modeling_hydronic_networks.20120520.4fb93a96c3d966.77448123

7/29/2019 modeling_hydronic_networks.20120520.4fb93a96c3d966.77448123

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Modeling Hydronic Networks Rev 1: Feb 15, 2009

V. Chandan Page 1 of 29

MODELING HYDRONIC NETWORKS

1. Introduction

1.1 Background

Space heating and cooling energy usage accounts for more than 20% of the total

energy consumption in both commercial and residential sectors [1]. The official end

usage energy consumption statistics for 2007 as published by the Energy Information

Administration (EIA) which reveals this fact is reproduced in Fig. 1.

Residential Commercial

Fig. 1: End use energy usage, United States, 2007 (Source: EIA/ Annual Energy Outlook 2009)

Given this background, it is important to note the rising popularity of centralizedheating and cooling systems, particularly in North America and Europe to meet

energy needs in different sectors. In 2003, nearly 25% of all commercial buildings in

the United States (with cooling infrastructure) used centralized air conditioning as the

primary means to achieve space cooling [2]. Similarly, the penetration of district

heating in a time window of 15 years for residential spaces in Denmark [3] can be

gauged from Fig. 2.

Fig. 1: Dwellings according to type of heat installation (Source: Statbank Denmark 2004)


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Of the various centralizedheating and cooling solutions, hydronic systems which use

hot or chilled water are quite popular, where water flows through piping that connects

a boiler, water heater or chiller to suitable terminal heat transfer units located at the

space or process. Most hydronic systems today are forced (i.e. use pumps to maintain

flow) and closed (recirculating). Energy requirements in conditioned spaces are often

uneven, depending on various factors such as occupancy levels, physical location ofthe space and ambient conditions. Moreover, the significantly large footprint of such

systems in the global energy consumption scenario dictates the need to meet such

performance requirements in more robust, energy efficient ways. These factors

provide the necessary impetus to understand, analyze and control such systems.

The objective of the present work is to develop a compact framework for analyzing

hydronic systems and subsequently provide control solutions to meet the issues

addressed above. In order for the tools to be useful, they must be generic and

applicable to networks of any scale, complexity and configuration. Practical

implementation of the control methodologies is well aided by technological

advancements such as variable speed pumps and fans and electronically controlled

valves, and also robust communication technologies such as BACnet [4]. The main

challenge, however, lies on the theoretical frontier primarily because of the arbitrary

complexity that hydronic networks may posses.

1.2 Modeling challenges

The development of a generic model to describe hydronic systems faces the following

challenges:

(i) Scalability:The size of the system can be arbitrarily large, meaning that theinput, state and output vector space can have very high orders. The modelmust be scalable accordingly.

(ii) Generality: The model must take into account the fact that the systems canhave arbitrarily complex architectures. This is particularly true for district

hydronic networks, where the complexity depends on numerous factors such

as layout and planning of the service district that could range from a small

university campus to a big city.

(iii) Timescale differences:The dynamics in hydronic systems can be grouped intotwo categories hydraulic and thermal. The hydraulic modes are

practically several orders of magnitude faster than the thermal modes. This

time-scale separation is more of an advantage than a constraint as it allows the

decomposition of system dynamics, thereby reducing complexity in the

representation.

1.3 Control challenges

The issues facing the control tasks for the system are described below:


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(i) Complex interactions:The states of the system are coupled to inputs and otherstates, therefore SISO loops would most likely yield dissatisfactory

performance and higher order controllers shall be needed. The selection of

such control architectures is a non-trivial task though several techniques have

been proposed in literature.

(ii)

Modeling uncertainties: Given the complexity associated with the modelingof these systems, it is highly likely that large immeasurable uncertainties may

be present in the representation, and the controllers need to be designed

suitably to meet such challenges.

(iii) Energy optimality:The goal of the control task is to meet the performancerequirements with maximum efficiency. This is particularly difficult to achieve

due to the interactions and uncertainties present in the model.

1.4 Scope of the present work

In this document, we propose a graph theoretical framework to analyze complex

hydronic systems which is scalable and generic. Based on this framework, a formal

derivation of the full order state space representation is accomplished, which is

subsequently used to yield a simpler reduced order representation. The procedure is

explained using an example system and the corresponding model is validated through

simulation experiments on this system. Lastly, control design ideas based on the

model have been proposed for implementation in future.

2. Literature SurveyMathematical models for describing hydronic system components are very wellknown and extensively reported in literature. They are of varying complexity ranging

from static lumped models to dynamic Finite Element models. However, a formal

procedure to integrate these models into a generic framework for understanding and

controlling the larger system is not very well developed.

