Comparison of Bayesian Parameter Estimation and Least Squares Minimization for Inverse Grey-Box Building Model Identification

Comparison of Bayesian Parameter Estimation and Least Squares Minimization for Inverse Grey-Box Building Model Identification Submitted to: Prof. R. Balaji Prepared by: Greg Pavlak December 14, 2011

Abstract Bayesian parameter estimation and nonlinear least squares minimization are used for Inverse Grey-Box model identification of a retail and large commercial office model. Detailed simulation engine EnergyPlus is used to generate surrogate data for estimation of parameters, and optimal parameters are compared through annual simulation of building zone temperature and thermal loads. A brief overview of Bayesian estimation techniques is provided, along with ideas for improvements and future work.

Acknowledgements I would like to acknowledge my colleague Anthony Florita for generously providing a MAT- LAB Bayesian parameter estimation routine and kindly offering his knowledge of Bayesian statistics.

Contents 1 Introduction 1 2 Detailed Building Models 1 3 Reduced Order Model Structure 2 4 Least Squares Parameter Estimation 4 5 Bayesian Parameter Estimation 5 6 Results 6 Retail Model Parameter Estimation.............................. 6 Large Office Model Parameter Estimation........................... 10 7 Conclusion 13 8 Future Work 14 List of Figures 1 5 Zone Retail Building Model............................... 1 2 15 Zone Office Building Model.............................. 1 3 5 Paramater Thermal RC Network............................ 2 4 Retail: R1R2.............................. 6 5 Retail: R1R3.............................. 6 6 Retail: R1Rw............................. 7 7 Retail: R1C.............................. 7 8 Retail: R2R3.............................. 7 9 Retail: R2Rw............................. 7 10 Retail: R2C.............................. 8 11 Retail: R3Rw............................. 8 12 Retail: R3C.............................. 8 13 Retail: RwC.............................. 8 14 Retail: Cooling Load Comparision............................ 9 15 Retail: Zone Mean Air Temperature Comparison.................... 9 16 Office: R1R2.............................. 10 17 Office: R1R3.............................. 10 18 Office: R1Rw............................. 10 19 Office: R1C.............................. 10 20 Office: R2R3.............................. 11 21 Office: R2Rw............................. 11 22 Office: R2C.............................. 11 23 Officel: R3Rw............................. 11 24 Office: R3C.............................. 11

25 Office: RwC.............................. 11 26 Office: Cooling Load Comparision............................ 12 27 Office: Zone Mean Air Temperature Comparison.................... 13 List of Tables 1 Selected EnergyPlus Model Details............................ 2 2 Reduced Order Model Parameter Bounds........................ 5 3 Retail Model Parameter Estimates............................ 8 4 Office Model Parameter Estimates............................ 12

1 Introduction Advanced building control and fault detection methods utilize building energy models to predict or estimate expected building performance. Online implementation of such methods requires light-weight, computationally efficient models that capture the critical system dynamics. Inverse Grey-Box models have shown the potential for blending the benefits of building physics knowledge with measured performance data. Inverse Grey-Box building models have been successfully used to predict cooling loads and energy consumption for optimal control strategy evaluation, as well as online next-day load predictions [1],[2], [8]. Extended Kalman Filters (EKF) have also been incorporated with similar model structures to improve real-time load estimates using available BAS data [5]. Various model identification techniques have been demonstrated that typically involve time or frequency domain least squares minimization via traditional (e.g. Gauss-Newton) or metaheuristic (e.g. genetic) algorithms. Lauret et al. demonstrated the use of Bayesian parameter estimation in determining better estimates of the inputs for roof-mounted radiant barrier system forward model [4]. This paper applies Bayesian parameter estimation methods to Inverse Grey-Box models and provides comparison with least squares model identification. 2 Detailed Building Models Detailed simulation engine EnergyPlus was used generate surrogate surrogate data for two building models: 1) a 5 zone retail building and 2) a 15 zone office building. The load calculations from these detailed simulations is used as measured training data for the reduced order models. Figures 1 and 2 illustrate model geometry, and Table 1 highlights selected model details. Figure 1: 5 Zone Retail Building Model Figure 2: 15 Zone Office Building Model 1

