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1

Decoupled building-to-transmission-network for1

frequency support in PV systems dominated grid2

Obaid Ur Rehman1,*, Shahid A. Khan1, and Nadeem Javaid2

3

1Department of Electrical and Computer Engineering, COMSATS University Islamabad, Pakistan4

2Department of Computer Science, COMSATS University Islamabad, Pakistan5

*Corresponding author: Obaid Ur Rehman, obaid.rehman@comsats.edu.pk6

Abstract7

Tendency towards installation of distributed photo-voltaic (PV) systems has led to an increased emphasis on grid frequency8

control. Grid-interactive buildings equipped with heating ventilation and air conditioning (HVAC) system has a great potential9

to regulate the frequency in renewable energy resources rich grids. This paper integrates distributed PV systems in a decoupled10

building-to-transmission-network (B2TN) and explicitly formulates the interaction between a grid and the buildings through11

real time pricing (RTP). As grid frequency conveys the information about operating conditions of a power system, therefore,12

frequency based RTP generators namely; linear, hyperbolic tangent and inverse hyperbolic tangent are used. A price responsive13

model predictive controller (MPC) for optimal scheduling of HVAC load is developed where reference temperature set point is14

dynamically adjusted subject to RTP. Moreover, the clear and cloud covered sky impacts of PV power generation on frequency15

and RTP deviations are investigated. Comparatively, inverse hyperbolic tangent model maps the frequency deviations on a wider16

RTP range which increases the scale of reference temperature set point resulting peak frequency deviation suppression up to 40%17

and load regulation beyond 12 MW without inherently affecting the building electricity cost when compared to ordinary MPC18

(without demand response service).19

Keywords— Building-to-transmission-network, frequency based real time pricing, Model predictive controller, Frequency regulation,20

demand response, Photo-voltaic systems21

NOMENCLATURE22

Abbreviations23

ANN Artiﬁcial neural network24

B2DN Building-to-distribution-network25

B2G Building-to-grid26

B2TN Building-to-transmission-network27

BEMS Building energy management system28

DERs Distributed energy resources29

DR Demand response30

DSO Distribution system operator31

HVAC Heating Ventilation and Air Conditioning System32

LP Linear Programming33

MPC Model Predictive Controller34

OPF Optimal Power Flow35

PV Photo-voltaic36

QP Quadrature Programming37

RERs Renewable energy resources38

RTP Real Time Pricing39

TSO Transmission system operator40

Parameters41

𝜖Slack relaxation on zone temperature bounds42

𝜔Grid frequency43

𝜔𝑑𝑒 𝑣 Deviations in grid frequency44

𝜌Penalty cost on occupant’s thermal comfort violation45

𝜃Node voltage angle46

𝐴𝑏, 𝐴𝑔Coefﬁcient matrices of building and grid states, respectively47

𝑏Slope between RTP and frequency48

𝐵𝑢ℎ , 𝐵𝑢 𝑔 Coefﬁcient matrices of building and grid control inputs, respectively49

𝐵𝑤𝑏, 𝐵𝑢𝑔 Coefﬁcient matrices of building and grid disturbances/uncontrollable inputs, respectively50

𝐶𝑤 𝑎𝑙𝑙 , 𝐶𝑧 𝑜𝑛𝑒 Thermal capacitance of wall and zone, respectively51

𝐷Damping coefﬁcient52

𝐺Solar irradiance53

𝐺𝑠𝑡 𝑑 Solar irradiance in standard environment54

𝐿Total number of buildings attached to load buses55

𝑀Inertia coefﬁcient56

𝑁Number of nodes/buses in the power system57

𝑃(𝜔)Frequency based RTP function58

𝑃𝑏Building’s base load59

𝑃ℎBuilding’s HVAC load60

𝑃𝑚Mechanical power of a generator61

𝑃𝑟Rated power of PV system62

𝑃𝜔𝑛Electricity price at nominal grid frequency63

𝑃𝑏𝑙 Net building load64

𝑃𝑒,𝑖 Net electric load at 𝑖𝑡ℎ node65

𝑃𝑛𝑙 Frequency insensitive nodal load66

𝑃𝑝𝑣 PV generation power67

𝑄ℎ𝑣 𝑎 𝑐 HVAC load68

𝑄𝑖𝑛𝑡 , 𝑄 𝑠𝑜𝑙 Buildings’ internal and solar heat gains69

𝑅1,𝑤 𝑎 𝑙𝑙 Thermal resistance of internal wall structure70

𝑅2,𝑤 𝑎 𝑙𝑙 Thermal resistance of external wall structure71

𝑅𝑤𝑖 𝑛 Building’s internal and solar heat gains72

𝑆𝑝Certain irradiance point73

𝑆ℎ𝑖𝑔 ℎ Gradient between desired temperature set point and maximum value of RTP74

𝑆𝑙𝑜 𝑤 Gradient between desired temperature set point and minimum value of RTP75

𝑇𝑑𝑒𝑠 𝑖𝑟 𝑒𝑑 Desired temperature set point76

𝑇𝑟𝑒 𝑓 Reference temperature set point77

𝑇𝑤 𝑎𝑙𝑙 Wall temperature78

𝑇𝑧𝑜𝑛𝑒 Zone temperature79

𝑢ℎ, 𝑢𝑔Building and grid control inputs80

𝑤𝑏𝑙 , 𝑤𝑔Building and grid disturbances81

𝑋𝑏, 𝑋𝑔Building and grid states82

𝑌Number of PV generation systems83

𝑍Number of thermal generators84

1. INTRODUCTION85

1.1. Back ground and motivation86

Ever increasing electric energy demand owing to growing population and depletion of fossil fuels resources forced the researchers to explore87

alternate energy resources. Among the several renewable energy resources (RERs), solar photovoltaic (PV) systems are being deployed at88

supply side [1]. However, increasing penetration of PV systems is exacerbating the supply-demand imbalance; decreasing the system inertia89

and increasing the reserve requirements [2, 3]. Frequent supply-demand mismatch due to intermittent renewable generation results in frequency90

deviations [4, 5]. Therefore, minute-level regulatory service is required to maintain frequency stability [6, 7]. Conventionally, generation side91

was used for frequency regulation. Provisioning of real-time frequency regulation through conventional generators is no more a sustainable92

solution owing to their slow response time, limited ramp up/down capabilities and maximum/minimum generation bounds [8].93

Many grid service operators, e.g., California independent system operator [9], Florida reliability coordinating council [10], New England94

independent system operator [11], and Australian energy market operator [12] are expected to face regulation issues in near future. It is95

believed that future electrical grids where supply side will remain the only source of RS may not be capable to incorporate high shares of96

PV generation. Unlike power generators, demand side resources can provide quick response to the sudden change in load or generation.97

The ﬂexible operations of both supply and demand sides can accommodate higher PV penetration than could be achieved by relying on98

conventional generation alone. Fortunately, ﬂexible loads with superior regulation capabilities can promote the renewable energy penetration99

through demand response (DR). Particularly, thermostatically controlled loads such as heating, ventilation and air-conditioning (HVAC)100

systems are ﬂexible loads and account for a signiﬁcant portion of commercial and residential buildings load [13]. For instance, demand101

side ﬂexibility of residential heating system was used to absorb surplus renewable power generation [14]. An optimization framework102

for optimal scheduling of hydro-thermal generation and demand ﬂexible loads was developed considering uncertainty and outage of wind103

turbine and solar PV power generation to ensure stable grid operation [15]. Likewise, a mixed integer linear programming (MILP) based104

optimization technique was adopted for combined scheduling of PV system, battery storage and deferrable appliances meanwhile ensuring the105

grid’s reliability [16]. Another study proposed a coordinated scheduling of demand ﬂexible heating system and distributed energy resources106

(DERS) including gas turbines, gas storage and solar PV system to increase the shares of renewable energy in a power system [17]. A mixed107

integer non-linear problem was formulated to improve the resiliency of an electrical distribution grid considering demand side management108

and optimal placement of solar PV system in a distribution network [18].109

In aforementioned studies [14-18], the potential of DR program was analyzed from grid’s perspective, however, the consumers’ concerns110

regarding electricity consumption cost and comfort were ignored. In this regard, several building-to-grid (B2G) integration frameworks have111

been developed for simultaneously optimizing building and grid objectives. The idea is to optimally use the building load ﬂexibility to112

support renewable energy integration in a power grid while ensuring the building and grid operational constraints.113

