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                                                                                                                                                                 DOI: 10.3303/CET2294084 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 08  April  2022; Revised: 27  July  2022; Accepted: 28  July  2022 
Please cite this article as: Chen W.-H., You F., 2022, An Energy Optimization Model for Commercial Buildings with Renewable Energy Systems 
via Nonlinear MPC, Chemical Engineering Transactions, 94, 505-510  DOI:10.3303/CET2294084 
  

A publication of 

 
 CHEMICAL ENGINEERING TRANSACTIONS  

 

VOL. 94, 2022 The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Petar S. Varbanov, Yee Van Fan, Jiří J. Klemeš, Sandro Nižetić 
Copyright © 2022, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-93-8; ISSN 2283-9216 

An Energy Optimization Model for Commercial Buildings 
with Renewable Energy Systems via Nonlinear MPC 

Wei-Han Chen*, Fengqi You 
Cornell University, Ithaca, New York, 14853, USA 
wc593@cornell.edu 

In this study, we develop a novel nonlinear model predictive control (NMPC) framework for climate control of 
buildings with renewable energy systems to minimize electricity costs. A nonlinear dynamic model of the building 
climate and renewable energy systems, including temperature, humidity, thermal comfort, geothermal heat 
pump, and solar panels, is first constructed based on mass and energy balance equations. The nonlinear 
dynamic model is then integrated into the proposed NMPC framework, which iteratively solves a nonlinear 
programming problem to generate the optimal control inputs, which minimize energy consumption and carbon 
footprints for sustainability. A simulation case study on controlling a building located on the Cornell University 
campus is conducted to demonstrate the capability of renewable energy sources to reduce building energy 
consumption using the proposed NMPC framework. The results show the NMPC framework could efficiently 
minimize total electricity cost and constraint violation for thermal comfort to 12.9 % with no more than 0.2 of 
violation on the predicted mean value index in different seasons. Implementing an electricity storage component 
could reduce the electricity cost by 19 %. The results indicate better sustainability for the smart building using 
sustainable energy sources and the NMPC framework. 

1. Introduction 
One possible way to cut down energy consumption while ensuring thermal comfort for occupants is by 
implementing renewable energy sources (e.g., solar energy and earth source heat energy) with a model 
predictive control (MPC) framework (Chen et al., 2021). In a hybrid energy system (Gong et al., 2015), 
renewable energy sources (e.g., earth source heat energy and solar energy) are utilized to reduce carbon 
footprints (Taebnia et al., 2020). Since the soil temperature is nearly constant under sufficient depth, geothermal 
energy can heat the building in winter and cool down in summer (Self et al., 2013). Solar energy can be captured 
by photovoltaic (PV) panels and stored in battery systems (Ogunjuyigbe et al., 2016). The electricity generated 
from solar energy can then be used for lighting and heating in the building (Tian et al., 2022). By adopting the 
hybrid renewable energy system, electricity costs can be significantly reduced (Noris et al., 2014). Because the 
building climate is a multi-input multi-output system (Hu et al., 2022), MPC has advantages over other classical 
control methods to gain the optimal control inputs for multiple control actuators (Trigkas et al., 2019). 
Some current studies investigated building climate control with a sustainable energy system (Deng et al., 2011). 
However, their building climate dynamic model is considered linear instead of nonlinear. Since the building 
climate has nonlinear dynamics, a linear model is not able to predict the future environment as accurately as a 
nonlinear model, leading to sub-optimal control performance (Chu, 2015). They only consider controlling indoor 
temperature instead of a more sophisticated thermal comfort index such as the predicted mean value (PMV) 
index. Thermal comfort includes multiple factors (e.g., air temperature, humidity, and air velocity), and thermal 
comfort cannot be satisfied if the temperature is controlled only. A nonlinear MPC (NMPC) framework for building 
climate control on renewable energy systems can be adopted to tackle this issue (Chen et al., 2022). A 
comprehensive configuration of renewable energy systems, including PV panels, earth source heating, and 
electricity storage battery, can be adopted through this framework. 
In this work, we propose an NMPC framework for smart and sustainable buildings’ climate regulation to minimize 
the total electricity cost and maintain thermal comfort for occupants. Building energy optimization has been 

