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Development of Control-Oriented Models for Model Predictive Control in Buildings.


In the United States, the buildings sector (commercial & residential) accounts for nearly 41 percent of the primary energy consumption (DOE, 2010). A significant amount of energy in buildings is consumed by their heating, ventilation, and air-conditioning (HVAC) systems. Efficient and automatic building control algorithms could help improve the energy efficiency of HVAC systems and thus reduce the overall energy consumption in buildings (Oldewurtel et al., 2012). Traditionally, building control algorithms are based on ad-hoc and heuristic rule-based approaches that typically require significant effort in the tuning process during the commissioning and retro-commissioning processes. Furthermore, the tuned control approaches may not be close to the optimal solution and control performance cannot be guaranteed if the actual operational conditions have drifted from these trail-and-error based tuning ranges.

Recently, model-based optimal control has been investigated for the control of full scale (or partially) HVAC systems and thermal mass in buildings (Braun, 2003, Sun and Reddy, 2005, Wang and Ma, 2007, Zavala, 2011, Oldewurtel et al., 2012). Among possible approaches, model predictive control (MPC), an optimization-based control strategy, could be one means to realize near optimal performance under any operating conditions. Most of the previous MPC studies have been simulation-based assessments with very few demonstrations in real-buildings. In the authors' opinion, one of the most attractive features of MPC for application to building energy systems lies in its capability of specifying multi-objective cost functions, e.g., energy consumption and thermal comfort, while handling constraints for states and control (actuator or control setpoint) variables in a systematic manner.

Modeling accuracy is a key enabler for effective and robust controller performance. Compared to the process industry, where the effectiveness and benefits of MPC have been successfully demonstrated (Qin and Badgwell, 2003), models for building HVAC systems have larger uncertainties due to cost constraints that limit the number and quality of sensors that are available for model training or tuning. In this paper, we present a low-order state-space model for the thermal zones in buildings that has low cost sensor requirements. The development of this control-oriented model is followed by evaluations in terms of model accuracy compared to TRNSYS (Klein, 1976) predictions.


Model-based optimal control for building HVAC systems is gaining significant interest recently (Zavala, 2011, Siroky, 2011, Oldewurtel et al., 2012, Kim et al., 2012, Ma et al., 2012, Li et al., 2012). Most of the literature focuses on controller development, implementation, and energy savings. There appears to be very little work that has focused on modeling approaches (Privara et al., 2012a) and their impact on controller performance.

Some general procedures and guidelines for selecting building models for predictive-based optimal control design were documented in Privara et al. (2012b). Building models incorporated within whole building simulation programs such EnergyPlus, TRNSYS and ESP-r are not explicit (Privara et al., 2012b) and would require significant computational effort when integrated with MPC algorithms for an on-line implementation in buildings. Typically, there are two types of building models that are used in MPC: 1) black-box input-output models such as ARX (Autoregressive model with eXogenous input), ARMAX (Autoregressive--moving-average model with eXogenous inputs), BJ (Box-Jenkins), OE (Output-Error), and state-space models and 2) grey-box models such as resistance and capacitance thermal networks that are based on a physical understanding of the building. It is worth noting that accurate parameters of thermal-network models are typically determined from actual measurements using parameter estimation methods. Another grey-box approach is probabilistic semi-physical modeling (Privara et al., 2012b). This approach typically starts from stochastic differential equations that describe the physical process, from which the parameters are determined based on statistical methods such as maximum likelihood (Bacher and Madsen, 2011)

For autoregressive models, Ma et al. (2012) adopted a 5th order ARX model to predict zone temperature dynamics and used an EnergyPlus model as a virtual testbed. The model was incorporated within a controller that used a sampling interval of 15 minutes to minimize the combined energy and demand charges. There were no results showing model accuracy compared to the EnergyPlus model, but the authors did note the difficulty in obtaining accurate predictions using a linear model. The effect of model error on controller performance is to be investigated in their future work.

Oldewurtela (2012) used a state-space model with uncertainty estimation and a sampling time of 1 hour. Results of model mismatch with data were not presented. Privara et al. (2012a) investigated details on model selection and identification methods based on certain statistical criteria. The impact of proposed modeling and identification approaches on the closed-loop predictive control performance was not investigated.


