexity.

## Lumped Parameter Models

Lumped parameter models perform basic balances of mass and energy regarding some control volume of interest. The dynamics within the control volume are ‘lumped’ together into a small number of variables of interest, hence the name. An example of such a lumped parameter system would be the control volume shown in Figure 2.

FIGURE 2 Lumped parameter representation of thermal energy change.

The balance of energy into and out of the volume would be

$mCpT˙=∑heat in flow=hin︸heat transfer coefficient⋅Ain︸cross sectional area⋅Tin−T−houtAoutT−Tout.$

This simplified model leads to 1st order system behavior. This approach is often sufficient to coarsely represent a large number of transient thermal problems, particularly where there is no phase change. The heat transfer coefficient is governed by the specific problem: convection or conduction. If there is mass accumulation in a system, then a conservation of mass calculation can be used to determine system characteristics such as pressure in a given volume.

FIGURE 3 Finite volume representation of heat conduction problem.

###### Finite Volume Models

An alternative to the lumped parameter approaches are finite volume modeling approaches. Conceptually, one can think of these as lumped parameter models but with a large degree of spatial discretization. A given system would be reduced to a collection of N individual subsystems, each with a fixed volume. Within each volume, the appropriate equations of state can be numerically integrated with the adjacent volumes providing the boundary conditions. Figure 3 demonstrates a 1-dimensional heat conduction example discretized into N=6 distinct volumes with heat flowing from hot to cold reservoirs.

Within each zone, the individual heat conduction equations can be determined across each interface with U representing the conductance.

$Cp_iT˙i=Q˙in_i−Q˙out_i i ∈1,⋯,6$

$Q˙in_i=U⋅A⋅Ti−1−Ti; Q˙out_i=U⋅A⋅Ti−Ti+1 i ∈1,⋯,6$

This results in a series of coupled equations with the number of dynamic variables equal to the number of volume elements. Figure 4 shows a 1 dimensional combined heat and mass transport problem where an energy carrying fluid interfaces with a heat exchanger which then rejects the heat to ambient. The number of states equals 2N where N is the number of individual volume elements. Each volume utilizes energy and mass conservation to track the thermal energy flowing from the fluid to the heat exchanger wall (Qc_w) and then from the wall to the ambient air (Qwvggb gbb_a).

FIGURE 4 Finite volume representation of heat conduction problem.

It is also possible to utilize conservation of momentum in each volume to determine changes in pressure as the fluid flows from one zone to the next. However, this may introduce numerical challenges due to incompressibility if the fluid is liquid. The resulting system will have fast pressure dynamics relative to the thermal dynamics making for a stiff set of differential equations to solve.

The advantage of the finite volume approach is a spatial resolution that the lumped parameter approach does not have.

FIGURE 5 One dimensional moving boundary modeling approach with 3 zones.

This resolution usually results in increased accuracy since the variations in parameters such as specific heat and/or heat transfer coefficients can be tailored to individual subsections of a given component. For example, a bend in a heat exchanger could be given a different pressure drop coefficient than the straight sections of the heat exchanger thereby resulting in a more precise system simulation. The primary drawback of the finite volume approach is the increased state dimensionality resulting from an increased number of volumes. This reduces its simulation speed and limits its use in real-time applications. For most modern computing platforms, this is not a serious concern. However, the high dimensionality of the approach creates many more dynamic states than can be measured or even estimated. Therefore, the finite volume approach is better suited for ‘desktop’ system simulation than it is for controller or estimator design4.

###### Moving Boundary Models

Moving boundary models attempt to bridge the behavior of both the lumped parameter and finite volume approaches5,6. Moving boundary methods have a lumped parameter formulation for a particular volume. However, the boundaries of that volume are allowed to move and vary as a function of the dynamics of the overall system. As shown in Figure 5, the N zone cooling system of Figure 4 is replaced by a 3 zone model, but the length of each zone, denoted by ζj (t)L, j ∈ {1,2,3}, can vary as a function of time. Here, ζj(t), j ∈ {1,2,3} is the fraction of the total volume, L, encompassed by the j-th zone. The dynamics of each zone are governed by their j individual conservation laws (mass, momentum, energy) which results in differential equations for each zone. For many classes of systems, particularly those involving fluid phase changes, the moving boundary approach allows for improved accuracy over low order lumped parameter models. However, it also avoids the high dimensionality of the finite volume approaches thereby proving amenable to controller or estimator design5, 6.