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shown to be effective in reducing carbon footprint (Sun et al., 2022). The control and renewable energy system 
configurations include heating, ventilation, and air conditioning (HVAC), geothermal heat pump, PV panel, and 
electricity storage battery. The nonlinear dynamic models of the building climate and sustainable energy 
systems, including temperature, relative humidity, thermal comfort, earth heating sources, PV panels, are first 
constructed using first-principal equations. The temperature dynamic model is built by the energy balance 
equations and multiple renewable energy models. A dynamic model for humidity is developed by using mass 
balance accompanied by the approximation of the respiration rate of humans. After constructing all dynamic 
models, historical weather data is gathered from the weather station. The upper and lower bounds for system 
states are set to ensure occupants’ healthy and comfortable environment. The upper and lower bounds for 
control inputs are determined according to the limitation of each control actuator (Chen et al, 2021). Lastly, a 
nonlinear optimization problem is formulated to minimize electricity consumption, leading to fewer carbon 
footprints. At each sampling time, the nonlinear optimization problem is solved to generate optimal control inputs 
for the sustainable building climate (Grüne and Pannek, 2017). A case study on controlling a real-world building 
with renewable energy systems is used to demonstrate the advantages of the proposed NMPC framework. 

2. Dynamic model formulations 
Figure 1 shows the schematic of the NMPC framework for sustainable building climate using renewable energy 
sources. The system states are temperature, relative humidity, and PMV index. The control and renewable 
energy system configurations include HVAC, geothermal heat pump, PV panel, and electricity storage battery. 
The disturbances are from weather forecast errors. The dynamic models for indoor temperature and relative 
humidity are developed from mass and energy balance equations. The PMV index, which can better gauge the 
thermal comfort of occupants, is estimated from factors, including indoor temperature and relative humidity. 
 

 

Figure 1: The NMPC framework for building climate control with renewable energy sources 

2.1 Temperature dynamic model 

An indoor air temperature dynamic model is required in the NMPC framework to control the indoor temperature 
better. Building thermodynamic models are often developed as a resistance-capacitance model, where building 
components are viewed as resistances and capacitances. In this work, we employ the Building Resistance-
Capacitance Modeling (BRCM) MATLAB (version R2022a) Toolbox to develop the dynamic model for building 
indoor temperatures (Sturzenegger et al., 2016). The BRCM Toolbox generates the system dynamics tailored 
to the MPC framework through building geometry, structure, and materials (Shang et al., 2019). The system 
dynamics are given by: 

,
1

1 , ,( )
u

k k u k k vu i k Tu i
i

i k

n

v kT AT B u B v B v B T u


      (1) 
where Tk denotes the temperatures of rooms or wall/floor/ceiling, uk the inputs, and vk the predicted disturbances 
at time step k. A, Bu, Bv, Bvu,i, and Bxu,i are matrices of appropriate sizes. 

2.2 Humidity dynamic model 

The humidity inside the building can be modeled by differential equations. The absolute humidity is first modeled 
by using the mass balance equation. The relative humidity is then calculated from absolute humidity and building 

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indoor temperature. The mass balance equation of water, including the net flow from ventilation, respiration of 
occupants, and the humidifier, is shown as, 

d
d m

i
ven res hutV

h
m m m

t
     (2) 

where   is the water density, V is the air volume, ih  is the absolute humidity, ventm  is the water net flow from 
ventilation, resm  is the water net flow from the respiration of the occupants, and humm  is the water net flow from 
the humidifier system.  
When ventilation draws outdoor air into the building, replacing and removing indoor air, the moisture in the air 
is also removed. The water net flow of the ventilation is the difference between indoor absolute humidity and 
outdoor absolute humidity. The net flow can be represented as, 

 , maxvent a fan i o fanm C hu h u   (3) 
where ho is the outdoor absolute humidity and Ca is the air specific heat. The control input of the fan affects both 
the building temperature and the building indoor humidity, which shows the importance of optimizing multiple 
system states simultaneously. 
The absolute humidity model can then be converted into relative humidity using the equation as follows, 

)
100

0.611 (
i

i
sat i

h P
H

p T
  (4) 

where iH  is the relative humidity, P is the atmospheric pressure, and satp  is the saturated atmospheric 
pressure. Eq(4) shows the dependence of the temperature and humidity models. 