1.2. Literature review114

Existing B2G integration frameworks use demand side ﬂexibility for grid services. In literature, B2G integration frameworks are classiﬁed115

as: (1) building-to-distribution-network (B2DN) mainly for voltage control (2) building-to-transmission-network (B2TN) mainly for frequency116

control.117

In a B2DN, the buildings and RERs are integrated to power distribution network. For instance, a techno-economical assessment performed118

for building integrated solar PV with battery storage system under power distribution grid constraints showed signiﬁcant reduction in grid119

energy supply and electricity bill [19]. A model predictive control (MPC) based control strategy was formulated to exploit the synergy of120

solar PV system, battery storage and ﬂexible heating load for electricity cost minimization, occupants’ thermal comfort and grid’s load factor121

maximization [20]. Similarly, a MPC based B2DN power ﬂow control mechanism was developed where the impact of solar PV-curtailment122

2

and demand ﬂexible cooling system as a DR program was analyzed from the building’s and distribution network’s perspectives in multiple123

solar PV generation scenarios [21]. The work in [22] integrated residential, commercial buildings and DERs including solar PV system and124

battery storage system in a distribution network where building’s and grid’s control variables were optimized simultaneously. In [23], a novel125

DR model was developed to support high PV penetration in a distribution network, aiming to reduce electricity cost and maximize the solar126

PV generation revenue for customers while ensuring the voltage stability of a grid.127

Regarding frequency control, which is the main focus of this study, the existing B2TN can be classiﬁed into: (1) fully integrated B2TN (2)128

decoupled B2TN. In fully integrated B2TN frameworks, a single controller is implied for optimizing the control decisions of building and129

grid. For instance, authors in [24] developed a mathematical model for B2TN that coupled building and transmission network dynamics along130

optimizing building and grid variables simultaneously using centralized MPC controller. This work was further extended by incorporating a131

Markov chain-based occupancy model where results validated signiﬁcant generation cost reduction as compared to decoupled MPC strategy132

[25]. It is worth mentioning that building load was the only source of uncertainty in refs [24, 25]. Adopting, fully integrated B2TN framework133

of [24], another study integrated distributed wind energy system and battery storage where MPC approach provided robustness in the presence134

of uncertain load and wind power generation, ensuring reliable grid while maintaining thermal comfort of the building occupants [26].135

On the contrary, a decoupled control approach deploys separate controllers to segregate building and grid control actions. Further, decoupled136

B2TN integration frameworks usually opt grid’s control signals to enable DR service. In refs. [27, 28], proportional-integral derivative based137

control strategies were adopted to modulate the power use of HVAC systems subject to grid’s automatic gain control signal for frequency138

control. Another study developed a MPC based frequency control strategy to make HVAC system price-responsive and grid-interactive [29].139

Aiming for transmission network operation control, an artiﬁcial neural network (ANN) based price generator was developed where time-140

varying prices were generated and broadcasted to obtain a certain reaction from building’s demand ﬂexible loads. ANN model generated daily141

price proﬁles by mapping non-linear inter-dependencies between historical consumption and electricity price. From the control perspectives, a142

well-known load frequency controller model was opted for real-time frequency regulation at transmission network level and a price-responsive143

proportional integral controller was developed at building level [30]. Likewise, a non-linear auto-regressive model with exogenous input was144

used by transmission system operator (TSO) to predict real time pricing (RTP) [31]. The price generated in [30, 31] requires training data,145

accurate prediction of energy consumption, and other factors such as weather forecast for appropriate prediction of price signals.146

Since frequency indicates the actual conditions (supply-demand imbalance) of a grid and remains the same in entire power system,147

therefore, the frequency dependent RTP can be generated across the grid without relying on high-speed communication infrastructure.148

Moreover, the historical data regarding grid’s frequency is readily available to construct data driven RTP prediction models. Fortunately,149

frequency measurement devices are easily available which further assist to compute time varying price signals without the requirement of150

historical data of grid frequency. These frequency measurement devices can be deployed at building level to reduce the communication151

requirements. In this regard, decentralized real-time frequency based pricing model was developed for PV system integrated grid [32, 33].152

However, the deployment of frequency measurement devices at larger scale for the application of frequency regulation provision seems to be153

an expensive solution [34] therefore, installation at TSO level is considered an economical choice. The adaption of a frequency-based RTP154

generator at TSO level is found in [35], where building’s DERs were explicitly modeled to provide ancillary service to the PV generation155

dominated grid.156

The above mentioned literature discusses a wide variety of B2TN, ranging from fully connected to decoupled. Each has its own advantages157

and limitations. Such as, a fully connected B2TN employs a single controller resulting in computational complexity as well as vulnerable158

to single point of failure. On the contrary, decoupled B2TN reduces the computational burden by employing separate controllers at building159

and grid levels and are also less prone to single point of failure [36]. Moreover, the decoupled B2TN can be implemented without major160

modiﬁcations in exiting power systems.161

1.3. Contributions and organization162

Motivated by the above mentioned remarks, this paper develops a decoupled B2TN integration mechanism based on frequency-based RTP163

signals that facilitates application of demand ﬂexibility and RERs integration for frequency regulation provision. In the proposed framework,164

TSO generates a frequency-based RTP signal that is broadcasted to the price responsive consumers. When building operators receive the165

RTP signals, they modulate their consumption patterns to maximize their thermal comfort and minimize energy consumption cost through166

the building energy management system (BEMS). The salient contributions of this study are as follows:167

1) A RTP based interaction of smart buildings and TSO is developed for frequency regulation provision by adopting the decoupled control168

strategy of base paper [24], where separate MPC controllers were developed for building and TSO operations control. In base paper,169

the buildings’ MPC was not price responsive. In order to make MPC controller price responsive, this study develops a consumer’s170

price response model at building level where reference temperature set point is adjusted dynamically subject to deviations in RTP.171

2) In order to assess the performance of TSO in the presence of renewable generation, the distributed PV systems are integrated in172

decoupled B2TN. Further, the grid’s MPC is explicitly modeled by considering uncertainties in both PV generation and energy173

demand. To the best of our knowledge, existing literature lacks the integration of PV generation in a decoupled B2TN.174

3) Three frequency based price generators namely; linear [37, 38], hyperbolic tangent and inverse hyperbolic tangent [39] are used to175

compute RTP as a function of grid frequency. Further, the solar energy generation patterns for clear sky and cloudy days are generated176

to investigate the performance of three frequency dependant price generators in terms of grid’s frequency deviations suppression.177

Although previous studies have developed frequency based RTP generators, none of the studies have analyzed the effects of various178

types of price generators in distributed PV systems penetrated grids.179

4) The proposed framework is validated in a standard IEEE 30-bus system with distributed PV power generation.180

The rest of the paper is organized as follows. In Section 2, decoupled B2TN is brieﬂy explained. Section 3 provides mathematical models181

to implement B2TN system. Section 4 discusses MPC based optimization for both building and grid. In Section 5, simulation results are182

presented. Concluding remarks are given in Section 6.183

2. A BRIEF DESCRIPTION OF B2TN INTERACTION MECHANISM184

B2TN interaction mechanism uses distributed control strategy to provide DR at temporal scale of the grid using RTP signal. The generation185

and broadcasting of RTP signal that depends on the frequency of the grid enables TSO to exploit the ﬂexibility of building load. DR is required186

when the disturbances occurs in the grid (e.g., PV generation variations, load changes). In case of any disturbance, the grid experiences with187

3

a frequency deviation. Once the power disturbance occurs, the TSO solves a MPC based control problem to quantify the required DR based188

on the frequency deviation and formulates the price signal denoted by 𝑃(𝑤)in Fig. 1. Upon receiving the RTP signal, the BEMS modiﬁes the189

energy consumption patterns to compensate for the power disturbance owing to intermittent PV generation and, therefore, stabilizes system’s190

frequency. The proposed framework performs two level of optimization between the building and a grid. The building’s MPC considers191

RTP signal, exogenous disturbances including solar heat gain, internal heat gain and outdoor ambient temperature to determine power use of192

HVAC load (𝑢ℎ). After performing optimization at the building level, the grid’s MPC adjusts its controllable input/generator power power193

(𝑢𝑔) by considering the following disturbances; intermittent PV generation (𝑃𝑝𝑣 ), 𝑈𝑏, and 𝑈ℎ. The dynamics of the building are modeled194

using 3R-2C network while the grid dynamics are modeled using swing equation. The building cost function is linear while the grid cost195

function is quadratic in nature. Accordingly, linear programming and quadratic programming based optimizations are executed in MATLAB196

to achieve building and grid objectives. The working mechanism of decoupled optimization framework is further discussed in more detail197

in Section 4.198

Building

dynamics model

(3R-2C Thermal

Network)

Building MPC

Grid dynamics

model

(Swing equa on)

Grid MPC

ug

Xg

Constraints on

buildings'

variables: uh , Xb

Disturbances

( Solar irradiance,

Internal heat

gains) Building op mizer Grid op mizer

uh

P( )

Constraints on grids’

variables:

ug , Xg

Xb : Building temperature states u h : HVAC power use /control input ub : Building base load Xg : Grid states

P( ) : Frequency based RTP func on u g : Generator output power /control input Ppv: PV output power

Xbuh

Building level TSO level

ub

Frequency

based RTP

generator

PV system PPV

Fig. 1: Schematic of decoupled B2TN optimization framework.