A significant attempt towards that has been reported in [5], where the authors propose

a graph based procedure to obtain a static matrix representation of the behavior of

heat exchanger networks (used in process industries). Static models have also been

developed to optimize the production and distribution schedules in district heating

networks [6] and for distributed control of such networks [7]. Though a static

representation is useful for estimating the steady state system response and to design

static controllers - as in the above examples, real time control design is difficult

without modeling the transient dynamical behavior of the system. The present work

aims to overcome this limitation by introducing a dynamic representation which is

both generic and scalable.

Graph theory is the tool of choice for analyzing complex systems. Several


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applications can be found in power systems, where graph theoretical methods have

been used to address scheduling [8], security [9] and observability [10] problems.

Linear graph theory has also been used for modeling the dynamical behavior of

multi-body electromechanical systems [11, 12]. The present work extends these ideas

to the modeling of complex hydronic systems based on a linear graph description,

similar to techniques used in above references. This approach is unifying and moreinsightful when compared to alternate modeling methodologies, such as bond graphs,

because the graph structure is closely reminiscent of the physical architecture of the

network.

The problem of controlling complex systems is highly intertwined with the control

structure selection and control objectives. The common practice is perfectly

decentralized control using SISO loops [13, 14, and 15]. However, it is impossible to

optimize overall system performance using such localized control architecture without

taking into account the interactions among the different physical states of the system.

On the other extreme, a single centralized controller is the ideal solution to this

problem, but is not practical, particularly for highly complex systems with large sizes

of the state, input and output vector spaces. To reduce computational effort, most

centralized schemes [6, 16] assume highly simplified, static models, which undermine

their accuracy in meeting the control objectives.

Since both centralized and decentralized schemes have limitations and practical issues

associated with them, intuitively the best approach would be to take a middle path.

Such an approach has been used for district heating networks in [7] where the

consumers are grouped into clusters, each of which is then supervised by an agent and

the objective is to optimally distribute the excess energy available in the network

among all components in a cluster. Static models were used and the clustering isperformed heuristically. The main thrust area that has been identified for the present

work is to develop a unified, formal procedure for coarsening the system into

subsystems and thereby use hierarchical control architecture to achieve the desired

objectives. This report proposes some simple ideas in that direction towards the end.

The organization of the report is as follows. The general architecture of hydronic

systems has been explained in section 3 using an example. The linear graph

theoretical representation for representing interconnectivity in the system has been

proposed in section 4. Section 5 describes the structure of linear models of the

components in the system and procedures for obtaining them, demonstrated using the

example system. The procedure to obtain the state space representation of the overall

system using the connectivity information from the graph and linear component

models has been described in section 6. The steps to obtain a reduced order system

representation using time-scale decomposition have also been presented. Section 7

shows validation results for the reduced order representation on the example system.

Finally, ideas for coarsening and controlling the system are proposed in section 8,

listing the future direction of research.


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3. Hydronic system architectureThe components of a hydronic system are chillers/boilers, air handling units, terminal

units, air ducts, fans/blowers, pumps, valves and piping. Some systems can also have

secondary (booster) pumps, apart from the primary ones to increase circulationthrough specific portions of the network. Typically, the fluid circulated is water which

may sometimes be mixed with additives to modify the freezing and boiling points, in

order to ensure safe operation. Physical details of these components can be found in

handbooks, e.g. [17].

The layout of the system depends on several factors, the primary one being the spatial

layout of the service zones and can be arbitrarily intertwined. Fig. 3 shows the

schematic layout of a cooling system which is used as example for demonstration and

validation purposes in this work. It emulates the architecture of the cooling network

for a 2-storeyed building with three clusters of zones in each floor. Each cluster is

assumed to be handled by a heat exchanger and cold air is then ducted to the terminal

units in the zones that constitute the cluster. Arrows indicate the direction of fluid

flow in the system. The inputs in the system are the valve opening factors, pump

speeds, chiller cooling rates, the air mass flow rates and inlet temperatures in the

Liquid Air Heat exchangers (LAHXs). The system outputs are the zonal heat transfer

rates, achieved by each of the LAHXs. The Pipes, junctions, chillers and LAHXs are

dynamical; hence the system also has several states.

The system was simulated using THERMOSYS, and the inputs and physical

parameters were iterated till a satisfactory operating condition was obtained. The

values of the inputs, states and outputs at the chosen operating condition have beenpresented in Appendix (A.1). The choice of these parameters is physically consistent.