MPC requires a dynamic model to predict the future response of a plant (e.g., building, its HVAC equipment and system) and make control decisions to minimize a predefined cost function. In order to have an effective MPC algorithm, the dynamic models need to have sufficient accuracy to achieve good MPC performance. A common practice for building MPC studies is to take advantage of the different dynamic time scales associated with the building envelope and HVAC equipment models. For supervisory level control optimization, it is often assumed that the HVAC equipment can be treated using quasi-steady-state models and only the dynamics associated with the building envelope structure and contents are considered. The goal in building modeling is to capture the dynamics of each zone temperature in response to external and control variables.

One building modeling approach involves use of a detailed physics-based (usually high-order) dynamic model such as a thermal-network model. One drawback of this approach is that it would generally require a significant amount of measurements for model parameter estimation, calibration and validation. In addition, a high-order model may lead to excessive computational requirements when embedded within a control optimizer. Alternatively, a low-order dynamic model based on system identification approaches seems to be more suitable for control development and field implementation of MPC. A data-driven approach has been recently adopted for modeling and control of building energy systems (Privara et al., 2011, Ma et al., 2012, Li et al., 2012). In this paper, we perform a simulation-based study to evaluate the effectiveness of data-driven based approaches and their applications in MPC.

Linear models are considered for control development in this study for the following reasons (Li et al., 2012):

* System identification for linear input-output parametric models is a well-established field. There exists many numerically robust and efficient tool such as the MATLAB System Identification Toolbox (Ljung and Singh, 2012), which allows effective and efficient model parameter estimation either offline or recursively during online system operation.

* Local linear models allow highly efficient real-time implementation of optimization-based algorithms and thus facilitate scalability of the optimal control approaches for HVAC systems (Privara et al., 2012b).

Modeling Approaches

According to ASHRAE (2009), modeling approaches can be classified within two categories: 1) forward (classical) modeling, and 2) data-driven (inverse) modeling. Forward modeling approach typically starts from exploiting physics of the system. For example, a very detailed physics-based building envelope model could be built by inputting information for building geometry, physical parameters of each wall (internal/external surfaces) and windows, and their connectivity graphs. One of the drawbacks of this approach is that it would require a large number of parameters to be determined, which may not be easy to obtain from existing measurements in practice. Data-driven (inverse) modeling approaches, on the other hand, typically start from processing measurement data from the system. However, some inverse modeling approaches utilize a physics-based model structure, e.g., thermal-network, where the data is mainly used to train the model parameters.

The major advantage of the forward modeling approach is that it can be applied to systems in the design phase prior to construction (ASHRAE, 2009). On the other hand, a data-driven modeling approach is easier to apply for an existing system and is often more accurate in terms of predicting system responses compared to forward models (ASHRAE, 2009).

In our MPC study, we adopted a data-driven modeling approach and applied system identification methods to obtain the input-output model parameters for building thermal zones. There are many model formats that could fit into this modeling category. A low-order state-space model was developed in this study. More details about system identification and the choice of model formats will be presented in the next section. This model was developed for MPC problems with a limited range of prediction horizons where zone temperature setpoints are prescribed (i.e., not part of the optimization) and the objective is to minimize energy consumption (not energy cost).


Building Description

A detailed whole-energy simulation model of an existing multi-zone building (Building 101) was developed with TRNSYS (Klein, 1976) and used as a virtual testbed in this study. Building 101 is located in the Philadelphia Navy Yard and has been extensively instrumented to enable field studies as part of the DOE Energy-Efficient Buildings HUB (EEB HUB, 2012) efforts. The building automation system for Building 101 is currently being upgraded to enable MPC studies in the near future.

Our case study is focusing on the north-wing of Building 101 served by Air Handler Unit 3 (AHU). The HVAC system includes a direct expansion (DX) coil for cooling and a gas-fired boiler for heating. The AHU is connected to 8 VAV boxes downstream with reheat coils. The 8 VAV boxes serve a total of 10 zones, in which VAV #1 and VAV #8 serve zones 1 & 10 and zones 8 & 9, respectively. Figure 1a shows the floor map and zone mapping for the north-wing of

Building 101. Figure 1b shows the outside view of Building 101 at the Philadelphia Navy Yard.