3. B2TN SYS TE M MO DELIN G199

Aiming for accessing the performance of B2TN, appropriate models of buildings and power transmission network are needed. In this200

paper, the transmission network issues are only limited to frequency regulation since it is one of the most challenging issues in the RERs201

rich grids.202

The behaviour of the power system frequency is described by swing equation at TSO level, accounting for changes in load and PV203

generation. DSO provides technical supports by performing optimal power ﬂow (OPF). In case of any change in consumption/generation204

occurs, the swing and power ﬂow equations are solved for determining new conditions of the grid. Furthermore, TSO generates and broadcasts205

the frequency based RTP signal to receive DR from the buildings. The different parts of B2TN in Fig.2 are explained in the subsequent206

subsections.207

3.1. Frequency based RTP generators208

This sub-section discusses the frequency based RTP generation mechanism. The electricity consumers provide demand ﬂexibility for209

grid’s frequency support in response to RTP signal received from the TSO. The main reason for adopting frequency based RTP is that grid210

frequency is an indicator of generation and demand mismatch. The frequency increases in case of excess generation and vice versa. In future211

grids, the frequency deviations occur frequently owing to intermittent nature of the renewable generation. For stable grid operation, the212

variations in grid frequency are allowed within certain limits. The allowable deviations from nominal frequency are 50 ±0.2 Hz in normal213

grid operations, however, frequency deviations upto 50 ±0.5 Hz are tolerable [40, 41].214

As discussed earlier, the electricity price variations are directly related to the variations in the grid frequency. The designing of an215

admissible price function 𝑃(𝜔)for electricity consumers can motivate them to participate in DR program. This study considers linear and216

nonlinear frequency based RTP functions namely linear, hyperbolic tangent (𝑡𝑎 𝑛ℎ) and inverse hyperbolic tangent (𝑡𝑎𝑛ℎ−1). Hyperbolic217

tangent is a rescaled logistic sigmoid function with output range from -1 to 1. Whereas, the range of inverse hyperbolic tangent is from −∞218

to ∞. The hyperbolic and inverse hyperbolic tangent functions generate zero centered output, therefore, these functions are used to model219

RTP as function of frequency deviations from nominal grid frequency. The price curves of under-study frequency based RTP functions are220

depicted in Fig. 3. The linear price curve is obtained by using Eq. (1).221

𝑃(𝜔)=𝑃𝜔𝑛−𝑏𝜔𝑑 𝑒𝑣 ,(1)

where 𝑃𝜔𝑛represents price at nominal frequency set as 15.39 ct/kWh. For a slope 𝑏= 10 𝑐 𝑡/𝑘 𝑊 ℎ

2𝜋Hz, the price curve should be bounded222

only to a price range (𝑃(𝜔) ∈ [10.4,20.4]cents/kWh) at a operational frequency range (𝜔𝑑𝑒𝑣 /2𝜋∈ [49.5,50.5]Hz). However, linear price223

curve generates unbounded price signals for the electricity consumers as shown in Fig. 3(a).224

For bounded price amplitudes, a hyperbolic tangent function is given in Eq. (2).225

𝑃(𝜔)=𝑃𝜔𝑛+𝑏

2𝑡𝑎 𝑛(𝜔𝑑 𝑒𝑣

𝜋𝛽 ).(2)

4

13

2

45

6

PV2

PV1

2- Generate frequency

based RTP: (t) P( )

TSO:

1-Solve swing equa on

DSO-1:

Solve Power ow

equa ons

DSO-2

Solve power ow

equa ons

123

1 3

2

BEMS:

Price-responsive

DR for frequency

regula on provision

Fig. 2: Conceptual representation of B2TN system including TSO, PV systems, DSO and buildings.

The graph of hyperbolic tangent function is shown in Fig. 3(b). The price curve varies linearly around nominal frequency; slope of the226

curve decreases away from the nominal frequency and ultimately becomes constant beyond [50 ±0.5] Hz. As a result, the hyperbolic tangent227

function maps the operational range (𝜔𝑑𝑒 𝑣 /2𝜋∈ [49.5,50.5]Hz) to a price range (𝑝(𝜔) ∈ [11.59,19.2]cents/kWh). The hyperbolic tangent228

function offers bounded but limited price range at operational frequencies as compared to linear price function. A limited range of price229

ultimately restricts the DR capability of demand side to participate in RS. Thus, hyperbolic tangent function is not suitable to implement230

effective price based DR strategy when variations in frequency are high.231

An inverse hyperbolic function is used to deal with the issue of limited price range at operational frequencies [39]. The function in Eq. 3232

allows the price amplitude to vary linearly around nominal frequency while the slope increases away from nominal frequency and goes to233

maximum at 50 ±0.5Hz as shown in Fig. 3(c). Conclusively, this function maps the operational frequency range [49.51 50.49]Hz to the234

price range [3.9 26.9]cents/kWh where a rigorous change in price amplitudes is observed at high and low frequencies. As a consequence,235

inverse hyperbolic function can be used to compensate high frequency deviations through low/high price signals. An inverse hyperbolic236

function generates price amplitudes of −∞ and ∞at points 49.5Hz and 50.5Hz, respectively. At this stage, emergency measures either load237

shedding or PV curtailment must be initiated.238

𝑃(𝜔)=𝑃𝜔𝑛−𝑏

2𝑡𝑎 𝑛−1(𝜔𝑑𝑒 𝑣

𝜋𝛽 ),(3)

In order to analyze the impact of frequency based RTP functions on demand ﬂexibility of electricity consumers, an example scenario239

is discussed here. It is assumed that power consumption of HVAC load is 50 kWh at nominal frequency of 50 Hz with electricity rate of240

15.39 cents/kWh. The total cost of electricity consumption is therefore equal to 769.5 cents (50 kWh ×15.39 cents/kWh). In case of linear241

price function, the price signal is bounded to [10.4 20.4] cents/kWh at operating frequency range. This price function allows the power use242

of HVAC system ranging from 37.72kWh to 73.95kWh without affecting the consumer’s electricity cost. It is understood that 73.9 kWh ×243

10.4 cents/kWh and 20.4 kWh ×37.72 cents/kWh result in same electricity cost which offers by (50 kWh ×15.39 cents/kWh) which is244

equal to 769.5 cents. In the same manners, hyperbolic tangent and inverse hyperbolic tangent functions allow building controller to adjust245

the power of HVAC load from 40.02 to 66.4 kWh and 28.61 to 197.4 kWh, respectively. Inverse hyperbolic tangent function can provide246

highest demand ﬂexibility amongst all three frequency based RTP functions. However, this ﬂexibility is subjected to the constraints on power247

use of HVAC load.248

3.2. Grid models249

In this sub-section, appropriate transmission network and building load models are developed to assess the performance of the power250

system.251

3.2.1. Transmission system model: Aiming for real-time frequency control at TSO level, the dynamics of transmission network is252

modeled using swing equation [24, 25] which consists of small signal model and synchronous generators to correct steady state frequency253

error by ramping up/down its generation within few minutes. The response of generator to the load/generation changes depends on the ramp254

rate limits. We extend the swing equation [24] by integrating the PV-power generation in the following form:255

𝑀𝑖

𝑑2𝜃𝑖

𝑑𝑡2+𝐷𝑖

𝑑𝜃𝑖

𝑑𝑡

=𝑃𝑚,𝑖 (𝑡) + 𝑃𝑃 𝑉 ,𝑖 (𝑡) − 𝑃𝑒, 𝑖 (𝑡)

−

𝑁

Õ

𝑗=1

𝑏𝑖 𝑗 𝑠𝑖𝑛 (𝜃𝑖(𝑡) − 𝜃𝑗(𝑡)),

(4)

5

49 49.5 50 50.5 51

Frequency (Hz)

5

10

15

20

25

30

Electricty price (Cents/kWh)

(a) Linear

49 49.5 50 50.5 51

Frequency (Hz)

5

10

15

20

25

30

Electricty price (Cents/kWh)

(b) Hyperbolic tangent

49 49.5 50 50.5 51

Frequency (Hz)

0

10

20

30

Electricty price (Cents/kWh)