Note that the dummy junctions 3, 10 and 12 in the system were introduced for

purely pathological purposes in order to ensure proper pump causalities.


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Fig. 3: Example system schematic

4. Graph representation of interconnectivity

In this section, a generic and scalable graph theoretic framework for representing a

hydronic system has been presented .The (di) graph maps the interconnectivity among

the various physical components in the system, and because its vertices and edges

correspond to physical variables, it ultimately describes the interactions among these

variables - quantified by some matrices that shall later be used in the state space

representation. The graph for the example system is shown in Fig. 4.


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Fig. 4: Graph representation of the example system

4.1 Features

The proposed representation has the following features:

(i) Vertices(a)Energy flow vertices : Heat Exchangers and chillers/boilers [shown by

bluedot](b)Mass flow vertices:Junctions [shown by blackdot]

The vertices have dynamics. The mass flow vertices have fast hydraulic and

thermal dynamics, whereas the energy flow vertices have slow thermal

dynamics only. The example system has 11 mass flow and 8 energy flow

vertices

(ii) EdgesEdges are directed along the direction of fluid flow. Each edge is associated

with two variables the temperature of the fluid and its mass flow rate. Thus,

edges carry mass flow and energy flow information to and from the vertices.

The example system has 25 edges (not numbered in its graph)

(iii) PipesA pipe is a directed path in the graph which originates and ends at mass flow

vertices. All other vertices in the pipe are energy flow vertices. Physically, a

pipe represents a section in the network having the same fluid mass flow rate


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dynamics. Each pipe is associated with one variable the fluid mass flow rate.

Pipes have directions and correspond to a pump or a hydraulic resistance in

the actual system. The example system has 17 pipes. It must be noted that the

entire set of pipes in a network decomposes it, since this set encompasses all

possible edges. This fact can be verified from Fig. 4.

4.2 Connectivity matrices

The graph representation described above can be used to formally obtain connectivity

information about the system in the form of matrices described here which shall be for

the state space representation.

Define the following variables:

Number of energy flow vertices in

Number of mass flow vertices jn

Number of edges kn

Number of pipes ln

Number of control valves vn

Number of pumps pn

Number of chillers cn

The connectivity matrices are now defined as follows

(i) Flow incidence matrix,j ln n

A defined as:

1 if pipe comes out of junction

1 if pipe enters junction

0 otherwise

pq

q p

a q p

=

(ii) Semi incidence matrix, ( )k j in n nB + defined as:1 if vertex is tail of edge

0 otherwisepq

q pb

=


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This Matrix is split into 2 sub-matrices, fB and tB , as:

|f tB B B =

(iii) Pipe decomposition matrix,k ln n

C defined as:

1 if edge is contained in pipe

0 otherwisepq

p qc

=

(iv) Flow incidence matrix,j kn n

D defined as:

1 if edge enters junction

-1 if edge leaves junction

0 otherwisepq

q p

d q p

=

(v)

Energy semi-incidence matrix, i kn nE defined as:

1 if edge enters energy flow vertex

0 otherwisepq

q pe

=

A MATLAB based routine is used to number the pipes and edges in the representation

and then generate the above matrices, the inputs being the graphs adjacency matrix

and labels for energy flow and mass flow vertices. The program uses the sparseness

properties of these matrices to reduce required memory and computation resources.

5. Component modelsThe general forms of the linear dynamical equations for the components of the

hydronic system have been presented in this section. It must be noted that the physical

variables that appear in these equations are deviation variables, describing

perturbations about nominal values.

5.1 Governing equations

5.1.1 Pumps and Hydraulic Resistances (Pipes)

Governing Equation

Conservation of momentum:

1 2 3 3 4 1,2,...l l l l ll

l in out l l

dma m a a p a p a AI l n

dt= + + + =

&& .(1)


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Here:

lm& : =deviation in mass flow rate through the pipe

: =deviation in angular speed of pump, if applicable

inp : =deviation in fluid pressure at inlet section of the pipe

outp : =deviation in fluid pressure at outlet section of the pipe

lAI : =deviation in isentropic area1 of valve, if applicable

Coefficient Matrices

The following matrices are to be constructed:

(i)11 1

{ }l

l

n nA diag a = ,

(ii) 2 l pn nA =Constructed algorithmically as per the following logic:

a) All entries of row l of 4A are zero if pipe l doesnt have apump.

b) If pipe l has a pump whose number isp, then all entries ofrowl , except the element 2( , )a l p are assigned zero.