Table 1: Selected EnergyPlus Model Details Property Retail Office Units Floors 1 32 floors Total Floor Area 2300 77000 m 2 Occupancy 7.11 51.8 m 2 /person Lighting 32.3 9.8 W/m 2 Appliance 5.23 4.63 W/m 2 3 Reduced Order Model Structure Inverse Grey-Box models are based on the approximation of heat transfer mechanisms with an analogous electrical lumped resistance-capacitance network. This approximation creates a flexible structure that allows the modeler to choose an appropriate level of abstraction. Model complexity can range from representing entire systems with a few parameters, to modeling each heat transfer surface with numerous parameters. Depending on the model structure and complexity, model parameters can approximate physical characteristics of the system. Model parameters are then identified thought a training period with measured data. For this work, a 5 parameter model, shown in Figure 3, based on ISO13790 was used to predict summer cooling loads for a small retail and large office building [3]. Heat transfer through the opaque building shell materials is represented by R 1, T a R 1 T a T m T s T z C Q g,r+sol,w Q gc Figure 3: 5 Paramater Thermal RC Network, and C. These elements link the ambient temperature node to an pseudo surface temperature node (T s ), accounting for potential heat storage of the mass materials. Glazing heat transfer is represented by a single resistance connecting the ambient temperature node to the surface temperature node, as thermal storage of glazing is typically neglected. represents a lumped convection/radiation coefficient between the surface temperature node and zone air temperature node T z. The convective portion of internal gains (lighting, occupants, and equipment) are applied as a direct heat source to the zone temperature node, shown as Q gc, and the radiant fraction along with glazing transmitted solar gains ( Q g,r+sol,w ) are applied to the surface node. Although, convective and radiative splits are made, radiative heat transfer mechanisms are lumped together 2

in.an energy balance can be written on the mass temperature node T m, as C dt m dt = T a T m R 1 + T s T m (1) Since no storage occurs at the surface node, flows entering and leaving the node sum to zero. T a T s + T z T s + T m T s + Q g,r+sol,w = 0 (2) The heat gain to the space is then represented by the total heat flow to the zone air node. Q sh = T s T z + Q gc (3) Equations 1-4 form a first order differential equation that can be rewritten in state space form with non-zero matrix elements ẋ = Ax + Bu y = cx + du A(1, 1) = 1 C B(1, 1) = 1 C B(1, 2) = 1 C ( 1 R 1 + 1 + ( ( B(1, 3) = 1 R 1 C B(1, 6) = 1 C B(1, 7) = 1 C B(1, 8) = 1 C ( ( ( + + + + + + + + + + + + ) ) ) ) ) ) C(1) = + + D(1) = 1 + ( + + ) D(2) = + + D(6) = D(7) = D(8) = D(9) = 1 + + ) + + ) + + ) and state and input vectors described as x T = [T m ] u T = [T z T a T g Q sol,c Q sol,e Q g,r,c Q g,r,e Q sol,w Q g,c ] 3