(c) Inverse hyperbolic tangent

Fig. 3: Frequency based RTP functions

where 𝜃represents voltage angle of rotor; 𝑀𝑖represents moment of inertia of generator; 𝑃𝑚, 𝑖 (𝑡)represents generator output power; 𝑃𝑃𝑉 ,𝑖 (𝑡)256

represents the solar power injection at bus 𝑖;𝑃𝑒,𝑖 (𝑡)represents load at any distribution network connected to bus 𝑖. The state of a generator257

is determined through rotor angle 𝜃(𝑡)reference to angular frequency (𝜔). The power ﬂow between buses 𝑖and 𝑗is assumed to be purely258

reactive which is determined by the line susceptance, 𝑏𝑖 𝑗 =𝑏𝑗 𝑖 , and the difference in voltage angle, 𝜃𝑖(𝑡)-𝜃𝑗(𝑡). The angular frequency259

deviations relative to reference (𝜔=2𝜋×50 Hz or 𝜔=2𝜋×60 Hz) are calculated by taking the ﬁrst derivative of rotor angle (𝜔=𝑑 𝜃

𝑑𝑡 ).260

According to Eq. (4), the variations in the power system dynamics depend on moment of inertia (𝑀𝑖)and damping constant (𝐷𝑖). For the261

purpose of simplicity, it is assumed that all the 𝑀𝑖and 𝐷𝑖are identical. The angular frequency deviation 𝜔=1

𝑁Í𝑖𝜔𝑖is then determined262

by the difference (𝑃𝑚,𝑖 (𝑡) + 𝑃𝑃𝑉 ,𝑖 (𝑡) − 𝑃𝑒,𝑖 (𝑡)).263

Proof

In case all 𝑀𝑖=𝑀and 𝐷𝑖=𝐷are identical then Eq. (1) can be expressed as:

𝑀𝑑

𝑑𝑡 𝜔+𝐷𝜔 =1

𝑁(𝑃𝑚(𝑡) + 𝑃𝑃𝑉 (𝑡) − 𝑃𝑒(𝑡) ), rearranging for 𝜔(𝑡)gives us:

𝜔(𝑡)=𝜔(𝑡0)𝑒−𝐷𝑡

𝑀+𝑃𝑚(𝑡)+𝑃𝑃𝑉 (𝑡)− 𝑃𝑒(𝑡)

𝑁 𝐷 (1−𝑒−𝐷𝑡

𝑀). As 𝑡→ ∞,𝜔(𝑡)converges to 𝑃𝑚(𝑡) +𝑃𝑃 𝑉 (𝑡)− 𝑃𝑒(𝑡)

𝑁 𝐷 , Hence, it is proved that

𝜔(𝑡)is directly proportional to the difference (𝑃𝑚(𝑡) + 𝑃𝑃𝑉 (𝑡) − 𝑃𝑒(𝑡)). The frequency deviations increase as the difference between

generation and load increases.

Electricity load 𝑃𝑒, 𝑖 in Eq. (4) is a composition of three components and expressed as:264

𝑃𝑒,𝑖 =𝐹𝑖

𝑑𝜃

𝑑𝑡 +𝑃𝑛𝑙 ,𝑖 +

𝐿𝑖

Õ

𝑙=1

𝑃𝑏𝑙, (𝑙 ,𝑖),(5)

where ﬁrst term at R.H.S represents frequency-sensitive nodal load, second term represents frequency-insensitive nodal load, and last term265

represents total power demand of buildings attached to 𝑖𝑡ℎ node.266

The building load (𝑃𝑏𝑙 ) in Eq. (5) is calculated as:267

𝑃𝑏𝑙 =𝑃ℎ+𝑃𝑏,(6)

where 𝑃ℎis building HVAC load that can provide demand ﬂexibility, and 𝑃𝑏is base load of a building. By combining Eqs. (4)–(6), the268

governing equation for grid is given as:269

𝑀𝑖

𝑑𝜃𝑖

𝑑𝑡

=−( 𝐷𝑖+𝐹𝑖)𝜔𝑖+𝑃𝑚,𝑖 +𝑃𝑃𝑉 , 𝑖 −

𝐿𝑖

Õ

𝑙=1

(𝑃ℎ+𝑃𝑏)

+

𝑛

Õ

𝑗=1

𝑏𝑖 𝑗 𝑠𝑖𝑛(𝜃𝑖−𝜃𝑗).

(7)

The transmission network dynamics from Eq. 7 are captured in the state-space model given in Eq. 8.270

¤𝑥𝑔(𝑡)=𝐴𝑔𝑥𝑔(𝑡) + 𝐵𝑢𝑔𝑢𝑔(𝑡) + 𝐵𝑤 𝑔𝑤𝑔(𝑡),(8)

where 𝑢𝑔=[𝑢𝑚+Δ𝑢𝑚]𝑇represents controllable mechanical power vector ; 𝑤𝑔=[𝑤𝑏𝑙, 𝑤 𝑛𝑙, 𝑤 𝑃𝑃𝑉 ]𝑇represents a disturbance/uncertain271

vector including building load, nodal load and PV generation power; 𝑥𝑔=[𝜃, 𝜔]𝑇represents grid’s state vector including voltage angle and272

frequency.273

3.2.2. Building load model: This subsection discusses the building thermal comfort model and consumers’ price response model.274

3.2.2.1. Building thermal comfort model: The commercial buildings are mainly equipped with base and demand ﬂexible loads.275

The building’s base load includes: lighting, computers and other electrical equipment whereas HVAC is a demand ﬂexible load. The power276

use of HVAC load can be modiﬁed when needed. Thermal dynamics of building can be modeled by using resistance and capacitance (RC)277

network. 3R-2C thermal network is widely adopted to model the dynamics of multi-zone commercial buildings. As this model gives the best278

estimation of thermal dynamics of commercial buildings, therefore, this study uses 3R-2C network to model thermal dynamics of buildings.279

6

R1,wall

R2,wall

Rwin

Cwall Czone

Tamb Tzone

Qint + Qhvac

Twall

Qsol

Fig. 4: 3R-2C thermal network [21].

The under-study thermal network has two temperature states: zone temperature (𝑇𝑧𝑜𝑛 𝑒)and wall temperature (𝑇𝑤 𝑎𝑙 𝑙 ). For 3R-2C thermal280

network given in Fig. 4, the temperature states are expressed as:281

𝐶𝑤 𝑎𝑙𝑙 𝑇𝑤 𝑎𝑙𝑙 =𝑇𝑤𝑎 𝑙𝑙 −𝑇𝑧𝑜 𝑛𝑒

𝑅1,𝑤 𝑎 𝑙𝑙

+𝑇𝑎𝑚𝑏 −𝑇𝑤 𝑎𝑙𝑙

𝑅2,𝑤 𝑎 𝑙𝑙

+𝑄𝑠𝑜𝑙 ,(9)

282

𝐶𝑧𝑜𝑛𝑒𝑇𝑧𝑜 𝑛𝑒 =𝑇𝑤 𝑎𝑙𝑙 −𝑇𝑧𝑜𝑛𝑒

𝑅2,𝑤 𝑎 𝑙𝑙

+𝑇𝑎𝑚𝑏 −𝑇𝑤 𝑎𝑙𝑙

𝑅𝑤𝑖 𝑛

+𝑄𝑖𝑛 +𝑄ℎ 𝑣 𝑎𝑐 ,(10)

where 𝑅𝑤𝑖 𝑛, 𝑅1,𝑤 𝑎 𝑙𝑙 , 𝑅2,𝑤 𝑎 𝑙𝑙 are thermal resistances of window, interior and exterior walls’ structures; 𝐶𝑤𝑎𝑙 𝑙, and 𝐶𝑧𝑜𝑛𝑒 are thermal283

capacitances of wall and zone, respectively; 𝑇𝑎𝑚 𝑏,𝑇𝑧 𝑜𝑛𝑒 , and 𝑇𝑤 𝑎 𝑙𝑙 are ambient outdoor, zone, and aggregated walls’ temperatures; 𝑄𝑖𝑛𝑡 ,284

𝑄𝑠𝑜𝑙 , and 𝑄ℎ 𝑣 𝑎𝑐 are internal heat gain, solar heat gain, and HVAC load, respectively.285

The building’s thermal dynamics in Eqs. (9, 10) are captured in the following state-space model:286

