Minimization of energy consumption in HVAC systems with data-driven models and an interior-point method

Energy Conversion and Management 85 (2014) 146–153

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Minimization of energy consumption in HVAC systems with data-driven models and an interior-point method Andrew Kusiak a,⇑, Guanglin Xu a, Zijun Zhang b a b

Department of Mechanical and Industrial Engineering, 3131 Seamans Center, The University of Iowa, Iowa City, IA 52242-1527, USA Department of Systems Engineering and Engineering Management, P6600, 6/F, Academic 1, City University of Hong Kong, Hong Kong

a r t i c l e

i n f o

Article history: Received 13 March 2014 Accepted 15 May 2014 Available online 13 June 2014 Keywords: HVAC Interior-point method Internal heat gain Multilayer perceptron Nonlinear optimization model Poisson process Time-series method

a b s t r a c t In this paper, a data-driven approach is applied to minimize energy consumption of a heating, ventilating, and air conditioning (HVAC) system while maintaining the thermal comfort of a building with uncertain occupancy level. The uncertainty of arrival and departure rate of occupants is modeled by the Poisson and uniform distributions, respectively. The internal heating gain is calculated from the stochastic process of the building occupancy. Based on the observed and simulated data, a multilayer perceptron algorithm is employed to model and simulate the HVAC system. The data-driven models accurately predict future performance of the HVAC system based on the control settings and the observed historical information. An optimization model is formulated and solved with the interior-point method. The optimization results are compared with the results produced by the simulation models. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Heating, ventilating and air conditioning (HVAC) systems have been recognized as major consumers of energy by residential and commercial buildings [1]. Thus, reducing the energy consumption of HVAC systems is desirable and has gained attention of research and industrial communities. The recent research on energy consumption of HVAC systems has focused on developing model-based control solutions [2]. Two main streams of model-based HVAC control research have been observed. The first one involves physics-based models and simulation tools. Bhaskoro et al. [3] studied energy saving of a centralized HVAC system with an adaptive cooling technique. A simulation program (TRNSYS) was utilized to model a building. Budaiwi and Abdou [4] utilized the Visual DOE building energy simulation program to model mosques and examined various HVAC simulation strategies. Lu et al. [5,6] formulated an overall model of a HVAC system by integrating the mathematical forms of its major components. Based on physics-based models, different HVAC controllers were designed and examined. Chu et al. [7] proposed a least enthalpy estimator based fan coil unit fuzzy control system for operating the HVAC system. Mossolly et al. [8] examined optimal control strategies of variable air volume air ⇑ Corresponding author. E-mail addresses: [email protected] (A. Kusiak), [email protected] (Z. Zhang). http://dx.doi.org/10.1016/j.enconman.2014.05.053 0196-8904/Ó 2014 Elsevier Ltd. All rights reserved.

conditioning system. Parameshwaran et al. [9] developed a genetic fuzzy optimization method to improve thermal comfort and indoor air quality requirements without compromising the energy savings potential. Physics-based models are explicit; however, usually they are abstract as they involve numerous assumptions. Control of HVAC systems with data-driven models is another active stream. The Neural Network (NN) algorithm [10] was often used to develop data-driven HVAC models. Jahedi and Ardehali [11] applied a wavelet based NN to identify the nonlinearity of HVAC system in studying its energy efficiency. Kusiak et al. [12,13] modeled the HVAC system with NN algorithm and studied opportunities of saving HVAC energy. In the past studies [11–13], NN models were treated as black boxes and lack of further analyses. In this paper, an in-depth study to control HVAC systems with a data-driven approach is presented. The objective is to minimize energy consumption while maintaining the indoor temperature within a specified range. Performance of the HVAC system is modeled with a NN algorithm. The topology of developed NN model is analyzed. The particular NN presented in this paper is differentiable. The uncertain occupancy level of the building is modeled. The Poisson and uniform distributions are applied to simulate the behavior of the occupants impacting the internal heat balance. Based on the NN model and the considered constraints, an optimization model is formulated. A nonlinear interior-point algorithm is used to solve the model. The interior-point algorithm was originally developed for linear programming and then