2.3 Ground source heating system dynamic model 

Ground source heat pump (GSHP) is the renewable energy source in this work. The heat extracted rate by the 
unit in the heating mode, 𝑄𝑒, can be modeled as (Ozgener and Hepbasli, 2007): 

, , ,( )out wa ie wa p wa n waQ m C T T    (5) 

The rate of heat rejection in the condenser is modeled as 

2 3( )rcond efQ m h h    (6) 

The rate of heat transfer in the evaporator can be calculated by: 

1 4( )revap efQ m h h    (7) 

The work input rate to the compressor is: 

,

1

,

2( )ref
i comp m comp

comp

m h h
W

 



  (8) 

In case the mass flow rate on the refrigerant side is not measured, the space heating load, 𝑄𝑠1, is estimated by: 

,,1 ,( )s air out air in rp air aiQ m C T T    (9) 

rair a iir am V   (10) 

where 𝑚𝑎𝑖𝑟 is the mass flow rate of air, 𝐶𝑝,𝑎𝑖𝑟is the specific heat of air, 𝑉𝑎𝑖𝑟 is the volumetric flow rate of air, ρair 
is the density of air, Tin,air and Tout,air are the average air temperatures entering and leaving the fan-coil units. The 
coefficient of performance (COP) of the GSHP can be calculated as: 

cond

comp

Q
cop

W



  (11) 

2.4 PMV index model 

Although some climate control studies evaluate thermal comfort only by temperature, occupants’ actual thermal 
comfort conditions can be better gauged using the PMV index. PMV index serves better by calculating 
occupants’ real feelings through the energy balance between the environment and occupants’ bodies. The 
factors considered in the PMV index include indoor temperature, indoor relative humidity, air velocity, mean 
radiant temperature, clothing insulation, and metabolic rate (Standard ASHRAE, 2010). The PMV index can be 
estimated by (Yang et al., 2018): 

( )bM diffPMV ae c Q   (12) 

where M is the metabolic rate of a human being and Qdiff is the difference between the internal heat production 
and loss that occurs in a human body, and it can be calculated by, 

diff work r pes se en vasQ M Q Q Q Q      (13) 

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( ) ( )res air vapQ dM g T hM g p     (14) 
4 4( ) ( )nclo msens clo cr airlo co v cloQ jf T T f h T T     (15) 

( ) [ ( ) ]work work v pvap aeQ k M Q l m n q M Q p        (16) 

where pvap is water vapor pressure, fclo is clothing factor, and Insclo is clothing insulation (1 clo = 0.155 m2-K/W), 
and the value of coefficients a, b, c, d, g, h, j, k, l, m, n, and q in Eqs(12)-(16) can be found in ISO 7730 by 
International Standards Organization (1994). In Eq, the cloth surface temperature can be estimated by: 

4 4[ ( )] [ ( )]clo skin clo clo conv c air clo mrlo clo cloT T R f h T T Ins jf T T      (17) 
There are two nonlinear items, radiative heat transfer and water vapor pressure, pvap, in the PMV model, which 
require an NMPC framework to control. 

3. Nonlinear model predictive control 
NMPC is used to control the building climate using renewable energy sources in this work. Due to its simplicity, 
the Euler method is adopted to discretize the building climate dynamic models (Butcher, 2016). The nonlinear 
system model can be expressed as the following after the discretization: 

 1 , ,k k k kx f x u v   (18) 
where system states xk include building indoor air temperature, relative humidity, and thermal comfort. Control 
inputs uk consist of entering signals to manipulate HVAC, geothermal heat pump (Lee et al., 2019), and 
electricity storage battery operations. Disturbances vk contain ambient temperature, ambient relative humidity, 
and solar radiation. f(∙) is the nonlinear function. The nonlinear building climate model in Eq(18) can then be 
shown compactly for a certain length of time step as: 

 0 , ,xx f u v  (19) 

where x, u, and v are the system state, control input, and disturbance sequences vectors, x0 is the initial system 
states, and f(∙) is the nonlinear vector function. After constructing the dynamic model for sustainable building 
climate and renewable energy systems, we could then develop the nonlinear optimization problem to be solved 
at each time step of the NMPC system. The nonlinear optimization problem includes constraints on system 
states and control inputs. The constraints are defined for control inputs and system states throughout the 
prediction horizon H. The control inputs u should be between the minimum and maximum values umin ≤ uk ≤
umax. The minimum and maximum values for building climate are also required to ensure the thermal comforts 
of occupants. The regulations on system states and control inputs are shown as: 

min max 0: 1,   k Hu u u k      (20) 

min, max, 1:,   k k k Hx x x k     (21) 

where umin and umax are the lower and upper bounds for control inputs. xmin,k and xmax,k represent the lower and 
upper bounds for thermal comfort. The k subscript indicates that the lower and upper bounds may vary 
depending on the occupancy or different settings (Jia et al., 2020). The compact form of Eq(20)-(21) is given by 