Model Development: System Identification

System identification typically requires an iterative procedure involving the following steps:

1. Selection of input and output signals.

2. Functional tests to excite system dynamics within certain input frequencies of interest.

3. Selection of model formats and parameter estimation.

4. Validation of models based on an independent data set.

Until the model reaches an acceptable accuracy, the above steps may be repeated for an incremental improvement of the model accuracy.

For the building model, we considered the control inputs of VAV supply air flow rate setpoints and zonal supply temperature with ambient temperature as measured disturbances. The outputs of the model are zone air temperatures. Although solar radiation and internal gains were not included as measured disturbances, the responses of the zones to these inputs are significantly slower in comparison to VAV supply conditions. However, this may limit the ability of the model to look ahead over longer time intervals and to properly consider the effects of energy storage within the building mass. Note that for a MPC implementation, an additional model is needed to compute the VAV supply air temperature from the supply air flow and reheat valve position. For simplicity, the same sampling time was used for all the zones. Step response tests were first conducted to evaluate the dominant time scale of the system, the sampling time (3 minutes) was then chosen to ensure that there are at least 4-10 samples within the rise time of the step response (Astrom and Wittenmark, 1997).

In the early development, solar radiation was included as an input. However, it was found that the effect of solar radiation on the dynamic response of the zones was quite small and not included in the final model. This may not be the case for other buildings or if longer time horizon predictions are desired. Figure 2 shows the input (flow setpoints and valve positions) and output signals (zone temperature) selected in the system identification experiments.

The linear state-space model for zone temperature dynamics is given by

[??](k + l) = A[??](k) + Bu(k) (1)

[[??].sub.model](k) = C[??](k) + Du(k) (2)

where k is the current time step, [??] is the state vector, u is the input vector, [[??].sub.model] is the predicted zone temperature, and A, B, C, D are system matrices. The convective coupling between zones was not considered in the current TRNSYS model and thus the state-space models, but the importance of this coupling will be considered in our future studies. We considered a scalable approach to train the model in a semi-automatic fashion. A simultaneous zone model training structure was established to inject signals and perform identification experiments for all zones. The data was collected and converted to specific format that is consistent with the MATLAB System Identification Toolbox (Ljung and Singh, 2012).

Figure 3a and 3b show the excitation signals for supply air flow setpoint and reheat coil valve position, respectively. The ambient temperature profile during the functional test period is shown in Figure 3c. Figure 3d shows the maximum temperature, minimum temperature and maximum temperature difference of each zone during the functional tests. Most of the zones had a zone temperature variation larger than 10[degrees]C (18[degrees]F), which is credited to the full-range input signals employed in the identification experiments.

Our preliminary study suggested that a low-order (3rd order) ARX model identified (1) with the same training data and the same number of inputs could not predict the zone dynamics well when the magnitude of excitation was large, and it only worked well for a small range of operation (Li et al., 2012). Example validation results are shown in Figure 4a for an ARX trained for zone 5. Through a trial-and-error process, we selected the low-order state-space model as it yielded superior performance over the ARX model for the full-range of input signals. Figure 4b provides validation results for the low-order state-space model of zone 5, which has significant better performance compared to the ARX model predictions shown in Figure 4a.

Model Development: State Estimation

In order to use the state-space model for online control, the initial states at each times step should be estimated. Since we opted for a data-driven approach, the states have no physical meaning and it would be difficult to guess their actual values. Alternatively, one could consider the system to start from an equilibrium condition, but the initial transients of system states will be large since it is expected that the initial states are not zero.

To tackle this problem, we applied a linear Luenberger observer for the purpose of state estimation. The outputs from the TRNSYS model (i.e., virtual test bed) were assumed to be the "measured" values. The observer equation is given by

[[??].sub.model](k) = C[??](k) + Du(k) (3)

[??](k + 1) = A[??](k) + Bu(k) + L([T.sub.trnsys](k) - [[??].sub.model](k)) (4)

For a given zone, the observer gain L was determined based on an offline optimization that minimizes the sum of squared errors.

[mathematical expression not reproducible] (5)

where [N.sub.p] is the model prediction horizon in MPC, and [N.sub.obs] is the number of historical data points used in the optimization to obtain L, and [T.sub.trnsys] is zone temperature "measured" from the TRNSYS model. Figure 5 shows the block diagram for the signal flows among the TRNSYS model, state observer, and MPC controller.