A. Kusiak et al. / Energy Conversion and Management 85 (2014) 146–153

147

Nomenclature ð1Þ

wj List of Roman letters a the input variable of activation functions in a neural network ATE a Jacobian matrix of cE() ATI cE() cI() d d e E[] f1() f2() f3() f4() g() g0 () h j k M1 M2 ð1Þ mj ð2Þ

mj

N n nd N(t) Q q S s T t Ti u ^ u U[a,b] W

a Jacobian matrix of cI() a set of equality constraints a set of inequality constraints the Newton direction a unit of time increment a size n vector, [1, 1, . . ., 1]T the expected value of a variable the data-driven model for predicting y at t + d the data-driven model for predicting T at t + d the data-driven model for simulating y at t the data-driven model for simulating T at t an activation function adopted by the hidden node in an MLP the identity function adopted by an output node in an MLP the heating load produced by each occupant per hour the index of nodes in the hidden layer the number of iterations for running interior-point method the number of hidden nodes in MLP based f1() the number of hidden nodes in MLP based f2() the matrix of input-hidden weights in an MLP model for predicting Tt+d the weight between hidden node j and output node in an MLP model for predicting Tt+d the number of occupants in the conditioned space the number of time increments of input parameters the number of data points the number of occupants at t a random variable describes the length of occupants’ stay in the conditioned space the value of the random variable Q a nn diagonal matrix contains all components of s a vector of nonnegative slack variables the indoor air temperature °C the current time a vector of selected historical states of T in fi(), i = 1, 2, 3, 4 the observed value of output parameter the predicted value of output parameter a uniform distribution to generate the random occupancy period of each occupant the Hessian matrix of the Lagrangian of a general nonlinear programming model

extended to nonlinear programming [14–17]. The solutions are the set points of the supply air static pressure and the supply air temperature. A case study is presented to demonstrate effectiveness of the proposed approach. 2. Model formulation and parameter selection 2.1. Description of the HVAC system A typical variable air volume (VAV) heating, ventilating, and air conditioning (HVAC) system includes a chiller, a chilled water

ð2Þ

wj x xi x1 x2

x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 y yi yE yG Z z

the matrix of input-hidden weights in an MLP model for predicting yt+d the weight between hidden node j and output node in an MLP model for predicting yt+d a vector of variables in the optimization model a vector of exogenous input parameters for fi(), i = 1, 2, 3, 4 the supply air duct static pressure set point, a controllable parameter the AHU supply air temperature set point, a controllable parameter the internal heating load the chilled water coil mixed water temperature the chilled water coil valve position the mixed air temperature the outside air flow rate the outside air inlet temperature the return air temperature the return air flow rate the return fan VFD speed the supply air flow rate the supply fan pressure differential the infrared radiation the outside air temperature the solar normal flux the variable air volume box damper position the variable air volume box velocity pressure differential the total energy consumption of the HVAC system (kW h) the vector of selected historical states of y in fi(), i = 1, 2, 3, 4 the energy consumption in the form of electricity (kW h) the energy consumption in the form of natural gas measured in kW h a nn diagonal matrix contains all components of z the vector of Lagrangian multiplier for cI()

List of Greek letters a the updating step size d() the optimality evaluation function eð1Þ the vector of hidden bias in an MLP model for predicting j yt+d e0ð1Þ the vector of hidden bias in an MLP model for predicting j Tt+d h() the nonlinear objective function k the vector of Lagrangian multipliers for cE() l the barrier parameters, l > 0 n the mean inter-arrival rate of occupants per hour

cooling coil, a heating coil, a mixing box, a supply fan, a return fan, pumps, dampers, and VAV boxes. A schematic diagram of such a system is illustrated in Fig. 1. In the HVAC system, the return fan circulates the air from the conditioned (heated) zones to the mixed air chamber. In the mixed air chamber, the return air is mixed with the outside air. The mixed air flows through the cooling coil (heating coil). In the cooling (heating) coil, the chilled (hot) water driven by pumps is used to remove (add) heat in the mixed air to a prespecified level. Next, the conditioned (heated) air is distributed to the conditioned (heated) zones by the supply fan. The chilled (hot) water after the heat exchange is then circulated by pumps

148

A. Kusiak et al. / Energy Conversion and Management 85 (2014) 146–153

Damper Exhaust air

Return air

Exhaust grille

Damper

VAV terminal Cooling coil

Heating coil

Outside air Mixed air

Supply air

Diffuser

Damper

Valve V-1

Valve

V-4

V-3

Valve

Fig. 1. Schematic diagram of a typical HVAC system.