¤𝑥𝑏(𝑡)=𝐴𝑏𝑥𝑏(𝑡) + 𝐵𝑢ℎ 𝑢ℎ(𝑡) + 𝐵𝑤 𝑏 𝑤𝑏(𝑡),(11)

where 𝑥𝑏(𝑡)=[𝑇𝑧𝑜𝑛𝑒 𝑇𝑤 𝑎𝑙𝑙 ]𝑇is the building’s thermal state vector, 𝑢ℎ=[𝜂𝑄ℎ𝑣 𝑎 𝑐 =𝑃ℎ 𝑣 𝑎𝑐 ]is the HVAC system control input vector287

and 𝜂is the coefﬁcient of performance, and 𝑤𝑏=[𝑇𝑎𝑚𝑏 𝑄𝑖𝑛𝑡 𝑄𝑠 𝑜𝑙 ]𝑇is the disturbance vector. For the explicit modeling of coefﬁcient288

metrics 𝐴𝑏,𝐵𝑢ℎ , and 𝐵𝑤 𝑏 , interested readers are referred to [20, 21].289

3.2.2.2. Consumers’ price response model: This research work develops a mechanism to determine cost-effective reference290

temperature set-points for MPC. The reference set-points are determined based on RTP as shown in Fig. 5. In general, a high reference291

temperature set-point assigned to high price signal and low reference temperature set-point assigned to low price signal guarantees a cost-292

efﬁcient thermal comfort for occupant. This strategy is developed in DR enabled MPC through a “dynamic price-to-temperature set point”293

model. RTP signal is transmitted to the building MPC after every 5 minutes. When RTP is equal to the price at nominal frequency then294

MPC sets reference temperature set point (𝑇𝑟 𝑒 𝑓 ) to the desired temperature (𝑇𝑑𝑒𝑠 𝑖𝑟 𝑒𝑑 ). When RTP and price at nominal frequency are not295

equal, then HVAC system controller shifts 𝑇𝑟 𝑒 𝑓 away from 𝑇𝑑𝑒𝑠 𝑖𝑟 𝑒𝑑 . In case RTP is lower than the price at nominal frequency, then MPC296

sets 𝑇𝑟𝑒 𝑓 higher than 𝑇𝑑𝑒 𝑠𝑖𝑟 𝑒 𝑑 and vice versa. The temperature deviations must not exceed maximum and minimum allowable temperature297

limits (𝑇𝑧𝑜𝑛𝑒,𝑚𝑎 𝑥 and 𝑇𝑧𝑜 𝑛𝑒, 𝑚𝑖𝑛 ). The difference between 𝑇𝑟 𝑒 𝑓 and 𝑇𝑑 𝑒𝑠𝑖𝑟 𝑒 𝑑 is represented with Δ𝑇𝑧𝑜𝑛𝑒 . The positive value of Δ𝑇𝑧𝑜𝑛𝑒

298

indicates that 𝑇𝑟𝑒 𝑓 is higher than 𝑇𝑑𝑒𝑠 𝑖𝑟 𝑒𝑑 and vice versa.299

𝑇𝑟𝑒 𝑓 =𝑇𝑑𝑒𝑠𝑖𝑟 𝑒 𝑑 +Δ𝑇𝑧𝑜𝑛𝑒 (12)

300

𝑇𝑧𝑜𝑛 𝑒,𝑚𝑖 𝑛 −𝑇𝑑𝑒𝑠 𝑖𝑟 𝑎𝑏𝑙 𝑒 ≤Δ𝑇𝑧𝑜𝑛𝑒 ≤𝑇𝑧𝑜 𝑛𝑒, 𝑚𝑎 𝑥 −𝑇𝑑𝑒𝑠𝑖 𝑟 𝑎𝑏𝑙 𝑒 (13)

A graphical overview of the correlation between RTP and reference temperature set point is shown in Fig. 5. The slopes 𝑆ℎ𝑖𝑔 ℎ and 𝑆𝑙𝑜 𝑤

301

of 𝑇𝑟𝑒 𝑓 can be found using Eqs. (14, 15).302

𝑆ℎ𝑖𝑔 ℎ =𝑅𝑇 𝑃𝑚𝑎 𝑥

𝑇𝑧𝑜𝑛𝑒,𝑚 𝑎𝑥 −𝑇𝑑 𝑒𝑠𝑖𝑟 𝑎 𝑏𝑙𝑒

𝑖 𝑓 𝑅𝑇 𝑃𝑖>0(14)

303

𝑆𝑙𝑜 𝑤 =𝑅𝑇 𝑃𝑚𝑖𝑛

𝑇𝑧𝑜𝑛 𝑒,𝑚 𝑎𝑥 −𝑇𝑑 𝑒𝑠𝑖𝑟 𝑎 𝑏𝑙𝑒

𝑖 𝑓 𝑅𝑇 𝑃𝑖<0(15)

The deviations in RTP can be calculated using Eq. 16.304

𝑅𝑇 𝑃𝑑𝑒 𝑣 =𝑅𝑇 𝑃𝑖−𝑅𝑇 𝑃𝑛𝑜 𝑚𝑖𝑛 𝑎𝑙 ,(16)

where 𝑅𝑇 𝑃𝑖is instantaneous RTP and 𝑅𝑇 𝑃𝑛𝑜𝑚𝑖 𝑛𝑎𝑙 is the RTP at nominal frequency. The values of Δ𝑇𝑧𝑜𝑛𝑒 for a cooling system subject to305

RTP deviations are given by Eqs. (17, 18)306

Δ𝑇𝑧𝑜𝑛𝑒 =𝑅𝑇 𝑃 𝑑𝑒𝑣

𝑆ℎ𝑖𝑔 ℎ

𝑖 𝑓 𝑅𝑇 𝑃𝑑𝑒 𝑣 >0,(17)

307

Δ𝑇𝑧𝑜𝑛𝑒 =𝑅𝑇 𝑃 𝑑𝑒𝑣

𝑆𝑙𝑜 𝑤

𝑖 𝑓 𝑅𝑇 𝑃𝑑𝑒 𝑣 <0.(18)

Combining Eqs. (12), (17) and (18) yield Eqs. (19) and (20) which shows dynamic price-to-reference temperature set point model.308

𝑇𝑟𝑒 𝑓 =𝑇𝑑𝑒𝑠𝑖𝑟 𝑒 𝑑 +𝑅𝑇 𝑃𝑑𝑒𝑣

𝑆ℎ𝑖𝑔 ℎ

𝑖 𝑓 𝑅𝑇 𝑃𝑖>0,(19)

309

𝑇𝑟𝑒 𝑓 =𝑇𝑑𝑒𝑠𝑖𝑟 𝑒 𝑑 +𝑅𝑇 𝑃𝑑𝑒𝑣

𝑆𝑙𝑜 𝑤

𝑖 𝑓 𝑅𝑇 𝑃𝑖<0.(20)

7

Tdesired

Shigh

RTPmax

Slow

Tzone

Tzone,min Tzone,max

¨Tzone=0

Thermal comfort range

RTPnominal

RTPmin

Frequency based RTP range

Fig. 5: Dynamic price-to-temperature set point model.

3.3. PV generation model310

Solar irradiance data with the resolution of 5-min is used to generate PV generation proﬁle. The mathematical expression to relate solar311

irradiance (G) and PV power generation (𝑃𝑝𝑣 ) is given in [42]:312

𝑃𝑝𝑣 (𝐺)=(𝑃𝑟𝐺2

𝐺𝑠𝑡 𝑑 𝑆𝑝if 0≤𝐺≤𝑆𝑝

𝑃𝑠𝑟 𝐺

𝐺𝑠𝑡 𝑑 if 𝐺 > 𝑆 𝑝,(21)

where, 𝑃𝑟represents the rated power of PV system; 𝐺𝑠𝑡 𝑑 represents solar irradiance in standard environment with the value 800 𝑊/𝑚2;313

𝑆𝑝represents a certain irradiance point with the value 120 𝑊/𝑚2.314

4. MODEL PREDICTIVE CONTROLLERS (MPCS)315

This section deals with the formulation of MPC based optimizers both at building and grid levels.316

4.1. Building MPC317

Two types of MPCs are designed at building level: (1) an ordinary MPC without considering grid dynamics (RTP) (2) a DR-enabled MPC318

responsive to RTP.319

4.1.1. Ordinary MPC: The ordinary MPC does not consider RTP signal to formulate an optimization problem. In short, it does not

provide DR service to the grid. The linear form of the ordinary MPC during the prediction horizon [0, 𝑡𝑝] for the building thermal dynamics

deﬁned in Eq. (11) is derived as:

min

𝑢ℎ

𝑡𝑝−1

Õ

𝑡=0

𝑢ℎ(𝑡) + 𝜌.𝜖 )(22a)

s.t. ¤𝑥𝑏(𝑡)=𝐴𝑏𝑥𝑏(𝑡) + 𝐵𝑢ℎ 𝑢ℎ(𝑡) + 𝐵𝑤 𝑏 𝑤𝑏(𝑡)(22b)

𝑇𝑧𝑜𝑛 𝑒,𝑚𝑖 𝑛 −𝜖≤𝑇𝑧𝑜 𝑛𝑒 (𝑡) ≤ 𝑇𝑧 𝑜𝑛𝑒 ,𝑚𝑎 𝑥 +𝜖(22c)