c) The element 2( , )a l p then is assigned the value 2la

(iii)13 3

{ }l

ln nA diag a =

(iv) 4 l vn nA : Constructed algorithmically as per the following logic:

d) All entries of row l of 4A are zero if pipe l doesnt have acontrol valve.

e) If pipe l has a control valve whose number isv, then allentries of rowl , except the element 4( , )a l v are assigned zero.

f) The element 4( , )a l v then is assigned the value 4la 1The isentropic area is linearly related to the valve opening factor


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5.1.2 Flow Junctions (mass flow vertices)

Governing Equations

Conservation of mass:

1 1,2....j j

inlets outlets j

dpb m m j n

dt = = & & (2)

Here:

jp : =deviation in pressure of the junction

inletsm & : =sum of deviations in inlet mass flow rates to the junction

outletsm & : =sum of deviations in outlet mass flow rates from the junction

Conservation of energy:

{ } .{ } { } .{ } { } .{ } { } .{ }j j j j j j j j j

inlets outlets inlets outlets

dTd m e m f T g T

dt= + & & 1,2.... jj n= .. (3)

Here:

jT : =deviation in temperature of the junction

{ }jinletsm& : =vector of deviations in inlet mass flow rates to the junction

{ }joutletsm& : =vector of deviations in outlet mass flow rates from the junction

{ }jinletsT : =vector of deviations in inlet fluid stream temperatures to the junction

{ }joutletsT : =vector of deviations in outlet fluid stream temperatures from the junction



(i) 1 j kn nW : In rowj of D (defined in section 3), replace all1s by the elements of

{ }jd and all 1s by the elements of{ }je . Doing this for all 1,2.... jj n= , gives the

matrix 1 j kn nW


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(ii) 2 j kn nW : In rowj of D, replace all1sby the elements of { }jf and all 1s by the

elements of{ }jg . Doing this for all 1,2.... jj n= , gives the matrix 2 j kn nW

5.1.3

Chillers/boilers and Heat Exchangers (Energy Flow Vertices)

Governing Equations

Fluid energy conservation

,1 , 2 , 3 , 4 , 1,2,...

L i i i i i

in i in i L i w i i

dTq T q m q T q T i n

dt= + + + =& (4)

Structure (wall) energy conservation

,1 , 2 , 3 , 4 , 5 , 6 , 7

w i i i i i i i iin i in i a in i a in i L i wi indT r m r T r m r T r T r T r Q

dt = + + + + + + && & 1,2,... ii n= ..... (5)

Here:

,L iT : = deviation in liquid temperature for the component (chiller/ boiler/ heat

exchanger)

,w iT : =deviation in wall temperature for the component

,in iT : =deviation in inlet liquid temperature for the component

,in im& : =deviation in liquid mass flow rate for the component

,a in im& : =deviation in air mass flow rate through heat exchangers

,a in iT : =deviation in heat exchanger inlet air temperature

inQ& : =deviation in inlet heat transfer rate to the chiller/ boiler



(i) 1 1{ }i ii

n nQ diag q = . Similarly construct 2 3 4, ,Q Q Q

(ii) 1 1{ }i ii

n nR diag r = . Similarly construct 2 3 4, ,R R R


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(iii) 5 ( )i i cn n nR is obtained by deleting the first cn columns from 5{ }idiag r . Similarly

construct 6R

(iii) 7 i cn nR : Constructed algorithmically as per the following logic:

g) First construct the matrix *7R as: 7{ }:i cdiag r i n h) Then construct *77

( )0i c

i c c

n n

n n n

RR

=

5.2 Procedure for obtaining coefficients

The coefficients appearing in equations (1) to (5) can be obtained by linearization of

non-linear models for the components or through experiments on an actual system.

For the example system in this work, THERMOSYS [18] based non-linear models of

the components were used. The techniques for linearizing them are described in table

1 below.

Table 1: Linearization techniques used for various components in the example system

Component Linearization technique

Pumps Simulation experiments

Hydraulic Resistances Equations

Flow Junctions Equations

Chillers System Identification tool

Liquid Air Heat Exchangers Equations +Simulation experiments

The following points must be noted:

Pumps are modeled in THEMOSY S using performance curves. These curveswere arbitrarily designed for the example system and might not qualitatively

capture the real characteristics of a pump

For the chillers, linearization using equations and simulations produced largeerrors, possibly because the assumption that the heat transfer coefficients do not

depend on inlet stream liquid temperatures was not quite true in the operatingregimes of these components. Therefore, the system identification tool of

MATLAB was used and the results were satisfactory.2

2As a long termsolution to this problem, however, the THERMOSYS heat source and heat exchanger models need

be changed, such that the heat transfer coefficients depend on the temperature of liquid inside these components

rather than that of the inlet stream.