to a centralized chiller for re-cooling (re-heating). The VAV boxes are utilized to reheat the supply air based on the comfort demand of various zones. The chiller, pumps, supply fan, return fan, and VAV reheating coils are the main consumers of the energy. In the HVAC system considered in this paper, the chillers, pumps, the supply fan, and the return fan consume electricity while the VAV reheating coils consume natural gas. Therefore, the total energy consumption of the HVAC system is formulated as (1):

y ¼ yE þ yG

ð1Þ

The goal of this research is to minimize the total energy consumption and maintain the indoor temperature (thermal comfort) at a desirable level by adjusting set points of two controllable parameters, the supply air static pressure and the supply air temperature. The indoor humidity is not considered as such data has not been collected in the experimental research facility. Moreover, humidity is not an issue due to the geographic location and the time period of the experiment. The internal heat gain plays a significant role in energy consumption of an HVAC system. It is significantly impacted by the behavior of occupants. In commercial buildings, the number of occupants can be described as a random variable and its expected value can be estimated. To determine its expected value, activities of the occupants need to be modeled. In this study, the arrival rate of the occupants is assumed as a Poisson process and their departure rate is assumed to follow a uniform distribution. Based on these assumptions, the number of occupants remaining in the conditioned zone at time t is expressed as (2).

E½NðtÞ ¼ n

Z

t

ð1  PðQ < qÞÞdq

ð2Þ

Data set Simulation model

Prediction model Inputs

Simulated data

Optimization model

Compare

Optimized result Report

Fig. 2. The relationship between the simulation model and the prediction model.

A data-driven approach is applied to develop prediction and simulation models. The data-driven prediction models of energy consumption and indoor temperature are formulated as (3) and (4).

ytþd ¼ f1 ðy1 ; T1 ; x1 Þ

ð3Þ

T tþd ¼ f2 ðy2 ; T2 ; x2 Þ

ð4Þ

Two data-driven models for simulating energy consumption and indoor temperature of a HVAC system are expressed as (5) and (6):

yt ¼ f3 ðy3 ; T3 ; x3 Þ

ð5Þ

T t ¼ f4 ðy4 ; T4 ; x4 Þ

ð6Þ

0

Since it is difficult to conduct experiments in an operating HVAC system with random number of occupants, simulation experiments are utilized. Simulation models are developed for running simulation experiments and generating data for prediction models. Prediction models are constructed for anticipatory control. Results of anticipatory control computed based on prediction models will be validated by the simulation results. Fig. 2 illustrates the relationship between the simulation model and the prediction model.

2.2. Experimental design and data collection The HVAC system in this study is installed in a one-floor energy research facility, the Energy Resource Station (ERS), located in Ankeny, Iowa. The ERS has two identical test areas, A and B, of four thermal zones and each area is equipped with an independent air