###### System Nonlinearity

The modeling relationships presented above are often valid only around an operating point. One of the key aspects of thermal systems is their large change in behavior as a function of operating conditions: i.e. nonlinearity. The mass flow elements, such as fans, compressors, or pumps, usually have an efficiency that varies with operating condition. For example, the nonlinear relationship between air flow rate and power consumption of a typical fan follows a 3rd order polynomial7. Similar nonlinear variations in efficiency occur in pumps/compressors and valves7.

Similar to the mass flow elements, the thermal energy transfer elements in these systems also vary in their behavior as a function of operating condition. Convective heat transfer coefficients (i.e. gains) change drastically as a function of Reynolds number. Should the working fluid also perform a change in phase, the resultant changes in heat flow gains are amplified.

The variations in critical component gains can make the transient performance of thermal management systems inherently nonlinear. Figure 6 illustrates a VCS operating at different compressor speeds where the valve and compressor are changed about their nominal operating point. Nonlinearity is evidenced by the large variation in evaporator superheat owing to 100 rpm step variations in compressor speed at different operating speeds8. There is a factor of 7 change in the steady state response as the operating condition changes by a factor of 2.5. This affects the ability to perform tight control of the system with a single controller. The current state of the art is to linearize the system behavior about critical operating points and switch system representations and controllers as a function of these points 9,10.

FIGURE 6 Time domain illustration of nonlinear gain in system step responses.

## System Modeling and Control Example: Refrigerated Transport

As a case study, the modeling techniques described previously will be applied to a truck transport refrigeration system (TTRS). This example is chosen here because it is susceptible to widely varying external transient loads, more so than a home or building. Additionally, it is composed of multiple sub-systems some of which, such as the refrigeration unit, have a sufficiently low thermal capacitance that transient analysis is necessary. A typical TTRS in operation is shown in Figure 7a. Here a VCS unit is attached to the cargo compartment of a trailer as it transports perishable food. Figure 7b gives a detailed view of the VCS unit that conditions the environment within the trailer.

FIGURE 7A VCS unit on a trailer.

Photo: Thermo King Corp

FIGURE 7B Standalone VCS unit.

Photo: Thermo King Corp

The schematic of the VCS in Figure 8 shows several components interconnected to form a complete system. The ambient side of the system rejects hot air to the external environment, and the refrigerated side pulls heat from the conditioned environment. There are several components in the physical system of Figure 7b that have been abstracted away in the schematic shown in Figure 8 including the prime mover operating the compressor. The modeling approach described in the section on Moving Boundary Models has been used to model and validate a number of different VCSs6, including this example TTRS. Figure 8 shows evaporator refrigerant pressure and evaporator air outlet temper-ature varying as the compressor starts and stops and as the thermostatic expansion valve (TXV) opens and closes13. Note the transients caused by the starting and stopping of the entire system; to properly capture these, a dynamic system representation is a necessity.

FIGURE 8 Validation of evaporator refrigerant pressure and evaporator air outlet temperature.

These systems often utilize a hysteretic on-off control logic, which turns the cooling system on or off when the temperature exceeds some maximum (Thigh) or minimum (Tlow) value respectively. The value of the nonlinear model is that it can perform an embedded online controller optimization, based on environmental conditions, and choose optimal (Thigh/Tlow) switching points for the controller logic. Figure 9a illustrates a locally convex cost function that for a specific condition, can be computed online. As conditions vary, the cost function minimum guides continuously varying switching conditions illustrated in 9b14. In reference 14, the online optimization approach demonstrated a 5-10% fuel savings over a well-tuned industrial baseline controller. The finite volume approach was too computationally intensive and the lumped parameter too inaccurate to achieve effective online optimization. The switched system moving boundary approach, documented in reference 15, achieved the right balance of complexity versus accuracy.

FIGURE 9A Locally convex cost function for online optimization.

FIGURE 9B Controller parameter (Thigh/Tlow) variation as a function of time.

NSF Sponsored Workshop on Building Systems

Buildings Currently Account for a Large Fraction of Global energy usage and contribute 40% of global greenhouse emissions. In the U.S. buildings consume approximately 40% of total energy usage, including 70% of electricity and 50% of natural gas. This is on par with all transportation systems and the carbon footprint is actually higher. As the world population becomes more urbanized, this cost will rapidly increase with the current pace of global urbanization.