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Linearization of the liquid-air heat exchangers using equations and simulationsalso produced errors larger which were comparatively larger than that for other

components. System identification as was used for chillers would possibly

increase accuracy of the estimated coefficients but it was not pursued since the

errors were less than 15% for small perturbations in inputs

6. State Space RepresentationThis section describes the procedure of obtaining the full order and reduced order

state space representation of the system from its connectivity information (section 4)

and the individual component models (section 5)

6.1 States, Inputs and Outputs

The hydronic system has the following states, inputs and outputs3 which are

deviations from nominal values

States

(i) Pipe mass flow rates(ii) Junction pressures(iii) Junction temperatures(iv) Liquid temperatures in the boilers/chillers and heat exchangers(v) Structure (wall) temperatures of the boilers/chillers and heat exchangersInputs(i) Valve isentropic areas(ii) Pump speeds(iii) Chiller/boiler external heat transfer rates4(iv) Air mass flow rates in the heat exchangers(v) Inlet air temperatures to the heat exchangersOutputs

Heat transfers (cooling/heating) achieved by the heat exchangers

3The outputs here refer to quantities which are of practical usefulness such as heat transfer rates. They may or not

correspond to the usual definition (fromcontrols perspective) as measurable quantities.

4These are assumed to be independently controllable inputs. This is a hypothetical assumption but is used here for

simplicity. Using more advanced formulations of the chiller/boiler models, this can be done away with.


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6.2 Full Order State Space Representation

The full order state space representation of the system is obtained by assembling the

connectivity and coefficient matrices obtained before in proper order.

1 3 4 2

1

1 2 2

, 2 1 1 3 4 , ,

, 1 2 2 5 6 , 7 3 4 , ,

'l l v

j j

j f t j in

L i f t L i a i

W i f t W i a in i

m A A A m A A AI

p B A pd

T WC WB WB T Qdt

T Q EC QEB QEB Q Q T m

T REC R EB R EB R R T R R R T

= +

+ +

& &

&

&

Some interesting observations about the system behavior can be made from the state

space representation:

(i) Interconnectivity of states:The hydraulic dynamics, which corresponds tomass flow rates and pressures, is decoupled from the thermal dynamics but

not vice versa. This can be observed from the state space matrix. It is

physically justified, because the hydronic fluid is assumed incompressible.

(ii) Connectivity between inputs and states:The role of each input on the states isevident from the representation. All inputs affect the thermal states

( , ,, ,j L i W iT T T ). The mechanism, in which the effect propagates, however, varies.

For example pump speeds, affect the thermal states through the hydraulic

states, but the inlet air temperatures affect them directly.

(iii) Singularity in the state space matrix: The state space matrix is singularbecause the hydraulic dynamical system is closed, leading to one equation perloop which is linearly dependent on the other hydraulic equations.

6.3 Reduced Order State Space Representation

The full order space representation describes all possible states of the system. In

practice, the most important states are the thermal states, particularly the wall

temperatures which directly affect the heat transfer rates achieved in the heat

exchangers. Fortunately, the time-scale variations of the dynamics associated with the

different states in the system allow the desired reduction of the system to a more

concise state space representation.

The time scales for the different dynamics in the example system, as estimated from

THERMOSYS simulations have been presented in Table 2 below. It can be observed

that the slowest modes correspond to the wall temperature changes, where the time

constants are much larger than those associated with the other modes. This would

generally occur for almost all hydronic systems, because the wall thermal capacity is


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much larger than the other compliances in the system. Therefore, it is reasonable to

assume that with the exception of the wall temperatures, all other states are static.

Table 2: Average time constant estimates for the modes in the example system

States Estimated Time constant (sec)Mass flow rates 0.05 1.7

Junction pressures 5~10

Junction temperatures ~0.01

Liquid temperatures for chillers/boilers and heat exchangers 1.5 5

Wall temperatures of chillers/boilers and heat exchangers 20 25

The static states in the representation can be solved algebraically and substituted in

the equations for the wall temperature dynamics. This leads to the following reduced

order state space representation:

{ } [ ]{ } [ ]

&

&

v

inW,i 33 W,i 31 32 33 34 35

a,i

a,in,i

AI

dQT = A T + B B B B B

dtm

T

(6)

The matrices that appear in this equation can be obtained by the following procedure:

(i) Define 1 2 1 2, , and ZY Y Z as follows:-1

1 1 3 1 2 2

-12 2 1 2 1

-1 -11 2 2 1 2 1

-1 -12 2 2 1 4

- ( )

- ( )

( ) -

( )

f f t

f f

f t

f t

Z QE Q QEB W B W B

Z Q E Q EB W B W C

Y W B W BZ Z WC

Y W B W BZ Q

= +

=

=

=

(ii) Then, compute 33A :-1

33 6 2 f 2 2 t 5 1 4A = R + R EB Y - (R EB + R )Z Q

(iii) Obtain 3Z as follows:


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1 3

p1

4 4

p

( 1)

13

after deleting last n rows and columns0

after deleting last n rows0 0

0

j j

j v j p

l l l j

T

fdn n

fdn n n n

n n n n

fd fd

A A AA

B A

A AB

P I

Z PA B

=

=

=

=

(iv) Obtain the matrices 31B and 32B as follows:[ ]

-1

31 32 1 2 f 1 2 t 5 i 2 3B | B = R EC + R EB Y - (R EB + R )Z Z Z

(v) Obtain the rest of the matrices 33 34 35, andB B B as follows:33 7

34 3

35 4

B = R

B = R

B = R

The above algebraic procedure to obtain the reduced representation can be formalized

and implemented in a way such that the computational and memory resource

requirements are minimal. It must also be noted that the state space matrix obtained is

full rank, because the singularity is eliminated in the process of eliminating the

hydraulic states. For the example system, the number of states in the actual

representation was 55 which were subsequently reduced to 8 after the above algebraic

manipulations.

6.4 Output relationships

Governing Equation

1 , 2 , 3 ,i i i

out a in i a in i wiQ s m s T s T = + +& & , 1,...c c ii n n n= + . (7)

Here:

outQ&

: =deviation in heat transfer rate from the heat exchanger

,a in im& : =deviation in air mass flow rate through heat exchangers

,a in iT : =deviation in heat exchanger inlet air temperature

,w iT : =deviation in wall temperature for the heat exchanger


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(i))1( ( ) 1

{ }i c i c

i

n n n nS diag s = . Similarly define 2S

(ii) 3( ) ( ) 30 ( )i c i i c ci

n n n n n nS diag s =

State-Input-Output Relationship

{ } [ ]

&&

&

i c v i c p i c c

v

inout 3 w,i (n -n )n (n -n )n (n -n )n 1 2

a,i

a,in,i

AI

QQ = S {T } + 0 0 0 S S

m

T

.. (8)

7. ValidationThe reduced order state space representation about the chosen operating point was

obtained for example system5 (Fig. 3). This model was then run through three test

cases, as described below and the results were compared with those obtained by

THERMOSYS simulations. In each of these test cases, the system is operating at the

chosen steady state at 5000 seconds when some transience is introduced and the

response obtained for the next 5000 seconds is obtained using both the reduced ordermodel and THERMOSYS. In the figures used to compare these responses, the cooling

rates achieved by the various liquid air heat exchangers (LAHXs) have been plotted

against this time window of 10000 seconds.

7.1 Test Case 1 (Disturbance propagation)

Here, the inlet air temperature in the first heat exchanger varies from the nominal

value of 35 deg C to 40 deg C. The transition begins at 5000 seconds and takes 3000

seconds to complete and is linear, as shown in Fig. 4. The responses for all the first

floor and second floor LAHXs have been presented in Fig. 5 and 6 respectively.

5Time Scale Factoring was used in the reduced order systemmodel as in THERMOSYS [18]. However, it can be

shown that this is not required, because the fast modes have already been eliminated in this representation.


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Fig. 4: LAHX 1 inlet air temperature transience used for Test Case 1

Fig. 5: Response of first floor LAHXs Test Case 1


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Fig. 6: Response of second floor LAHXs Test Case 1

7.2 Test Case 2 (Valve adjustments)

In this case, the valves feeding the first floor heat exchangers were manipulated about

the design opening factors as 5000 seconds. The output responses have been shown in

Fig. 7 and 8.


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7.3 Test Case 3 (Perturbations in thermal inputs)

In this case, the inlet air mass flow rate to LAHX 6 was increased by 10%, and

simultaneously the chiller cooling rates were increased by 5% each. These

perturbations were all introduced at 5000 seconds. The output responses have beenshown in Fig. 9 and 10.



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7.4 Observations

Following remarks can be made from the validation results:

(i) The reduced order model is a fairly accurate model of the system at the steadystate. The steady state deviation from THERMOSY S results for all the test

cases was within 15%.