149

A. Kusiak et al. / Energy Conversion and Management 85 (2014) 146–153

handling unit (AHU). The weather data was collected from sensors installed around the ERS facility. The HVAC condition data was obtained from an experiment designed to investigate the impacts of two crucial AHU set points, the supply air static pressure (SASPSPT) and the supply air temperature set point (SAT-SPT), on the total energy consumption. The settings of SA-SPSPT and SATSPT impact the energy consumption of HVAC. Since the designed experiment aimed at generation of sufficient data to serve the development of data-driven models, the set points of both AHUs were randomly adjusted. In particular, the SA-SPSPT varied from 0.4 in. WG (0.1 kPa) to 1.8 in. WG (0.45 kPa). Each increment/decrement is 0.2 in. WG (0.05 kPa). The SAT-SPT varied from 50 °F (10 °C) to 65 °F (18.33 °C) at 1 °F (0.556 °C) increments/decrements. To simulate the internal heating load of occupants, electric energy was utilized. More than 300 parameters, including weather conditions, energy consumption, and indoor temperature, were collected at 1-min intervals by sensors. The data was collected from June 22 to July 16, 2011. The original data set with 1-min sampling interval was transformed to 1-h data. After transformation, a data set with 837 instances was obtained. Three processes, training, testing and validation, were employed to develop prediction and simulation models based on the transformed data. Table 1 described the number of data points utilized in training, testing and validation process. 2.3. Parameter selection The original data set included over 300 parameters and most of them were irrelevant to modeling the HVAC system. The presence of irrelevant parameters could negatively impact the accuracy of data-driven models. Therefore, it was essential to perform parameter selection to develop accurate, scalable, and comprehensive models [17]. Boosting tree regression [18,19] is an algorithm for ranking parameters in prediction. The boosting tree regression algorithm computes a sequence of trees in which each successive tree is built to predict residuals of the preceding tree. At each step of the boosting algorithm, the data is partitioned into two subsets at every split node. The best partitioning is determined, and the regression errors are computed. Then, the successive tree is fitted to reduce the error. In the process of generating successive trees, the statistical significance of input parameters at each split of every tree is accumulated and normalized. Parameters with a higher rank indicate a larger contribution to the predicted output. Performance of the boosting tree algorithm in parameter selection has been demonstrated in many studies [18,19]. Therefore, it is also considered in this research for parameter selection. Input parameters selected for prediction models of energy consumption and indoor temperature are described as a group of vectors, yi, Ti, and xi, i = 1 and 2, where y1 = [yt]T, y2 = [yt]T, T1 = [Tt]T, T2 = [Tt, TtT d] , x1 = [x1,t+d, x2,t+d, x3,t+d, x4,t, x5,t, x6,t, x7,t, x8,t, x8,td, x9,t, x11,t, x12,t, x13,t, x14,t, x14,td, x15,t, x16,t, x16,td, x17,t, x18,t]T and x2 = [x1,t+d, x2,t+d, x3,t+d, x6,t, x8,t, x9,t, x10,t, x12,t, x13,t, x14,t, x15,t, x16,t, x16,td, x17,t, x18,t]T. In input parameters, xi,t+d, i = 1 and 2, is the controllable parameters and x3,t+d is the simulated parameter. They are utilized to predict yt+d and Tt+d with other parameters contained in yi, Ti and xi, i = 1 and 2. The prediction model will be applied to compute value of x1,t+d and x2,t+d based on other parameters observed at time

t so that the energy consumption and indoor temperature at t + d can be optimized. Vectors, yi, Ti, xi, i = 3 and 4, where y3 = [ytd]T, T3 = [Tt]T, x3 = [x1,t, x2,t, x3,t, x4,t, x5,t, x6,t, x7,t, x8,t, x8,td, x9,t, x11,t, x12,t, x13,t, x14,t, x14,td, x15,t, x16,t, x16,td, x17,t, x18,t]T, y4 = [yt]T, T4 = [TtT d, Tt2d] , x4 = [x1,t, x2,t, x3,t, x6,t, x8,t, x9,t, x10,t, x12,t, x14,t, x15,t, x16,t, x16,td, x17,t, x18,t]T, present the input parameters selected for two simulation models of energy consumption, yt, and indoor temperature, Tt. The internal heating load, x3, in prediction and simulation models is estimated based on a linear model in (7).

x3;t ¼ h  EðNðtÞÞ

ð7Þ

3. HVAC modeling 3.1. Model construction Multilayer perceptron (MLP) [10] has been widely applied. Kalogirou et al. [20] applied neural networks to model solar water heating systems. Zhang et al. [21] discussed scheduling operations of a pump system with a neural network. Zhang and Huang [22] surveyed various applications of neural networks in manufacturing. Zhang et al. [23] applied neural networks to model the power generation of a wind turbine. Besides MLP, other algorithms including support vector machine regression [24], regression trees [25], k nearest neighbor [26], and autoregressive algorithms [27] can also be utilized to build data-driven models. Comparison of MLP and other algorithms in modeling the components of HVAC system has been studied in our previous work [28,29]. The MLP has been considered as more suitable algorithm for modeling HVAC system components [28,29]. The MLP is employed in this research to develop fi(), i = 1, 2, 3 and 4, based on training, testing and validation dataset in Table 1. The MLP is a feed-forward neural network model which maps input data accurately to an output. In this study, MLPs with three layers, input, hidden, and output layers, are considered. The number of nodes in the hidden layer is a random number between 1 and two times of the number of input parameters. The output layer contains only one node since the number of output parameter in a prediction or simulation model is one. One of the four activation functions, the hyperbolic tangent function, logistic function, exponential function, and identity function, is randomly selected by the nodes in the hidden layer of a MLP in training. The node in the output layer adopts the identity function as the activation function. The Broyden– Fletcher–Goldfarb–Shanno (BFGS) method [30,31] is applied to train the network. One hundred networks are developed in the training process and the best network is retained. The maximal number of epochs to train a network of a MLP is 100; however, the training is suspended earlier if the testing error stops decreasing. The explicit expressions of the MLP models to predict energy consumption and the indoor temperature at t + d are expressed in (8) and (9).

ytþd ¼ g 01

M1 X ð2Þ ð1Þ ð1Þ wj g 1;j ðwj x1 þ ej Þ

!