𝑢ℎ, 𝑚𝑖𝑛 ≤𝑢ℎ(𝑡) ≤ 𝑢ℎ,𝑚 𝑎𝑥 ,(22d)

where Eq. 22(a) represents an objective function for HVAC power consumption minimization; Eqs. 22(c, d) specify the constraints on building320

temperature state 𝑇𝑧𝑜 𝑛𝑒 and control input 𝑢ℎ, respectively. The temperature limits may be violated during extreme weather conditions even321

if HVAC system runs at its full capacity. The soft bounds on temperature may relax the MPC problem. Therefore, a slack variable (𝜖)322

is introduced in Eq. 22(a) to ensure the feasibility of optimization problem. A large penalty cost (𝜌) on slack variable is imposed which323

enforces the optimizer to take small value of 𝜖.324

4.1.2. DR enabled MPC: Unlike ordinary MPC, DR enabled MPC considers RTP to formulate optimization problem. The objective

function for DR enabled MPC over the prediction horizon [0, 𝑡𝑝] is formulated in Eqs. (23a)-(23b).

min

𝑢ℎ(𝑡)

𝑡𝑝−1

Õ

𝑡=0

𝑤1(𝑢ℎ(𝑡).𝑅𝑇 𝑃 (𝑡) + 𝜌.𝜖 )2+𝑤2(𝑇𝑧 𝑜𝑛𝑒 (𝑡) − 𝑇𝑟𝑒 𝑓 (𝑡))2(23a)

s.t. 𝑇𝑧𝑜𝑛 𝑒 (𝑡), 𝑇𝑟 𝑒 𝑓 (𝑡) ∈ [𝑇𝑧𝑜𝑛𝑒,𝑚𝑖𝑛 𝑇𝑧𝑜𝑛𝑒 ,𝑚 𝑎𝑥 ].(23b)

Eq. 23(a) deﬁnes a multi-objective optimization problem which compute the squared sum of energy consumption and temperature deviation325

from a reference temperature set point, respectively. The weights 𝑤1and 𝑤2can be assigned based on the user’s preference either one’s326

interested in energy cost minimization or user comfort maximization. The weighted-sum method is used for assigning weights to both327

objectives. This multi-objective problem can be transformed into a single objective problem by assigning zero value to either 𝑤1or 𝑤2.328

A high value of 𝑤2guarantees less temperature deviations from reference temperature set point which ultimately results in high electricity329

cost. With dynamic reference temperature set-points, the MPC can effectively manage the trade-off between electricity cost and occupant’s330

thermal comfort. Constraint 23(b) enforces the MPC controller to maintain 𝑇𝑧𝑜𝑛𝑒 and 𝑇𝑟 𝑒 𝑓 within thermal limits.331

8

4.2. Grid MPC332

The linear form of generator cost function during the prediction horizon [0, 𝑇𝑝] for the grid dynamics deﬁned in Eq. (8) is derived as:

min

𝑢𝑔={𝑢𝑔𝑖}𝑛𝑔

𝑖=1

𝑡𝑝−1

Õ

𝑡=0

𝐽(𝑢𝑔)=𝑢𝑇

𝑔𝐴𝑢𝑔𝑢𝑔+𝑏𝑇

𝑢𝑔𝑢𝑔+𝑐𝑢𝑔(24a)

s.t. ¤𝑥𝑔(𝑡)=𝐴𝑔𝑥𝑔(𝑡) + 𝐵𝑢𝑔𝑢𝑔(𝑡) + 𝐵𝑤 𝑔𝑤𝑔(𝑡)(24b)

𝑢𝑔≤𝑢𝑔≤𝑢𝑔(24c)

Δ𝑢𝑔≤Δ𝑢𝑔≤Δ𝑢𝑔(24d)

𝑥𝑔≤𝑥𝑔≤𝑥𝑔,(24e)

where Eq. 24(a) represents the cost function of a generator; Constraint 24. (c) speciﬁes the limits on generator power 𝑢𝑔; constraint 24. (d)333

sets the ramp up and down limits on generator power; ﬁnally constraint 24. (e) ensures the grid’s MPC controller to maintain grid states334

(frequency) within operational range. 𝐴𝑢𝑔=

𝑎1

𝑢𝑔

...

𝑎𝑛𝑔

𝑢𝑔

and 𝐵𝑢𝑔=

𝑏1

𝑢−𝑔

.

.

.

𝑏𝑛𝑔

𝑢𝑔

are generator cost coefﬁcients with units ($/𝑀𝑊 2ℎ)and335

($/𝑀𝑊 ℎ), respectively.336

The ﬂowchart shown in Fig. 6 summarises the simulation model of the decoupled B2TN that was shown in Fig. 1. The reported solution337

method controls B2TN operation through two levels of optimization between the buildings and a grid. Initially, the optimal power ﬂow (OPF)338

problem is solved without the power regulation of HVAC load. Then, frequency based RTP is calculated using one of the functions given339

in Eqs.(1-3). The B2TN operation starts with the dynamic temperature set points allocation based on RTP. The DR-enabled MPC performs340

optimization to attain the real time temperature set points by considering exogenous inputs; solar irradiance and internal heat gains. Then,341

the optimized HVAC load (𝑢ℎ) and building base load (𝑢𝑏) are sent to the grid. The grid level optimization is performed to conﬁrm that342

building load and distributed PV generation proﬁles satisfy grid’s operational constraints. The states of the buildings and a grid are updated at343

each optimization period 𝑡and set as initial conditions for subsequent optimization interval 𝑡+1. This procedure is repeated until 𝑡becomes344

equal to 𝑡𝑝.345

Start

Solve optimal power flow problem

without the power regulation of HVAC

system and calculate control input and

state (ug0 and Xg0) of the grid

Calculate RTP based pricing (P( )) by

selecting one of the methods given in

Eq. (1), Eq. (2), Eq. (3).

Perform MPC based optimization at

building level:

Objective: Eq. (23a)

Subjected to: Eqs. 22( b, d) and (23b)

Save the results u h(t)

Set u h(t) as an

initial value

Set initial values: (ug0, xg0, u h(t) )

Perform MPC based optimization

at grid level:

Objective: Eq. (24a)

Subjected to: Eqs. 24(b-e)

Save the results (ug(t), xg(t))

t = t + 1

t = tp

End

Yes

No

Set initial values:

(ug(t) , xg(t))

Fig. 6: Proposed decoupled B2TN optimization model ﬂowchart.

5. SIMULATION RESULTS346

5.1. Simulation testbed347

5.1.1. Building testbed: A prototype commercial three story building from one of the referred studies is used [24]. The building348

dynamics are modeled using 3R-2C thermal network. Building thermal parameters which include resistance and capacitance are determined349

through American Society of Heating and Air-Conditioning Engineers (ASHRAE) standard 90.1. The bounds on room temperature are also350

deﬁned based on ASHRAE standards. Temperature bounds are as follows: (a) for ofﬁce hours 21.5-23.5 °C (b) for non-ofﬁce hours 23.5-25351

°C. Ofﬁce hours are speciﬁed from 8:00 AM to 8:00 PM while non-ofﬁce hours are speciﬁed from 8:00 PM to 8:00 AM. The maximum352

9

cooling load is 320 kW for each building and coefﬁcient of performance (COP) of HVAC system is set to 3. The time series data of weather353

forecast, internal and solar heat gains and building load are obtained from Github link [43] which is also used by one of the referred study354

[24].355

TABLE I: Generation limits and cost coefﬁcients of standard IEEE-30 bus system

Generator Bus number 𝑃𝑚𝑎 𝑥 Cost coefﬁcients

Quadratic $/𝑀 𝑤 ℎ2Linear $/𝑀 𝑤 ℎ Constant

1 1 80 0.02 2 0

2 2 80 0.0175 1.75 0

3 22 50 0.0625 1 0

4 27 55 0.00834 3.25 0

5 23 30 0.025 3 0

6 13 40 556 3 0

5.1.2. Grid system: The standard IEEE 30-bus system is used to perform simulations. Six generators are connected to the generation356

buses (1, 2, 22, 27, 23 and 13). The details of under-study test system are listed in Table I. Total of 385 buildings are connected in form357

of clusters to the speciﬁed load buses. Number of buildings are decided by keeping in view nominal bus load and generation capacities.358

In order to evaluate the impact of PV generation on power system operation, the thermal generators at buses 2 and 5 are replaced with359

distributed PV systems. This study assumes that PV systems are located in nearby vicinity, therefore, patterns of solar irradiance remains the360

same for both generators. Solar irradiance data of cloudy and clear sky days with the resolution of 5 mins is obtained from [44] and is used361

to generate PV generation patterns as shown in Fig. 7. The electricity rate for commercial sector in New England is adopted to generate362

frequency based RTP [45]. The grid parameters include: maximum and minimum generation limits, generator cost coefﬁcients, damping and363

inertia coefﬁcients for all buses are obtained from case-30 ﬁle of Matpower [46].364

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hour)

0

10

20

30

40

50

60

70

80

90

PV generation power (MW)

Cloudy day

PV generation profile with rated power 30MW

PV generation profile with rated power 80MW

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hour)

0

10

20

30

40

50

60

70

80

90

PV generation power (MW)

Clear Sky Day

PV generation profile with rated power 30MW

PV generation profile with rated power 80MW

Fig. 7: Generation proﬁles of under-study PV systems.