(ii) The transient characteristics of the responses also match well with theTHERMOSYS responses and the deviation in their time constants is within

10%. This suggests

(iii) Performance of the overall system model is strongly determined by accuracyof the component models. Therefore, to obtain best results, the individual

components must be modeled precisely. This is sometimes not possible, for

example a linear model may be highly inaccurate in certain regimes of some

components.

8. Control design ideasA linear state space representation of the system was developed which was validated

for the example system using simulation experiments. The first step for control design


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shall be to identify the control inputs and disturbance inputs from the entire set of

inputs in that representation. The perturbations in inlet air temperatures to the heat

exchangers are certainly measurable disturbances dictated by environmental

conditions and cannot be controlled. However, the rest of the inputs pump speeds,

valve opening factors, chiller/boiler heat transfer rates and heat exchanger air mass

flow rates classify as viable control input candidates, because they can beindependently manipulated. Though any number of control inputs (such that the

overall system is controllable) can be chosen for the system, it is preferable to choose

the same number of inputs as states, because the techniques for coarsening the system

that have been discussed later are based on the assumption that the transfer function

matrix from the inputs to states is a square matrix. Methods such as Single Input

Effectiveness [19] can be explored together with controllability tests to obtain the

best choice of control inputs satisfying these requirements.

The next strategy would be to coarsen the system into small clusters of states and

inputs based for block decentralized control, which is the control structure of choice

as described in section 2. Two most popular techniques for the block structure

selection, which have been reported in literature, are the Block Relative Gain (BRG)

[20] and the Structured Singular Value Interaction Measure (SSVIM) [21]. The

drawback of both BRG and SSVIM and also of other such techniques is that they are

intensively combinatorial and therefore difficult to implement for high order systems,

such as the hydronic system in the example (8 states, 20 control input candidates). To

avoid this problem, we are trying to extend data clustering methodologies to

dynamical systems. The area of spectral clustering [22] appears very promising as the

algorithms therein are matrix based which falls in line with the matrix representation

of the system. A suitable affinity matrix needs to be designed for spectral clustering

for which some affinity measures such as the steady state gain matrix or theparticipation matrix[23] can be useful.

After coarsening the system, the next objective is to design MIMO controllers for

each cluster. We propose a hierarchical scheme, where a supervisory controller

decides the optimal setpoints for the system outputs, based on zone models. Then,

each of the MIMO controllers can be designed to achieve these setpoints and also

satisfy other performance requirements such as disturbance and uncertainty

compensation. The control performance shall be tested on THERMOSYS models of

the system, in the absence of experimental facilities.


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9. References[1] Energy Information Administration, Annual Energy Outlook 2009. Retrieved February15,

2008, fromhttp://www.eia.doe.gov/oiaf/aeo/pdf/appa.pdf[2] Energy Information Administration, 2003 Commercial Buildings Energy Consumption

Survey. Retrieved February15, 2008, from

http://www.eia.doe.gov/emeu/cbecs/cbecs2003/detailed_tables_2003/2003set8/2003pdf/b

40.pdf

[3] Dwellings according to type of heat installations. Retrieved Feb15, 2008, fromhttp://dbdh.dk/images/uploads/statisticsbilleder/2-stat-2004.jpg

[4] BACnet Today.ASHRAE Journal,November 2008, pp. B4 B44.[5] L. Filho, .E Queiroz, A. Costa, A matrix approach for steady-state simulation of heat

exchanger networks, Applied Thermal Engineering 27, 2007, pp. 2385-2393

[6] G. Sandou, S. Font, S. Tebbani, et al, Predictive Control of a Complex District HeatingNetwork, Proceedings of the 44th IEEE Conference on Decision and Control, and the

European Control Conference 2005, December 2005, pp. 7372-7377.

[7] D. Paul, W. Fredrik, Embedded agents for district heating management, Proceedings of theThird International J oint Conference on Autonomous Agents and Multiagent systems,

2004, pp. 1148 1155

[8] M. Shukla, G. Radman, Selection of key buses for voltage scheduling using Graph Theory,Proceedings of the thirty seventh annual North American power symposium,2005, pp.

353 357.

[9] P. Oman, A. Krings, D. Leon, et al, Analyzing the security and survivability of real-timecontrol systems, Proceedings of the 2004 IEEE workshop on information assurance,

2004, pp. 342 349[10]A. Jain, R. Balasubramanian, S. Tripathy, et al, Power network observability: A fast

solution technique using graph theory, Proceedings of the 2004 International Conference

on Power Systems Technology,2004, pp. 1839 1844.