ð8Þ

j¼0

T tþd ¼ g 02

M2 X ð2Þ ð1Þ 0ð1Þ mj g 2 ðmj x2 þ ej Þ

! ð9Þ

j¼0

Table 1 Data description. No.

Data set type

Time period

Number of instances

1 2 3 4

Entire data set Training data set Test data set Validation data set

06/22 – 07/16/2011 Randomly selected from 06/22 to 07/14/2011 Randomly selected from 06/22 to 07/14/2011 07/15 – 07/16/2011

837 670 119 48

A. Kusiak et al. / Energy Conversion and Management 85 (2014) 146–153

Training and testing dataset are utilized to train 100 networks of MLP and determine the best one for fi(), i = 1, 2. The topologies of the best MLP based f1() and f2() are mathematically formulated as (10) and (11). In the hidden layer of f1(), all nodes select the logistic function as the activation function while exponential function is selected by hidden nodes of f2().

ytþd ¼

9 X

ð2Þ

wj

h

ð1Þ

ð1Þ

ðwj x1 þej Þ

1=ð1 þ e

Þ

i

ð10Þ

j¼0

T tþd ¼

20.00 18.00 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00

Energy consumption (kwh)

150

1

12 X ð1Þ 0ð1Þ ð2Þ m x þe mj e j 2 j

21

ð11Þ

61

Observed value

The simulation models, f3() and f4(), are also developed by MLP using on the same procedure.

!, nd

ð12Þ

Indoor temperature (°F)

nd X ^ i  ui j ju i¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X h Xn i2  nd d ^ i  ui j  ^i  ui j=nd ðnd  1Þ Std AE ¼ j u u j i¼1 i¼1

75.00 74.00 73.00 72.00 71.00 70.00 69.00 68.00 67.00 66.00 1.00

ð14Þ

i¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n hXn .  io2  nd d ^ i  ui j=ui Þ  ^ i  ui j ui =nd ðnd  1Þ Std APE ¼ ðj j u u i¼1 i¼1 ð15Þ

The validation dataset in Table 1 is applied to validate the quality of developed MLP models. Table 2 illustrates the performance of f1() and f2(). In Table 2, we can observe that the prediction accuracy of energy consumption is above 90% and that of indoor temperature is 99.7%. Fig. 3 and 4 demonstrate the predicted and observed values of the energy consumption and indoor temperature on 119 data points with sampling interval = 1-h. As shown in Figs. 3 and 4, the predicted values of the energy consumption and the indoor temperature closely follow the observed values. Table 3 lists the accuracy of f3() and f4() on generating simulation data of yt and Tt.

Predicted value

21.00

41.00

61.00

81.00

101.00

Time (1-h interval) Observed value

ð13Þ "n #, d X ^ MAPE ¼ ðjui  ui j=ui Þ nd

101

Fig. 3. Observed and predicted values of yt+d based on f1().

3.2. Model validation To evaluate performance of fi(), i = 1, 2, 3, 4, four metrics presented in (12-15) are utilized, the mean absolute error (MAE), the standard deviation of absolute error (Std_AE), the mean absolute percentage error (MAPE), and the standard deviation of absolute percentage error (Std_APE) [32].

81

Time (1-h interval)

j¼0

MAE ¼

41

Predicted value

Fig. 4. Observed and predicted values of Tt+d based on f2().

parameters, x1,t+d and x2,t+d, need to be computed to achieve the goal of optimization. The constraints considered in this optimization problem describe the feasible ranges of x1,t+d, x2,t+d and Tt+d. Due to the design of the HVAC system and preferences of the occupants, the x1 is only allowed to vary from 50 °F (10 °C) to 64 °F (17.7 °C). The x2 can vary between 0.4 in. WG (0.1 kPa) to 1.8 in. WG (0.45 kPa). The T should be maintained between 68°F (20.0 °C) and 73°F (22.8 °C). Based on the objective function in (10) and the considered constraints, the optimization model is formulated in (16).

min ytþd

x1;tþd ;x2;tþd

s:t: ytþd ¼

9 h i X ð1Þ ð1Þ ðw x þe Þ ð2Þ wj 1=ð1 þ e j 1 j Þ j¼0

T tþd

ð16Þ

12 X ð1Þ 0ð1Þ ð2Þ m x þe ¼ mj e j 2 j j¼0

68  T tþd  73

4. Optimization model

0:4  x1;tþd  1:8

4.1. Model formulation

50  x2;tþd  65

To optimize the energy consumption of the HVAC system, the objective, yt+d, is minimized while maintaining the value of Tt+d within an acceptable range. Optimal values of the controllable

Based on this optimization model, optimal settings of x1,t+d and x2,t+d can be computed for the operator once the system condition at t has been observed so that the system delay can be mitigated.