Simulation work is carried out in MATLAB and building and grid optimization problems are solved using linear and quadratic programming365

MPC Toolboxes. The optimization is performed after every 5 mins by following the procedure given in Fig. 6.366

5.2. Performance comparison of bang-bang and ordinary MPC367

The B2TN performance under bang-bang and ordinary MPC control methods are ﬁrst compared without considering the distributed PV368

generation systems. In order to verify the effectiveness of MPC control strategy, this study compares two schemes:369

(1) Bang-bang control at building level and MPC at grid level. (2) MPC at both building and grid levels370

The ﬁrst scheme assumes that building operator use bang-bang control. This is a rule based strategy where building controller enforces the371

HVAC system to follow a strict temperature bound (e.g., 23±0.2◦𝐶). The HVAC system is only allowed to switch on when zone temperature372

exceeds a certain threshold of ±0.2◦𝐶; otherwise, it remains switched off. The bang-bang control input (𝑢ℎ) is then guided to the grid’s373

optimizer. On the other hand, second scheme develops a MPC based optimizer which allows HVAC system to operate in cost effective374

manners while satisfying the occupant’s thermal comfort. The optimized HVAC’s system control input (𝑢ℎ) is sent to the grid controller375

where MPC based optimization is performed for real time generation dispatch.376

More precisely, bang-bang is an un-optimized control strategy which is simply an on-off control. Therefore, this strategy cannot be377

used to provide real time DR service to the grid. On the other hand, MPC is an optimized control strategy, which can be modiﬁed as378

demand responsive. The importance of DR-enabled MPC is realized particularly in distributed PV generation dominated grids where real379

time supply-demand balancing is required to maintain frequency stability.380

10

0 2 4 6 8 10 12 14 16 18 20 22 24

25

30

35

40

Outdoor air

temperature

(°C)

0

200

400

600

Solar radiation and

internal heat gains

(W)

Ambient Temperature (Tamb)

Internal heat gain (Qint)

Solar heat gain (Qsol)

0 2 4 6 8 10 12 14 16 18 20 22 24

20

20.5

21

21.5

22

22.5

23

23.5

24

Zone

temperature

(°C)

Zone temperature (Tzone)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hour)

0

100

200

300

400

Power

consumption

(kW)

HVAC power consumption (uh)

(a) Performance of bang-bang controller

0 2 4 6 8 10 12 14 16 18 20 22 24

25

30

35

40

Outdoor air

temperature

(°C)

0

200

400

600

Solar radiation and

internal heat gains

(W)

Ambient temperature (Tambt)

Internal heat gain (Qint)

Solar heat gain (Qsol)

0 2 4 6 8 10 12 14 16 18 20 22 24

21

21.5

22

22.5

23

23.5

24

24.5

25

Zone

temperature

(°C)

Zone temperature (Tzone)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hour)

0

50

100

150

200

250

300

Power

consumption

(kW)

HVAC power consumption (uh)

(b) Performance of ordinary MPC

Fig. 8: Building states: performance comparison of bang-bang controller and ordinary MPC

5.2.1. Building thermal behavior: Fig. 8 shows the 24 hours proﬁle of zone temperature and HVAC power consumption for bang-bang381

controller and MPC. The upper portions of Figs. 8(a) and 8(b) illustrate the time series data of exogenous disturbances including outdoor382

ambient temperature, and solar and internal heat gains. For efﬁcient control of zone temperature, these variables must be considered.383

Middle portions of Figs 8(a) and 8(b) depict zone temperature of bang-bang controller and MPC, respectively. The zone temperature set384

point is 23.5◦𝐶for non-ofﬁce hours while temperature set point for ofﬁce hours is 21.5◦𝐶. The zone temperature of the building gradually385

reaches to set point (23.5◦𝐶) from initial set point (22◦𝐶) during non-ofﬁce hours. A swift change in zone temperature is observed at 7am386

because the set point for ofﬁce hours is set to 21.5◦𝐶. The zone temperatures are mostly observed within limits; however, higher deviations in387

zone temperature are noticed in ofﬁce hours owing to exogenous disturbances when compared to non-ofﬁce hours. In contrast to bang-bang388

controller, the low deviations in zone temperature are observed in case of MPC as shown in the middle portion of Fig. 8(b). This is due to389

the reason that MPC maintains the temperature at upper temperature bound (23.5◦𝐶during ofﬁce hours which is 2◦𝐶higher when compared390

to bang-bang controller. Therefore, exogenous disturbances do not inherently effect the zone temperature.391

Bottom parts of Figs. 8(a) and 8(b) show the cooling power of HVAC system for bang-bang controller and MPC, respectively. In case of392

bang-bang controller, cooling demand of HVAC system changes frequently owing to variations in exogenous disturbances. The bang-bang393

controller enforces the HVAC load to turn-on and turn-off on frequent basis to meet the strict temperature bounds 21.5±2◦𝐶. Hence a394

considerable spikes in HVAC power consumption are noticed through out the time horizon. In comparison with bang-bang controller, MPC395

generates lower spikes of HVAC power conumption around 7am and these spikes are unnoticeable for rest of the optimization period.396

11

Conclusively, amongst the two controllers under-study, MPC is energy efﬁcient while bang-bang is not.397

5.2.2. Grid performance: In general, the objective of grid operator is to minimize the electricity generation cost while maintaining the398

frequency stability. As discussed earlier, building HVAC load experiences variations throughout the day, which causes deviations in both the399

power generation and grid’s frequency. The load variations are higher in case of bang-bang controller as compared to MPC. Hence, higher400

variations in power generation and grid frequency are noticed in case of bang-bang controller in Fig. 9(a) as compared to MPC in Fig. 9(b).401

0 2 4 6 8 10 12 14 16 18 20 22 24

0

50

100

150

200

250

300

Power

Generation (MW)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hour)

49.8

49.85

49.9

49.95

50

50.05

50.1

50.15

50.2

Frequency (Hz)

(a) Performance of Bang-bang controller

0 2 4 6 8 10 12 14 16 18 20 22 24

0

50

100

150

200

250

300

Power

Generation (MW)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hour)

49.8

49.85

49.9

49.95

50

50.05

50.1

50.15

50.2

Frequency (Hz)

(b) Performance of ordinary MPC

Fig. 9: Grid states: performance comparison of bang-bang controller and ordinary MPC

5.3. Impact of solar PV power generation patterns on B2TN operations402

This section investigates the cloud covered sky and clear sky impacts of solar PV power generation on B2TN operations. The performance403

of proposed framework is evaluated in terms of frequency deviations suppression, load regulation capability and building’s electricity cost404

for both clear sky and cloudy days.405

5.3.1. Clear sky impact of solar PV power generation: In order to mitigate the effects of uncertain PV generation in power system,406

the regulatory service is provided through ﬂexible operation of HVAC load. In this regard, a dynamic price to temperature set point model is407

formulated for DR-enabled MPC. As dynamic reference temperature set point is a function of frequency based RTP pricing, therefore, this408

study investigates the impact of three frequency based RTP functions on dynamic reference temperature set points allocation. The ordinary409

MPC is considered as reference case. Following cases of frequency based RTP functions are considered for the formulation of dynamic410

reference temperature set points for DR-enabled MPC.411

•Case 1: 𝑇 𝑎𝑛ℎ function based RTP.412

•Case 2: Linear function based RTP.413

•Case 3: 𝑇 𝑎𝑛ℎ−1function based RTP.414

The DR-enabled MPC follows the RTP signal for dynamic allocation of reference temperature set points. Eqs. (19, 20) are used to415

set dynamic reference temperature set points according to RTP signals generated using Eqs. (1-3). The DR-enabled MPC modiﬁes the416

power consumption of HVAC load according to the dynamic reference set points using objective function deﬁned in Eq. (23a) while the417