T z is the zone temperature setpoint, T a is the ambient external temperature, Q sol,c is the external solar gains incident on the roof, Q sol,e is the solar radiation incident on exterior walls, Q g,r,c is the radiative portion of internal gains applied to the ceiling surface node, Q g,r,e is the radiative portion of internal gains applied to the wall surface node, Q sol,w is the solar radiation transmitted through glazing, and Q g,c is the total convective internal gains. The state space equations are then converted to the following heat transfer function presented by Braun [1], and the conversion process is described by Seem [6]. Q sh,t = n S T k u t k τ k=0 m e k Q sh,t k τ (4) The transfer function method is an efficient calculation method as it relates the sensible heat gains to the space ( Q sh ) at time t to the inputs of n and heat gains of m previous timesteps. Equation 4 is used to perform load calculations for the zone that include the effects of dual temperature setpoints with deadbands and system capacity limitations. Zone temperature predictions are also made using an inverse form of Equation 4. k=1 T z = 9 S 0 (l)u t (l) + 8 S j u t j τ 8 l=2 j=1 j=1 e k Q sh,t j τ + 2 Cz τ T z,t τ + ṁ inf C p u t (2) + Q zs,t 2 Cz τ S 0(1) + ṁ inf C p (5) T z,t = 2 T z,t T z,t τ 4 Least Squares Parameter Estimation As a first approach to model identification, least squares minimization was used to identify model parameters that minimize the root-mean-squared error (RMSE), defined by Equation 6, between the reduced order model and EnergyPlus load calculations. A two stage optimization was implemented that first performs a direct search over the parameter space to identify a starting point for local refinement. The direct search is performed on p uniform random points located within the bounds of the parameter space. The local refinement, subject to the same parameter constraints, is performed via nonlinear least squares minimization implemented using a built-in MATLAB optimizer based on trust-region methods [7]. N ( Q d,i Q zs,i ) 2 i=1 J = (6) N For this study, 500 direct search points were used within the bounds listed in Table 2. The parameter estimation was repeated 1000 times, beginning each iteration with a new set of randomly generated direct search points. 4

Table 2: Reduced Order Model Parameter Bounds R1 R2 R3 Rw C1 Min 0 0 0.03030303 78.008832 Max 4.988662132 4.988662132 33.33366667 3 536659.2 Units m 2 -K/W m 2 -K/W m 2 -K/W m 2 -K/W J/m 2 -K 5 Bayesian Parameter Estimation Bayesian methods benefit over traditional methods in that an entire distribution of parameter probabilities is developed and prior knowledge of the system can be incorporated in to the estimation task. Specifically, the individual probabilities of events A and B are related to their conditional probabilities by Bayes Theorem (Eq. 7). p(a), p(b) p(a B), p(b A) p(a B) = p(b A)p(A) p(b) (7) From a parameter estimation perspective, the probability of parameters Θ given measured data D and a knowledge base of the system K can be written as posterior probability p(θ DK). Bayes Theorem then allows the conditional probability p(θ DK) to be computed from p(θ K), p(d ΘK), and p(d K) as in Equation 8, p(θ DK) = p(θ K) p(d ΘK) p(d K) (8) where p(θ K) represents prior knowledge about parameter values, p(d ΘK) represents the likelihood of observing the measured dataset D given a particular parameter set and knowledge of the system, and p(d K) is the probability of randomly observing the dataset. The relation can be written in alternate form where the numerator remains the product of likelihood and prior, and denominator is a normalization factor so that posterior probabilities sum to unity. p(θ D) = p(θ)p(d Θ) p(θ i )p(d Θ i ) i (9) Assuming random Gaussian noise about the measured data, the probability of an observation can be determined from its location under the normal distribution centered at µ equal to the measured value, with standard deviation σ ɛ. p(o i Θ) = 1 exp( (O i M i ) 2 ) (10) σ ɛ 2π 2σ 2 ɛ 5

Assuming independent errors, the likelihood of the entire dataset is simply the product of likelihoods of all individual points. p(o Θ) = p(o 1, O 2,..., O n Θ) = p(o 1 Θ) p(o 2 Θ)... p(o n Θ) (11) Computing the likelihood requires having generated a sufficient knowledge base so that the probability of combinations within the parameter space can adequately be determined. To generate the knowledge base for this study, 100,000 simulations were performed by randomly sampling a parameter set from uniform prior distributions. This results in 100,000 datasets, generated from known parameters, to which our measured data can be compared. The uniform priors represent the belief that any parameter within the bounds is equally likely. 6 Results To compare results between least squares parameter estimation and the Bayesian methods. The 1,000 solutions from the least squares training Monte Carlo were plotted on against 2-D contour slices of the posterior distribution. This allows for comparison of the probability of a least squares solution from the Bayesian perspective. Results are provided for the retail and large office model in the following sections. Retail Model Parameter Estimation Figures 4 and 5 show that the least squares algorithm generally finds R1 values that are fairly probable. However, for R2, the optimizer sticks to the lower bound every time. (As a side note: If the local optimizer is unbounded it finds the optimal R2 value in the negative region.) The optimizer also seems to pick values very near the probable region for R3. 0.07 5P Retail Model 2 5P Retail Model 0.75 2 8 8 6 0.7 6 0.03 4 2 5 4 2 0.08 0.08 0.01 5 0 0.7 0.75 R 1 [m 2 0.8 0.85 0.7 0.75 0.8 R 1 [m 2 0.85 0.9 0.95 Figure 4: Retail: R1R2 Figure 5: Retail: R1R3 6