Table 2 Performance of the models predicting HVAC system energy consumption and indoor temperature. Objective

Data set

MAE

MAPE (%)

Std_AE

Std_APE (%)

yt+d

Training Testing

0.362 0.454

6.6 10.0

0.353 0.452

5.5 19.4

Tt+d

Training Testing

0.157 0.210

0.2 0.3

0.159 0.218

0.2 0.3

151

A. Kusiak et al. / Energy Conversion and Management 85 (2014) 146–153 Table 3 Performance of the simulation models of energy consumption and indoor temperature. Objective

Data set

MAE

MAPE (%)

Std_AE

Std_APE (%)

yt

Training Testing

0.214 0.247

4.3 5.4

0.176 0.205

4.1 5.2

Tt

Training Testing

0.091 0.114

0.13 0.16

0.085 0.11

0.12 0.16

4.2. The interior-point algorithm The interior-point method [14–17] was initially proposed by John von Neumann for linear programming and was then extended to nonlinear programming. To solve (16), the interior-point method is applied. Since the objective function and constraints in (16) are continuously differentiable, solutions satisfy the Karush–Kuhn–Tucker (KKT) conditions [33] will imply the optimality. Consider a general nonlinear programming model [34] shown in (17).

minhðxÞ x;s

s:t: cE ðxÞ ¼ 0

ð17Þ

Step.1 Initialization: Randomly generate initial value of x0, s0, k0, z0, and l0, where x0, s0, k0, z0, l0 > 0; Randomly generate value of two parameters r, s, where r, s e (0,1). Set k = 0. Step.2 Loop while stopping criterion is not satisfied If k > 0, then choose lk e (0, rlk); Loop while dðxk ; sk ; kk ; zk ; lk Þ  lk Step.2.1 Obtain the search direction (dx, ds, dk, dz) by solving (22); Step.2.2 Compute amax ; amax ; s z k k Step.2.3 Update (x , s , kk, zk) based on (23); Step.2.4 Set lk+1 = lk; Step.2.5 Set k = k + 1 4.3. Case study

cI ðxÞ  s ¼ 0 s0 The Karush–Kuhn–Tucker (KKT) conditions derived from (17) are formulated as (18-21).

rhðxÞ  ATE ðxÞk  ATI ðxÞz ¼ 0

ð18Þ

SZ  le ¼ 0

ð19Þ

cE ðxÞ ¼ 0

ð20Þ

cI ðxÞ  s ¼ 0

ð21Þ

The Newton’s method [34] is applied to solve the optimality conditions. According to the set of variables, (x, s, k, z), the optimality conditions in (18-21) can be written as (22).

2

W 6 6 0 6 6 A ðxÞ 4 E AI ðxÞ

32 3 2 3 0 ATE ðxÞ ATI ðxÞ dx rhðxÞ  ATE ðxÞk  ATI ðxÞz 76 7 6 7 7 ds 7 6 Z 0 S 7 SZ  le 76 7 6 ¼ 7 6 7 6 cE ðxÞ 0 0 0 7 54 dk 5 4 5 I

0

0

dz

Once the value of (dx, ds, dk, dz) is determined by Newton’s method, values of (x, s, k, z) can be updated as (23).

k ¼ k þ amax dk ; z ¼ z þ amax dz z z

ð23Þ

where

amax ¼ max fa 2 ð0; 1 : s þ ads  ð1  sÞsg s amax ¼ max fa 2 ð0; 1 : z þ adz  ð1  sÞzg z

Assumption 1. The occupied schedule is from 8:00 am to 12:00 am; Assumption 2. n is set to 15 per hour; Assumption 3. Q follows uniform distribution U[0, 1]; Assumption 4. The arrival and leaving processes are independent; Assumption 5. h is set to 400 BTU (421,740 J), including sensible and latent heating load. Based on the assumptions, the average number of occupants staying in the conditioned zone at time t can be estimated as (26).

cI ðxÞ  s

ð22Þ

x ¼ x þ amax dx ; s ¼ s þ amax ds s s

In this section, the validation dataset (from 07/15/2011 to 07/ 16/2011) in Table 1 is selected for case studies. To investigate the impact of the stochastic building occupancy, a simulation program is developed based on the following five assumptions. The uncertainty of occupants’ arrival and departure rates is modeled with the Poisson and uniform distributions, commonly utilized in queuing theory [35].