12

ordinary-MPC uses Eq. (22a) in the optimization process. In short, DR-enabled MPC enables the active participation of HVAC load in DR418

by responding to RTP signals while ordinary MPC is not price responsive.419

The grid does not experiences with noticeable frequency deviations during clear sky conditions, therefore, all three under-study frequency-420

enabled RTP functions generate same pricing signal as shown in the upper portion of Fig. 10. The grid frequency remains close to the421

nominal frequency (50 Hz), hence the RTP signal remains ﬂat over the course of the day. Moreover, the reference temperature set point422

remains unchanged thus no signiﬁcant variations in HVAC power consumption are observed in lower middle potion of Fig. 10. Finally, grid423

frequency remains close to the nominal frequency as shown in bottom portion of Fig. 10.424

0 2 4 6 8 10 12 14 16 18 20 22 24

49.6

49.8

50

50.2

Frequency (Hz)

5

7.5

10

12.5

15

17.5

20

22.5

25

Electricty price

(Cents/kWh)

Grid frequency with ordinary MPC controller

RTP remains same for all three under-study frequency-enabled pricing functions

0 2 4 6 8 10 12 14 16 18 20 22 24

21.5

22

22.5

23

23.5

24

24.5

Tzone (°C)

Zone temperature (Tzone)

0 2 4 6 8 10 12 14 16 18 20 22 24

0

50

100

150

200

250

300

350

Power

consumption

(kW)

HVAC power consumption (Uh)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (hour)

49.5

50

50.5

Frequency (Hz)

Grid frequency with DR-enabled MPC controller

Fig. 10: Clear sky impact of PV generation on B2TN operations

5.3.2. Cloud covered sky impact of solar PV power generation: In cloudy conditions, the grid frequency experiences with signiﬁcant425

deviations from its nominal frequency owing to variations in solar PV power generation. The frequency-enabled pricing generators map the426

frequency variations into RTP signal. The system performance under 𝑇𝑎𝑛ℎ function based RTP is shown in Fig. 11. The results of B2TN427

are investigated in terms of zone temperature set points, power use of HVAC system, and grid frequency deviations suppression. Top portion428

of Fig. 11 portrays the variations in RTP w.r.t grid frequency. Upper middle portion shows the dynamic reference temperature set points.429

Whereas, HVAC load responses to dynamic reference temperature set points are shown in the lower middle part of Fig. 11. Bottom portion430

of Fig. 11 shows grid’s frequency deviations which are observed less as compared to ordinary MPC control (as shown in top portion of431

Fig. 11). Same patterns of simulation results are shown in Fig. 12 for Case 2. The variations in RTP w.r.t grid frequency are shown in top432

portion of Fig. 12. The results revealed more variations in RTP and zoom temperature set points which result in more ﬂuctuations in power433

use of the HVAC load. As a result, the grid experiences less frequency deviations as compared to Case 1.434

Fig. 13 shows the impact of 𝑡𝑎𝑛 ℎ−1relation between grid frequency and RTP as expressed in Eq. (3). High variations in RTP are noticed435

especially at peak dips and spikes of grid frequency. As, dynamic temperature set point assignment is based on RTP, therefore, high variations436

in dynamic reference temperature set points are observed leading to more ﬂexible operations of HVAC load compared to Case 1 and Case 2.437

For more clear overview of price and frequency deviations, the histograms of RTP and grids’s frequency are generated in Figs. 14 and438

15, respectively. The results in the form of histograms depict the inverse relationship between RTP and frequency deviations. 𝑇𝑎 𝑛ℎ−1based439

RTP function experiences high price and low frequency deviations. The ﬁndings of proposed decoupled B2TN in terms of RTP deviations,440

load regulation capability and peak frequency deviation suppression are given in Table II for both cloud covered and clear sky days.441

13

0 2 4 6 8 10 12 14 16 18 20 22 24

49.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5

Frequency (Hz)

5

7.5

10

12.5

15

17.5

20

22.5

25

Electricty price

(Cents/kWh)

Grid frequency with ordinary MPC Controller

Tanh function based RTP

0 2 4 6 8 10 12 14 16 18 20 22 24

21.5

22

22.5

23

23.5

24

24.5

Zone

temperature

(°C)

Zone temperature (Tzone)

0 2 4 6 8 10 12 14 16 18 20 22 24

0

50

100

150

200

250

300

350

HVAC power

consumption

(kW)

HVAC power consumption (uh)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hour)

49.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5

Frequency (Hz)

Grid frequency with DR-enabled MPC controller

Fig. 11: Performance of tanh function based price-to-temperature set point model

14

0 2 4 6 8 10 12 14 16 18 20 22 24

49.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5

Frequency (Hz)

5

7.5

10

12.5

15

17.5

20

22.5

25

Electricty price

(Cents/kWh)

Grid frequency with ordinary MPC controller

Linear price function based RTP

0 2 4 6 8 10 12 14 16 18 20 22 24

21.5

22

22.5

23

23.5

24

24.5

Zone

temperature

(°C)

Zone temperature (Tzone )

0 2 4 6 8 10 12 14 16 18 20 22 24

0

50

100

150

200

250

300

350

HVAC power

consumption

(kW)

HVAC power consumption (uh)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (hour)

49.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5

Frequency (Hz)

Grid frequency with DR-enabled MPC controller)

Fig. 12: Performance of linear function based price-to-temperature set point model

15

0 2 4 6 8 10 12 14 16 18 20 22 24

49.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5

Frequency (Hz)

5

7.5

10

12.5

15

17.5

20

22.5

25

Electricty price

(Cents/kWh)

Grid frequency with ordinary MPC controller

Tanh-1 function based RTP

0 2 4 6 8 10 12 14 16 18 20 22 24

21.5

22

22.5

23

23.5

24

24.5

Zone

temperature

(°C)

Zone temperature (Tzone)

0 2 4 6 8 10 12 14 16 18 20 22 24

0

50

100

150

200

250

300

350

HVAC power

consumption

(kW)

HVAC power consumption (uh)

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (Hour)

49.5

49.6

49.7

49.8

49.9

50

50.1

50.2

50.3

50.4

50.5

Frequency (Hz)

Grid frequency with DR-enabled MPC controller

Fig. 13: Performance of 𝑡𝑎𝑛ℎ−1function based price-to-temperature set point model

16

Fig. 14: Impact of under-study RTP models on RTP deviations

Fig. 15: Impact of under-study RTP models on grid’s frequency deviations

17

TABLE II: Clear sky and cloud cover impacts of PV power generation on B2TN

B2TG system’s performance parameters Clear sky day Cloud cover day

Tanh based RTP Linear function based RTP 𝑇 𝑎𝑛 ℎ−1based RTP

Peak spike 18 19.03 20.01 23.6

RTP deviations (Cents/kWh) Peak dip 14.8 11.72 10.72 6.9

Peak spike 0.52 0.37 0.33 0.28

Frequency deviations (Hz) Peak dip -O.3 -0.38 -0.35 -0.3

Power regulation (MW) Total 0 7 9 12

Generation cost ($) Total 4230 4444.8 4352 4259.6

Building electricity cost ($) Total 120 122.2 122.2 122.2

6. CONCLUSION442

This paper develops a decoupled B2TN framework integrating smart buildings, transmission network and distributed PV systems for power443

system frequency support. A decoupled design implies separate MPC controllers to segregate building and grid control decisions. The grid’s444

MPC is explicitly modeled by considering uncertainties in both PV generation and energy demand. Further, a price responsive MPC strategy445

for optimal scheduling of HVAC load is developed at building level where reference temperature set point is adjusted dynamically subject to446

RTP. As, frequency is an indicator of grid states (supply-demand imbalance), therefore RTP is computed as a function of real-time frequency.447

TSO generates and broadcasts frequency dependent time varying signals to modulate the consumers’ response for frequency support. Three448

frequency dependant RTP generators namely; linear, hyperbolic tangent and inverse hyperbolic tangent are used where each generator maps449

the frequency deviations at different price range. The following ﬁndings are observed with these three price generators for a cloud covered450

day.451

•Linear, hyperbolic tangent and inverse hyperbolic tangent functions map the operation frequency range [49.5 50.5] Hz to price ranges452

[10.4, 20.4], [11.59, 19.2] and [3.9 26] cents/kWh, respectively.453

•A price responsive MPC controller responded to linear, hyperbolic tangent and inverse hyperbolic tangent based RTP signals suppressed454

the peak frequency deviations up to 29.7%, 21%and 40%, respectively when compared to ordinary MPC (without DR service).455

•A price responsive MPC controller responded to linear, hyperbolic tangent and inverse hyperbolic tangent based RTP signals provided456

load regulation up to 9 MW, 7 MW and 12 MW, respectively.457

The synergy of HVAC load and other building’s demand ﬂexible energy resources; e.g., battery storage and photo hybrid electric vehicles458

can be considered to further extend the demand ﬂexibility to counter the challenges of TSO in PV systems penetrated grids.459

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