[11] M. Scherrer, J McPhee, Dynamic modeling of Electromechanical Multibody Systems,Multibody SystemDynamics 9,2003, pp. 87 115.

[12] K. Cannon, D. Schrage, S. Sarathy, et al, A Vector Graph Object based modelingtechnique for complex physical systems, IEEE Proceedings IEEE Southeast Con, 2002,

pp. 294 299

[13] M. Zaheer-uddin, V. Patel, A. Al-Assadi, The design of decentralized robust controllersfor multi-zone space heating systems, IEEE Transactions on Control Systems Technology

1, 1993, pp. 246-261.

[14] Al-Assadi, V. Patel, M. Zaheer-uddin, et al, Robust decentralized control of HVACsystems using

H performance measures,J ournal of the Franklin Institute341, 2004, pp.

543-567.

[15] W. Franco, M. Sen, K. Yang, et al, Comparison of thermal-hydraulic network controlstrategies, Proceedings of Inst. Of Mechanical Engineers, Part 1: J ournal of Systems and

Control Engineering, v 217, n1, 2003, pp. 35-47.


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[16] A. Gonzalez, D. Odloak, J. Marchetti, et al, Infinite Horizon MPC of a Heat-ExchangerNetwork, Chemical Engineering Research and Design, v 84, n11, 2006, pp. 1041-1050.

[17] Hydronic Heating and Cooling System Design, 2008 ASHRAE Handbook, HVAC Systemsand Equipments. Pp. 12.1 12.25.

[18] T. McKinley, A. Alleyne, Real time modeling of liquid cooling networks in vehiclethermal management systems,SAE Paper 2008-01-0386, 2008, pp. 1 18.

[19] Y. Cao, D. Rossiter, An input pre-screening technique for control structure selection,Computers and Chemical Engineering,v 21, n6, 1997,pp. 563 569.

[20] V. Manousiouthakis, R. Savage and Y Arkun, Synthesis of Decentralized Process ControlStructures Using the Concept of Block Relative Gain, AIChE J ournal, v 32, n6, 1986, pp.

991-1003.

[21] P. Grosdidier and M. Morari, Interaction measures for systems under decentralizedcontrol, Automatica, v 22, n3, 1986, pp. 309 320.

[22] Y. Weiss, Segmentation using eigenvectors,Proceedings IEEE International Conferenceon Computer Vision, 1999, pp 975-982.

[23] M. Salgado and A. Conley, MIMO Interaction Measure and Controller StructureSelection, Int. J . Control, v 77, n4, 2004, pp 367-383.


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APPENDIX

A.1 Operating condition parameters for example system

Inputs

Valve Opening Factors Pump Speeds

Chiller Cooling Rates

Heat Exchanger Inlet Air Properties

LAHX Number Air mass flow rate (kg/s) Inlet air temp (deg C)

1 2.85 35

2 1.99 35

3 3.75 35

4 3.15 35

5 2.27 35

6 4.05 35

Valve Number Opening Factor (%)

1 40

2 15

7 16

8 11

9 22

10 27

11 16

12 96

Pump Number Speed (rad/s)

PP1 800

PP2 700

BP1 950

BP2 900

Chiller Number Cooling Rate (kW)

1 107.9

2 87.7


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States

Mass Flow rates Pressures

Chiller Temperatures

LAHX Temperatures

Pipe Number Mass flow rate (kg/s)1 3.799

2 3.09

3 6.889

4 3.33

5 3.56

6 3.56

7 1.116

8 0.8724

9 1.341

10 1.19411 0.9552

12 1.41

13 3.33

14 3.56

15 6.889

16 3.799

17 3.09

Junction Number Pressure (kPa)1 107.3

2 104.7

3 98.37

4 434.1

5 373

6 26.57

7 27.04

8 20.72

9 22.46

10 201.912 170.8

Chiller Number Liquid temp (deg C) Wall temp (deg C)

1 7.002 -7.856

2 7.011 -4.318

LAHX Number Liquid temp (deg C) Wall temp (deg C)

1 15 15.66

2 15 15.57

3 15 15.73

4 15 15.68

5 15 15.6

6 15 15.75


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Outputs

LAHX Cooling Loads

LAHX Number Cooling load (kW)1 31.00

2 24.22

3 37.22

4 33.14

5 26.51

6 39.14

modeling_hydronic_networks.20120520.4fb93a96c3d966.77448123

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