ð24Þ

with s e (0,1). Based on (18-20), the optimality evaluation function is expressed in (25).  n o   dðx; s;k;z; lÞ ¼ max rhðxÞ  ATE ðxÞk  ATI ðxÞz; kSZ  lek; kcE ðxÞk; kcI ðxÞ  sk ð25Þ

where k  k is the vector norm. The basic steps of the interior-point method [34] considered in this research are presented as IP-M Algorithm.

E½NðtÞ ¼ n

Z 0

(

t

ð1  PðQ < qÞÞdq ¼

nt  12 nt 2 ; 0  t  1 1 n; 1  t  16 2

ð26Þ

Since n is 15, the expected number of occupants during [8:00 a.m., R1 9:00 a.m.] is 0 ðnt  12 nt 2 Þdt ¼ 5 and the expected number of occupants during an hour after 9:00 a.m. is 7.5. According to the expected value, arrival rate and departure pattern, a MATLAB program is coded to simulate the average number of occupants. Based on the generated average number of occupants, the model (7) is employed to estimate the internal heating load from 07/15/2011 to 07/16/2011. The details of the simulated average number of occupants and internal heating load are shown in Table 4. Based on the simulated internal heating load as well as the observed set points of supply air duct static pressure and the supply air temperature, the simulation models in (5) and (6) can produce the simulated energy consumption and indoor temperature under the simulated internal heating load during 07/15/ 2011 and 07/16/2011. A sample of the simulated energy consumption and indoor temperature is demonstrated in Figs. 5 and 6. Based on the information provided in Table 4 and the observed data in validation dataset, the optimization model (21) is solved.

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Table 4 The simulated internal load of the HVAC system. Occupants’ heating (Wh)

Lighting (Wh)

Total internal load (Wh)

0:00–8:00 7/15/2011 8:00–9:00 7/15/2011 9:00–10:00 7/15/2011 10:00–11:00 7/15/2011 11:00–12:00 7/15/2011 12:00–13:00 7/15/2011 13:00–14:00 7/15/2011 14:00–15:00 7/15/2011 15:00–16:00 7/15/2011 16:00–17:00 7/15/2011 17:00–18:00 7/15/2011 18:00–19:00 7/15/2011 19:00–20:00 7/15/2011 20:00–21:00 7/15/2011 21:00–22:00 7/15/2011 22:00–23:00 7/15/2011 23:00–0:00 7/15/2011 0:00–8:00 7/16/2011 8:00–9:00 7/16/2011 9:00–10:00 7/16/2011 10:00–11:00 7/16/2011 11:00–12:00 7/16/2011 12:00–13:00 7/16/2011 13:00–14:00 7/16/2011 14:00–15:00 7/16/2011 15:00–16:00 7/16/2011 16:00–17:00 7/16/2011 17:00–18:00 7/16/2011 18:00–19:00 7/16/2011 19:00–20:00 7/16/2011 20:00–21:00 7/16/2011 21:00–22:00 7/16/2011 22:00–23:00 7/16/2011 23:00–0:00 7/16/2011

0 4.68 8.97 5.95 9.35 9.09 7.47 10.79 6.27 10.36 10.01 9.21 6.42 5.75 10.38 9.47 6.86 0.00 3.05 7.61 14.56 11.40 6.06 7.23 6.51 8.67 11.93 7.56 7.82 10.06 7.70 7.66 8.62 9.96

0 524.55 1004.25 666.9 1047.15 1017.9 836.55 1209 702 1160.25 1121.25 1031.55 719.55 643.5 1162.2 1060.8 768.3 0 341.25 852.15 1630.2 1277.25 678.6 809.25 729.3 971.1 1335.75 846.3 875.55 1127.1 861.9 858 965.25 1115.4

0 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 0 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