The National Science Foundation (NSF) sponsored a Workshop on Building Systems at the University of Illinois Urbana-Champaign on 24-25 May 2010. The goals of this workshop were (1) to identify priority areas for research to improve energy efficiency of buildings and provide comfort, safety, and productivity of their occupants, and (2) to raise awareness in the controls community of the research issues in this area. Three dozen leading researchers from industry, academia and government convened to determine future directions for building ‘science’ from a systems and controls perspective. The participants were from industry, government, and academia. Four major issues were deemed fundamental to making progress:

Modeling Current modeling tools are not sufficient for predicting the actual behavior of buildings across the relevant range of dynamics, lack validation data, and are challenging to use and integrate into decisionmaking algorithms.

Estimation/Diagnostics The sensor networks in large buildings are extremely complex and while it is easy to get data, it is harder to obtain useful knowledge for modeling, diagnostics and control.

Information and communication Building network configuration and maintenance is a labor-intensive endeavor where the design, specification and deployment of networked sensors are challenging, particularly for retrofits. Integration and interoperability of building information sources could be greatly improved.

Controls and Optimization Controls systems must be developed that optimize the number of required sensors, provide building services at minimal cost, and are robust to variability and faults in equipment. This is required because buildings have a long lifetime and are not as carefully maintained as high precision equipment. Also, there is tremendous opportunity in integration with the power grid for demand response.

Since much of the energy use within buildings falls into the domain of the mechanical engineer, it is important that the ASME community, along with partner communities such as ASHRAE, IBPSA, and IEEE, turn significant attention to these systems.

## Advanced Transient Thermal Control Systems

The simple thermostatic approach given above is suitable for many systems where only one control degree of freedom is utilized. With increased electrification of modern systems, additional control degrees of freedom are available. These include electronically variable mass flow devices such as variable speed pumps, fans, and compressors as well as electronically variable valves. This electrification provides significant flexibility in system operation. A typical approach is to incorporate proportional-integral-derivative (PID) loops around individual pairs of inputs and outputs. For example, within a VCS such as the one shown in Section 3, it is important to control the compressor speed providing cooling capacity. At the same time, evaporator superheat should be controlled for system efficiency plus safety. Two separate PID controllers can be incorporated on each of these feedback loops whereby the compressor controls the cooling capacity and an electronic expansion valve controls evaporator superheat. Figure 10 illustrates an example of the type of ‘fighting’ that commonly arises between the two control loops as they both try to command the use of a common resource: refrigerant mass flow rate. One approach to this problem is to significantly detune the response of each individual controller and, for systems that operate at steady state, this is usually an acceptable option. The potential downside is the decrease in efficiency that may need to be absorbed in order to accommodate a lower performing controller.

FIGURE 10 Demonstration of controller fighting between individual feedback loops in a VCS system.

Note: Evaporator pressure is a surrogate for cooling capacity.

For systems that have multiple inputs and outputs, a direct approach is multiinput multi-output (MIMO) control. Since the majority of thermal systems have temperature time constants on the order of minutes to hours, they are relatively slow from the perspective of embedded processors. Additionally, the desired system performance often incorporates constraints into the formulation. Model-based optimization approaches lend themselves naturally to the solution of these types of problems. These include mixed integer linear/non-linear programming (MILP/MINLP)16 as well as more compact Model Predictive Control (MPC) approaches17,18. The use of more sophisticated constrained optimization approaches tends to scale with the associated thermal system. For district-level systems supplied by power plants, a static solution may be sufficient since the relevant time dynamics are slow relative to exogenous signals such as disturbances and setpoint changes. This means that MILP approaches via CPLEX, or other numerical methods, can be readily applied16. For smaller systems, such as buildings, supermarkets, or manufacturing facilities, an MPC approach can be highly effective once an overall system model capable of prediction has been created18. The compact system representations afforded by the switched systems moving boundary models15 provide an excellent balance of complexity versus accuracy for model based controls.

Figure 11 illustrates the performance of an MPC and its improvement over Figure 10 in tracking evaporator cooling capacity without ‘fighting.’ In addition to capacity tracking, this MPC also handles actuator constraints and minimizes transient compressor power consumption by incorporating a simple penalty on the actuation term. An MPC gives the designer of the controls the flexibility of easily considering multiple objectives, particularly in the presence of actuator constraints.

FIGURE 11(A) Control input signals generated by MPC and (B) feedback variable tracking performance for different penalties on compressor power consumption.