Figures 6 and 7 show that several solutions are found within the probable region of Rw, however there does not appear to be great consistency. Capacitance values appear cluster between regions of lower probability. 5P Retail Model 6 Retail Model 5.5 x 105 5 1.3 4 1.2 2 5 1.1 1 0.08 C J/[m 2 K] 4.5 5 0.9 4 0.8 0.4 R 1 [m 2 0.7 0.8 0.9 3.5 1 1.5 2 2.5 R 1 [m 2 3 3.5 4 4.5 Figure 6: Retail: R1Rw Figure 7: Retail: R1C Figure 8 again shows the lower boundary local optimizer choice for R2, and that R3 values are within the probable range. Figure 9 again shows the range of window resistances in and outside of probable ranges. 0.75 Retail Model 1.1 Retail Model 5 1 0.7 5 0.9 0.8 5 5 0.7 0.035 0.03 0.4 5 5 0.3 0 0.01 0.03 [m 2 0.07 0 0.01 0.03 [m 2 0.07 0.08 0.09 0.015 0.01 Figure 8: Retail: R2R3 Figure 9: Retail: R2Rw Figure 10-13 highlights the above results in a different manner by plotting the remaining combinations of posterior slices. Table 3 compares the median estimate from the 1000 nonlinear least squares trainings to the most probable Bayesian estimate. 7

x 10 5 5.4 5.2 Retail Model Retail Model 1.2 8 5 1 6 C [J/m 2 K] 4.8 4.6 4.4 5 0.8 4 2 4.2 4 0.4 0.08 3.8 3.6 0 0.08 [m 2 2 4 6 8 0.48 2 4 6 [m 2 8 2 4 6 Figure 10: Retail: R2C Figure 11: Retail: R3Rw 5.2 5 x 10 5 Retail Model 5 4.5 x 10 5 Retail Model 0.09 4.8 4 0.08 4.6 5 3.5 0.07 C [J/m 2 K] 4.4 4.2 4 3.8 3.6 3.4 3.2 C [J/m 2 K] 3 2.5 2 1.5 1 0.03 0 1 1.5 2 2.5 3 1 1.5 2 2.5 0.01 Figure 12: Retail: R3C Figure 13: Retail: RwC Table 3: Retail Model Parameter Estimates R1 R2 R3 Rw C Upper Bound 4.989 4.989 33.334 3.000 536659 NLSQ 0.849 0.000 77 84 504866 Bayes 0.784 0.018 13 0.795 352421 Lower Bound 0.000 0.000 0.030 0 78 Units m 2 -K/W m 2 -K/W m 2 -K/W m 2 -K/W J/m 2 -K The best parameter set from each method were used to perform an annual simulation of the building heating and cooling loads. The predicted thermal loads and zone temperatures are shown in Figures 8

14 and 15, respectively. Overall, the performance is virtually the same for the period shown. The annual RMSE of the load profiles increased by 4% with the Bayesian solution. x 10 4 0 Nonlinear Least Squares vs. Bayesian Parameter Estimation: Cooling Loads 1 2 Zone Sensible Load[W] 3 4 5 6 7 8 9 10 EnergyPlus 05P (least squares) 05P (Bayesian) 3840 3860 3880 3900 3920 3940 3960 3980 4000 4020 4040 Time [hours] Figure 14: Retail: Cooling Load Comparision 30 Nonlinear Least Squares vs. Bayesian Parameter Estimation: Zone Temperatures Zone Mean Air Temperature [C] 25 20 15 3840 3860 3880 3900 3920 3940 3960 3980 4000 4020 4040 Time [hours] EnergyPlus 05P (least squares) 05P (Bayesian) Heating SP Cooling SP Figure 15: Retail: Zone Mean Air Temperature Comparison 9