0 1524.55 2004.25 1666.9 2047.15 2017.9 1836.55 2209 1702 2160.25 2121.25 2031.55 1719.55 1643.5 2162.2 2060.8 1768.3 0 1341.25 1852.15 2630.2 2277.25 1678.6 1809.25 1729.3 1971.1 2335.75 1846.3 1875.55 2127.1 1861.9 1858 1965.25 2115.4

20 15 10 5 0

1

9

17

25

33

41

Static pressure setpoint (in.WG)

Avg. no. of occupants

Energy consumption (kwh)

Time

2 1.5 1 0.5 0

Time (1-h inerval) Simulated value

Indoor temperature (°F)

Fig. 5. Comparison between the optimized and simulated HVAC system energy consumption.

74 72 70 68 66 64

1

9

17

25

33

41

Time (1-h interval) Optimized value

9

17

25

33

41

Time (1-h interval)

Simulated value

Fig. 6. The optimized and simulated HVAC system indoor temperature.

The optimized and simulated energy consumption for 48 data points is shown in Fig. 5. Optimization results of the indoor temperature are illustrated in Fig. 6. Fig. 5 demonstrates that the optimized values are always below the simulated values. This result indicates the optimization model saves energy. Fig. 6 shows

Optimized value

Observed value

Fig. 7. Comparison between the optimized and observed values of supply air duct static pressure set point.

Supply air temperature setpoint (°F)

Optimized value

1

70 65 60 55 50 45 1

9

17

25

33

41

Time (1-h interval) Optimized value

Observed value

Fig. 8. Comparison of the optimized and observed values of supply air temperature set point.

the optimized indoor temperature is maintained between 68 °F and 73 °F in most cases. Figs. 7 and 8 compare the optimized and observed set points of supply air duct static pressure and supply

A. Kusiak et al. / Energy Conversion and Management 85 (2014) 146–153

Total energy consumption (kwh)

400

368.92

350 300

153

improving the model accuracy and better handling of user preferences.

294.66

References

250 200 150 100 50 0 Optimized total energy consumption

Simulated total energy consumption

Fig. 9. Comparison of the optimized and simulated total energy consumption for the 48 data points.

air temperature based on the same 48 data points. The optimized values are the recommended control points for the system. Based on these values over the two test days, 20% energy savings are achieved (see Fig. 9). Thus, the proposed approach reduces energy consumption while maintaining the indoor temperature at an acceptable level. Based on the computational results demonstrated in Figs. 5–8, several HVAC operational insights can be discovered. On 07/15/ 2011 and 07/16/2011, the minimum, mean, and maximum outside air temperature was about 70 °F, 83 °F, and 93 °F, respectively. The weather was clear and the maximum temperature, 93 °F, indicated that HVAC was operating in a cooling mode. As shown in Fig. 5, significant energy savings were obtained from the off-peak hours, especially during the night. The results of control settings (the SA-SPSPT is set to 0.4 in. WG and the SAT-SPT is set to 65 °F) demonstrated in Figs. 7 and 8 indicate that the HVAC cooling capacity should be operated at its lowest level during the night period. This is reasonable as the minimum outside air temperature is about 70 °F which is already within the pre-specified comfort range, 68 °F–73 °F. It is no longer necessary to continuously cool the supply air to 58 °F and then re-heat the supply air to the comfort level. The original control strategy has led to wasting of energy at night. Moreover, it can be observed that the pattern of energy consumption in Fig. 5 is similar to the pattern of internal load generated by occupants and lighting systems. Such observation indicates that operating HVAC based on the building occupancy plan is extremely important to energy saving since the internal load can significantly impact the energy consumption of the HVAC system. 5. Conclusion In this paper, MLP models with time-series were developed to model energy consumption of the HVAC system and the indoor temperature of the conditioned zones. A validation dataset was utilized to assess quality of the MLP models in prediction output parameters. The stochastic arrival of the occupants was considered in order to simulate the internal heating load in the conditioned zones. An optimization model was then constructed by incorporating the MLP models and considered constraints. Based on the optimization model, optimal control settings of the HVAC system could be produced. The interior-point method was applied to solve the optimization model. A case study illustrating the effectiveness of the proposed model and the algorithm was presented. Numerical results indicated that the proposed approach is effective in optimization of the HVAC system energy consumption. Energy savings of 20.15% were achieved while the comfort level was maintained based on a validation data set. Future research will focus on

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