## Conclusions

Thermal systems are critically important to nearly all domains of energy conversion, and controls are vital to extracting maximal efficiency from the overall system. Clearly, understanding the dynamics of transient thermal systems is the first step towards effective control design. While a great deal of understanding of steady-state performance of an overall system already exists, the combined performance of coupled and interconnected systems during transients is still not well described or understood. This becomes more important with increased system complexity (number of components, operational modes, etc.) or increased transient relevance (e.g., power electronics). Continued improvement in control-oriented modeling will be very valuable in terms of accuracy, speed, etc. As we develop better transient understanding, it will be important to simultaneous introduce new control approaches to monitor, diagnose and optimize these systems. For larger scale systems (e.g. buildings, districts), a nonlinear, hybrid complex systems representation becomes appropriate. Common SISO control tools will be insufficient for extracting the performance and efficiency gains necessary. Model based optimization tools are an excellent option provided the appropriate models exist. Since many thermal systems have predictable loads (e.g. diurnal cycle), it is often possible to introduce a predictive model element to the optimization. These models should all be compact, low-dimensional, and consistent with the information gathering used to validate them. Since viable instrumentation will comprise sensors located at particular points in space, it is important for the associated system models to be able to predict thermal phenomena at individual points rather than over an entire spatial domain.

With energy as a crucial theme for a sustainable future, it is clear that the Mechanical Engineering community must play a key leadership role in achieving this potential since the thermal energy domain is one with which we are most familiar. Utilizing our understanding of the physical phenomena as well as control theoretic tools is the best way to ensure we maximize the use of key resources in the future.

The authors in the lab:Andrew Alleyne (foreground), in discussion with Neera Jain (background).

## References

F. Incropera and D. DeWitt , Fundamentals of Heat Transfer, Wiley, 1981.
J. Howell and R. Buckius , Fundamentals of Engineering Thermodynamics, McGraw-Hill, 1992.
W. Minkowycz , E. Sparrow and Murthy, J.Y., Eds., Handbook of Numerical Heat Transfer, 2nd ed., Wiley, 2006.
B. S. J. Braun and E. Groll , “A comparison of moving-boundary and finite-volume formulations for transients in centrifugal chillers,” International Journal of Refrigeration, vol. 31, no. 8, pp. 1437–1452, December 2008.
B. Rasmussen , “Dynamic modeling for vapor compression systems—Part I: Literature review,” HVAC&R Research, vol. 18, no. 5, pp. 934–955, September 2012.
B. Rasmussen and B. Shenoy , “Dynamic modeling for vapor compression systems—Part II: Simulation tutorial,” HVAC&R Research, vol. 18, no. 5, pp. 956–973, September 2012.
ASHRAE Hand book: HVAC Systems and Equipment, American Society of Heating Refrigerating and Air-Conditioning Engineers, 2012.
A. Alleyne , V. Chandan , N. Jain , Otten and R. Otten , “Modeling and Control of Air Conditioning and Refrigeration Systems,” The Control Handbook, Salem, MA, CRC Press, 2010.
B. P. Rasmussen and A. G. Alleyne , “Gain sheduled control of an air conditioning system using the Youla parameterization,” IEEE Transactions on Control Systems Technology, vol. 18, no. 5, pp. 1216–1225, 2010.
B. Hencey and A. G. Alleyne , “A robust controller interpolation design technique,” IEEE Transactions on Control Systems Technology, vol. 18, no. 1, pp. 1–10, 2010.
S. Bendapudi , J. E. Braun and E. A. Groll , “A comparison of moving-boundary and finite-volume formulations for transients in centrifugal chillers,” International Journal of Refrigeration, vol. 31, no. 8, pp. 1437–1452, 2008.
B. Li , N. Jain , W. Mohs , S. Munns , V. Patnaik , J. Berge and A. Alleyne , “Dynamic Modeling of Refrigerated Transport Systems with Cooling/Heating Mode Switch Operations,” HVAC&R Research, vol. 18, no. 5, pp. 974–996, 2012.
B. Li , Dynamic Modeling, Simulation, and Control of Transportation HVAC Systems, Urbana, IL: University of Illinois at Urbana-Champaign, Ph.D. Thesis, 2013.
B. Li , R. Otten , V. Chandan , W. Mohs , J. Berge and A. Alleyne , “Optimal On-Off Control of Refrigerated Transport Systems,” IFAC Control Engineering Practice, vol. 18, no. 12, pp. 1406–1417, 2010.
B. Li and A. Alleyne , “A Dynamic Model of a Vapor Compression Cycle with Shut-down and Start-up Operations,” International Journal of Refrigeration, vol. 33, pp. 538–552, 2010.
M. Carrion and J. Arroyo , “A Computationally Efficient Mixed-Integer Linear Formulation for the Thermal Unit Committment Problem,” IEEE Transactions on Power Systems, vol. 21, pp. 1371–1378, 2006.
F. Oldewurtel , A. Parisio , C. Jones , D. Gyalistras , M. Gwerder , V. Stauch , B. Lehmann and M. Morari , “Use of model predictive control and weather forecasts for energy efficient building climate control,” Energy and Buildings, vol. 45, pp. 15–27, 2012.
Y. Ma , A. Kelman , A. Daly and F. Borrelli , “Predictive Control for Energy Efficient Buildings with Thermal Storage: Modeling, Simulation, and Experiments,” IEEE Control Systems, vol. 32, no. 1, pp. 44–64, 2012.
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## References