Simulation Results

For system identification and validation, we performed functional tests during the 1st week of August using virtual weather data from 2011 (Weather Analytics, 2012). Data from the first three days was used for model identification, and data from the last two days was used for model validation. Figure 6a shows comparisons of 2nd order state-space model predictions with data for all 10 zones served by AHU3 based on validation data. To evaluate the effectiveness of model performance for MPC implementation, Figure 6b and 6c show comparisons of root mean squared error (RMSE) for the 10 zones with predictions of 1 hr., 2 hr., 4 hr., 8 hr., and open-loop scenarios based on functional test and TRNSYS baseline input-output data, respectively. The scenarios for predicting 1 hr., 2 hr., 4 hr., and 8 hr. ahead were realized by reinitializing the system states every 1hr., 2 hr., 4 hr., and 8 hr., respectively. The open-loop scenario did not involve any reinitialization of system states.

Note that we performed very aggressive functional tests (see Figure 3) to fully excite the system dynamics with predictive results in Figure 6b that appear to be quite good for control design. For the additional model validation results in Figure 6c, baseline feedback controllers were implemented within the TRNSYS testbed that adjusted air flow rates or reheat valve to maintain zone temperature setpoints (2). The TRNSYS zone air flow rates and supply air temperatures were then fed into the simplified model to determine the zone temperature responses that were compared with TRNSYS zone temperatures. Overall errors in the zone temperature predictions for this test case are presented in Figure 6c. This control input scenario is much different than that employed during the model training periods shown in Figure 3a and 3b. Overall, the model yielded much worse results for both short-term and long-term predictions than those presented in Figure 6c. The degraded model performance under this validation scenario is probably due to the fact that the effects of solar radiation and internal loads become more important relative to the control inputs (zone supply air flow rate and temperature), as compared with the validation scenarios based on functional test data. Future study will be conducted to address the aforementioned limitations. In particular, better models may be needed to approach optimal MPC performance in the presence of variable utility rates and demand charges because longer prediction horizons are required. Uncertainty analysis for internal and solar heat gain predictions should be carried out before these gains are added as inputs to the predictive model.


We have investigated data-driven modeling approaches for MPC in buildings and have demonstrated that reasonable model accuracy for zone temperature predictions could be obtained with low-order state-space models with full-range excitation signals to the dynamic system in the absence of any feedback control. The results were not nearly as good when using control inputs from baseline control (2) of zone temperature, particularly for longer prediction horizons. Additional work is necessary to assess the utility of the proposed modeling approach within MPC implementations compared to alternative modeling approaches that include internal gains and solar radiation as inputs, particularly for longer-term prediction horizons needed to account for building thermal mass effects. Future work will include an investigation of the modeling approaches and accuracy for Building 101 at the Philadelphia Navy Yard in 2013 using real-time measurements.


This work is funded by Energy Efficient Buildings Hub, sponsored by the Department of Energy under Award Number DE-EE0004261. The authors are thankful to Draguna Vrabie and Miroslav Baric for their team work, Russell Taylor for the development of the initial TRNSYS model, and grateful to Satish Narayanan, Trevor Bailey and Timothy Wagner for their project management and supervision of the work.


This paper was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.


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Pengfei Li, PhD

Associate Member ASHRAE

Zheng D. O'Neill, PhD, PE


James E. Braun, PhD


Pengfei Li is a senior research scientist. Zheng D. O'Neill is a principal investigator/staff research scientist at United Technologies Research Center, East Hartford, CT. James E. Braun is the Herrick Professor of Engineering, Purdue University, West Lafayette, IN.

(1) The ARX model was identified based on least-squares estimation method implemented in the MATLAB System Identification Toolbox.

(2) The baseline controller will either adjust VAV reheat valve position with minimum supply air flow rate when the zone temperature is below a prescribed heating setpoint or adjust supply air flow rate with reheat valve closed when the zone temperature is above a prescribed cooling setpoint. The supply air flow rate will be kept as minimum with reheat valve closed if the zone temperature could be maintained between the heating and cooling setpoints (i.e., no feedback control).
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Author:Li, Pengfei; O'Neill, Zheng D.; Braun, James E.
Publication:ASHRAE Conference Papers
Date:Jun 22, 2013
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