, x x u u G x g G u g  (22) 
where Gx, Gu, gx, and gu help define the compact form of system states and control inputs constraints. After the 
dynamic model and constraints are prepared, the nonlinear optimization problem can then be written out as: 

 0

0

min  

s.t.  , ,

T

u

x x

u u

TJ

x

 



 





cc u

 x f u v
G
G

ε S

g ε
u

ε

ε

x
g

 (23) 

where cc is the electricity pricing, ε  represents slack variables, and S  is the penalty weight. The objective 
function is to minimize the total electricity cost. Electricity is used for multiple actuators, including water pumps 
and lightings. A vector of slack variables ε  is added to the objective function because there are limitations on 
control inputs which could cause the nonlinear optimization problem to become infeasible. In order to penalize 
the constraint violation, the penalty weight S  is added. Since the slack variables are always positive, the state 
constraints are softened (Lu et al., 2020). The receding horizon approach is adopted in this smart and 
sustainable building climate control NMPC framework. At the beginning of each time step, the data of current 
system states and the forecasted weather disturbances are gathered (Shang et al., 2020). The nonlinear 
optimization problem in Eq can be solved based on the information of current system states and weather 

508



disturbances, given by x0 and v in Eq., to obtain the optimal control inputs. The first control input of the horizon 
is then implemented for the current time step, while the rest control inputs are discarded via an online 
implementation. In the next time step, the same process is repeated, starting from collecting the data of current 
system states and the forecasted weather disturbances. 

4. Case studies on building with renewable systems 
In this work, a building on the Cornell University campus in Ithaca, New York, is simulated for closed-loop 
thermal comfort control under the proposed NMPC control framework using renewable energy sources (Tian et 
al., 2019). The dimension of this building is 43.28 m × 24.38 m. The system states controlled in this work is 
thermal comfort, a combination of temperature and humidity. Ambient temperature, relative humidity, and solar 
radiation are considered disturbances. The simulation is performed for one week in winter during January 1-7, 
2020, one week in spring during 1-7 April 2020, and one week in summer during 1-7 July 2020. The weeks in 
winter and summer have extreme ambient temperatures, which can test the performance of the NMPC 
framework under harsh weather conditions. The weather forecast data from 1-7 January 2020, 1-7 April 2020, 
and 1-7 July 2020, are collected for the NMPC simulations. The actual measurement data at the same period 
are also collected to reveal the system states at the next time step.  
Figure 2 shows the combination of PMV index profile and total electricity consumption in winter during 1-7 
January 2020. The lower and upper bounds are set as -0.5 and 0.5. When the PMV index is less than -0.5, the 
climate is too cold for the occupants. Due to the cold weather in winter, heat pumps operate most of the time to 
heat the building’s indoor environment. The PMV index is mainly maintained at the lower bound. However, 
constraint violations still occur sometimes due to forecast errors. As for the electricity consumption profile, there 
is a clear diurnal pattern throughout the week. The peak of the electricity consumption is around 70 kWh, and 
the valley is about 40 kWh. The peak occurs when the occupants are inside the building during working hours, 
requiring greater internal loads. The weather also affects the energy consumption profile, so each day would 
not be the same. 

  

Figure 2: The PMV Index and electricity consumption profile of the sustainable building on the Cornell University 
campus with renewable energy systems in winter during 1-7 January 2020 (left), and electricity consumption 
comparisons between winter, spring, and summer of the building on the Cornell University campus (right) 

Conclusions 
In this work, we developed an NMPC framework that could simultaneously control multiple system states for the 
indoor climate of a building with renewable energy systems. Energy and mass balance equations were utilized 
to generate nonlinear dynamic models for the building environment. Renewable energy sources such as PV 
panels and geothermal heat pumps were adopted to reduce non-renewable energy consumption. We presented 
a case study of simultaneously controlling the indoor temperature and relative humidity of a building on Cornell 
Campus. The sensitivity analysis on electricity storage capacity showed that the electricity storage component 
could reduce the electricity cost by 19 % and ensure better sustainability for buildings. 

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