F. Incropera and D. DeWitt , Fundamentals of Heat Transfer, Wiley, 1981.
J. Howell and R. Buckius , Fundamentals of Engineering Thermodynamics, McGraw-Hill, 1992.
W. Minkowycz , E. Sparrow and Murthy, J.Y., Eds., Handbook of Numerical Heat Transfer, 2nd ed., Wiley, 2006.
B. S. J. Braun and E. Groll , “A comparison of moving-boundary and finite-volume formulations for transients in centrifugal chillers,” International Journal of Refrigeration, vol. 31, no. 8, pp. 1437–1452, December 2008.
B. Rasmussen , “Dynamic modeling for vapor compression systems—Part I: Literature review,” HVAC&R Research, vol. 18, no. 5, pp. 934–955, September 2012.
B. Rasmussen and B. Shenoy , “Dynamic modeling for vapor compression systems—Part II: Simulation tutorial,” HVAC&R Research, vol. 18, no. 5, pp. 956–973, September 2012.
ASHRAE Hand book: HVAC Systems and Equipment, American Society of Heating Refrigerating and Air-Conditioning Engineers, 2012.
A. Alleyne , V. Chandan , N. Jain , Otten and R. Otten , “Modeling and Control of Air Conditioning and Refrigeration Systems,” The Control Handbook, Salem, MA, CRC Press, 2010.
B. P. Rasmussen and A. G. Alleyne , “Gain sheduled control of an air conditioning system using the Youla parameterization,” IEEE Transactions on Control Systems Technology, vol. 18, no. 5, pp. 1216–1225, 2010.
B. Hencey and A. G. Alleyne , “A robust controller interpolation design technique,” IEEE Transactions on Control Systems Technology, vol. 18, no. 1, pp. 1–10, 2010.
S. Bendapudi , J. E. Braun and E. A. Groll , “A comparison of moving-boundary and finite-volume formulations for transients in centrifugal chillers,” International Journal of Refrigeration, vol. 31, no. 8, pp. 1437–1452, 2008.
B. Li , N. Jain , W. Mohs , S. Munns , V. Patnaik , J. Berge and A. Alleyne , “Dynamic Modeling of Refrigerated Transport Systems with Cooling/Heating Mode Switch Operations,” HVAC&R Research, vol. 18, no. 5, pp. 974–996, 2012.
B. Li , Dynamic Modeling, Simulation, and Control of Transportation HVAC Systems, Urbana, IL: University of Illinois at Urbana-Champaign, Ph.D. Thesis, 2013.
B. Li , R. Otten , V. Chandan , W. Mohs , J. Berge and A. Alleyne , “Optimal On-Off Control of Refrigerated Transport Systems,” IFAC Control Engineering Practice, vol. 18, no. 12, pp. 1406–1417, 2010.
B. Li and A. Alleyne , “A Dynamic Model of a Vapor Compression Cycle with Shut-down and Start-up Operations,” International Journal of Refrigeration, vol. 33, pp. 538–552, 2010.
M. Carrion and J. Arroyo , “A Computationally Efficient Mixed-Integer Linear Formulation for the Thermal Unit Committment Problem,” IEEE Transactions on Power Systems, vol. 21, pp. 1371–1378, 2006.
F. Oldewurtel , A. Parisio , C. Jones , D. Gyalistras , M. Gwerder , V. Stauch , B. Lehmann and M. Morari , “Use of model predictive control and weather forecasts for energy efficient building climate control,” Energy and Buildings, vol. 45, pp. 15–27, 2012.
Y. Ma , A. Kelman , A. Daly and F. Borrelli , “Predictive Control for Energy Efficient Buildings with Thermal Storage: Modeling, Simulation, and Experiments,” IEEE Control Systems, vol. 32, no. 1, pp. 44–64, 2012.

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