Large Office Model Parameter Estimation The same analysis was repeated, training the 5 parameter model to data from a large 32 story office building. It should be noted that some level of mismatch inherently exists between the simple 5 parameter model and complex office building. It seems that the 5 parameter model may be overly simple for this type of building, regardless of parameter estimation technique. Values for R2 and R3 are chosen near Bayesian probable locations, however the remaining parameter selection end up quite different. Figures 16-25 plot combinations of parameter slices for visualizing the estimation results. 35 Large Office Model 4 5 0.45 4 2 Large Office Model 4 2 3 25 0.4 0.35 0.3 5 0.08 0.08 2 15 5 1 0.73 0.74 0.75 0.76 0.77 R 1 [m 2 0.78 0.79 0.8 0.81 0.82 0 0.8 1 1.2 R 1 [m 2 1.4 1.6 1.8 Figure 16: Office: R1R2 Figure 17: Office: R1R3 1.6 1.5 Large Office Model 5 5.2 x 10 5 Large Office 5 1.4 0.45 5.1 1.3 0.4 5 0.45 1.2 1.1 1 0.35 0.3 5 C [J/m 2 K] 4.9 4.8 0.4 0.35 0.3 0.9 4.7 5 0.8 5 4.6 0.7 4.5 5 0.8 1 1.2 1.4 1.6 1.8 2 R 1 1 1.5 2 2.5 R 1 [m 2 3 3.5 4 4.5 Figure 18: Office: R1Rw Figure 19: Office: R1C 10

0.08 Large Office Model 0.09 5 Large Office Model 1.5 1.4 5 0.07 0.45 1.3 1.2 0.45 0.4 0.35 0.3 1.1 1 0.9 0.4 0.35 0.3 5 0.8 5 0.7 0.03 5 5 0.08 0.09 1 2 3 0.08 0.09 [m 2 1 2 3 Figure 20: Office: R2R3 Figure 21: Office: R2Rw 5 x 10 5 Large Office 1 1.6 Large Office Model 5 0.09 4.5 0.08 1.4 0.45 C [J/m 2 K] 4 3.5 0.07 1.2 1 0.4 0.35 0.3 3 0.03 0.8 5 2.5 5 5 5 [m 2 0.3 0.35 0.4 0.45 0.01 0.03 0.07 0.08 Figure 22: Office: R2C Figure 23: Officel: R3Rw 5.2 5.1 x 10 5 Large Office Model 5 x 10 5 5.3 5.2 5.1 Large Office Model 0.4 0.35 5 0.45 5 0.3 C [J/m 2 K] 4.9 4.8 4.7 0.4 0.35 0.3 C [J/m 2 K] 4.9 4.8 4.7 5 4.6 5 4.6 5 4.5 4.4 5 4.5 4.4 0 1 1.5 2 0.8 1 1.2 [m 2 1.4 1.6 1.8 2 2.2 Figure 24: Office: R3C Figure 25: Office: RwC 11

Table 4 shows the median values from the nonlinear least squares optimizations and the expected value of the Bayesian estimation. These best parameter sets were used for an annual simulation of building cooling loads and zone air temperatures. The results in Figures 26 and 27 show that despite several large differences in parameter selections, the performance is very similar for cooling load estimation. The temperature predictions are quite different for the models, the Bayesian tending to over predict floating temperatures, and the least squares tending to under predict. Table 4: Office Model Parameter Estimates R1 R2 R3 Rw C Upper Bound 4.989 4.989 33.334 3.000 536659 NLSQ 4.988 0.083 0.030 36 503638 Bayes 0.776 26 0 1.554 447401 Lower Bound 0.000 0.000 0.030 0 78 Units m 2 -K/W m 2 -K/W m 2 -K/W m 2 -K/W J/m 2 -K 0 Nonlinear Least Squares vs. Bayesian Parameter Estimation Cooling Load x 10 6 Cooling Load [W] 1 1.5 2 EnergyPlus 05P (least squares) 05P (Bayesian) 2.5 2900 2950 3000 3050 3100 3150 Time [hours] Figure 26: Office: Cooling Load Comparision 12

Nonlinear Least Squares vs Bayesian Parameter Estimation: Zone Temp 28 Zone Mean Air Temperature [C] 26 24 22 20 E+ 05P (least squares) 05P (Bayesian) Heating SP Cooling SP 18 2900 2950 3000 3050 3100 3150 Time [hours] Figure 27: Office: Zone Mean Air Temperature Comparison 7 Conclusion Overall, the Bayesian parameter estimation selected parameter values that performed similarly to the least squares optimization for both models. The fact that similar performance was observed despite several differences in parameter selection may suggest that the model is relatively insensitive to parameter values in the range of interest. This is somewhat intuitive due to the fact that commercial buildings tend to be driven by internal gains (lights, occupants, equipment, etc.), rather than external influences (ambient temperature, ground temperature, etc.). Also, the training was performed for a three week summer period. During the cooling season, average ambient temperatures tend to be nearer to the desired internal zone temperature than in the heating season. This further reduces the influence of building shell parameters since heat transfer is driven by this temperature gradient. In both models, the R3 resistance that participates in transferring the radiative fraction of internal gains to the zone node seemed to be estimated the best, while the window resistance showed some of the largest variations. This suggests that the best training data set to identify building shell parameters may be from winter night-time operation. Posterior contours were very useful for gaining further insight to the solution space and observing patterns in the least squares solutions. Computationally the Bayesian methods are much more demanding, however the above results should be useful in improving the least squares minimization. 13

8 Future Work Similar analysis could be repeated for models of higher complexity. Currently six Inverse Grey-Box models exist ranging from 5 to 21 parameters. As model complexity increase, several parameters can model similar characteristics. It would be interesting to see if parameter tradeoffs are observed in the posterior distributions by areas of equally high probability. The estimation could also be extended to incorporate hierarchical models to investigate parameter covariance quantitatively. The results are also sensitive to the measurement error σ ɛ. Lauret et al. provide some insight into dealing with this by incorporating it as a free parameter in the estimation [4]. This technique could be implemented as the current environment requires trial and error to find an appropriate value. (If σ ɛ is too large, then everything is similarly likely and the posterior is useless. If σ ɛ is to small, nothing is likely and the posterior is zero everywhere.) Improvements in the least squares training method are also planned. The total least squares could be minimized between the temperature and load profiles, and the effects of various training period durations and seasons could be explored as well. State correction as a function of past deviations may also be explored. 14

References [1] J. E. Braun, Ph.D. and N. Chaturvedi, An Inverse Gray-Box Model for Transient Building Load Prediction, HVAC&R Research, 8 (2002), pp. 73 100. [2] J. E. Braun, Ph.D., K. W. Montgomery, and N. Chaturvedi, Evaluating the Performance of Building Thermal Mass Control Strategies, HVAC&R Research, 7 (2001), pp. 403 428. [3] ISO, INTERNATIONAL STANDARD ISO 13790, tech. rep., International Organization for Standardization, 2008. [4] P. Lauret, Bayesian parameter estimation of convective heat transfer coefficients of a roofmounted radiant barrier system, Journal of Solar Energy Engineering, 128 (2006), pp. 213 225. [5] Z. O Neill, S. Narayanan, and R. Brahme, MODEL-BASED THERMAL LOAD ESTI- MATION IN BUILDINGS, in Fourth National Conference of IBPSA-USA, New York, 2010. [6] J. Seem, Modeling of Heat Transfer in Buildings, PhD thesis, University of Wisconsin-Madison, 1987. [7] I. The MathWorks, lsqnonlin, tech. rep., MATLAB help files, 2011. [8] Q. Zhou, S. Wang, X. Xu, and F. Xiao, A grey-box model of next-day building thermal load prediction for energy-efficient control, International Journal of Energy Research, (2008), pp. 1418 1431. 15