transfer functions of solar heating systems for dynamic analysis

8/9/2019 Transfer Functions of Solar Heating Systems for Dynamic Analysis

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Transfer functions of solar heating systems for dynamic analysisand control design

Rich ard Kicsiny*

Department of Mathematics, Institute for Mathematics and Informatics, Szent Istv an University, P ater K. u. 1., 2100 G od oll}o, Hungary

a r t i c l e i n f o

Article history:Received 6 August 2014Accepted 1 December 2014Available online

Keywords:Solar heating systemsTransfer functionsDynamic analysisControl designPI control

a b s t r a c t

Mathematical modelling is the theoretically established tool for developing solar heating systems, e.g.with using transfer functions. If we know the transfer functions of the system, the outlet temperature canbe predicted as a function of the input variables (solar irradiance, inlet temperature, environmenttemperatures), dynamic analysis can be carried out, and stable system control can be effectively designedbased on the well-tried methods of control engineering. For these purposes, new, validated transferfunctions for solar heating systems are worked out in this study based on a mathematical model, whichcan be found in the literature and has been applied successfully in the eld. The transfer functions areused for dynamic analysis and control design of solar heating systems. The dynamic analysis is presentedand the ef ciency of the proposed stable control is demonstrated with respect to a real solar heatingsystem.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

Mathematical modelling is the theoretically established tool fordeveloping solarheating systems, e.g. with using transfer functions.

Ordinary differential equation (ODE) models are widely used asthey are relatively simple and easy to handle. Among the currentcollector models, the nonlinear ODE model proposed by Perersand Bales [1] may be the most widely used one, which is theimproved version of the quasi-dynamic model from the standardEN 12975 [2].

If there is an external heat exchanger in the system, it can bemodelled with the well-known effectiveness-NTU method [3], orthe separate sides of the heat exchanger can be assumed to havehomogeneous temperatures and can be modelled with ODEs [4] .

The pipes of the system may be modelled with ODEs (assuminghomogeneous pipe temperatures), or partial differential equations(PDEs) (with the one-dimensional linear heat transfer PDE) [5,6] .

Solar storages can be also modelled with ODEs. See Ref. [7] forODEs of mixed storages and strati ed storages.

Georgiev [8] connected a distributed (PDE) collector model anda mixed (ODE) storage model to describe a collector-storage

system. After connecting the models of the working components,solar heating systems are generally modelled with ODEs [9e 11] .

In Refs. [4,12] , collector-heat exchanger-storage systems aremodelled with a linear (multidimensional) ODE, which will be usedin the present paper. This model is validated [4,12,13] or partlyvalidated [14] and accurate enough for different successful appli-cations [10,12,13,15] as well as its improved version in Ref. [13] ,where the pipes of the system are also modelled with ODEs. Theadvantage of the basic linear model of Ref. [4] is that it is simplerand easier to use than its extended linear version [13] , its nonlinearversion [16] or the delay differential equation model of Ref. [15] .The extendedlinear model of Ref. [13] and the model of Ref. [15] areroughly the same precise and they are more precise than the basicmodel of Ref. [4] . On the other hand, the extended linear model [13]is simpler and easier to use than its nonlinear version [13] or themodel of [15] . Furthermore, the mentioned nonlinear models areapproximately the same precise as their linear versions (see Ref.[13] , cf [12,16] or see Ref. [17] ). Thus the models of [13,15] are themost advantageous in view of accuracy while the linear model of [4] is the most advantageous in view of simplicity. In addition, thelatter model is the basis for all other models in Refs. [13,15,16] .

From the model of Buz as et al. [9] , collector transfer functionshave been determined and applied for the dynamic analysis of realcollectors [18,19] . The present work extends these results bydetermining transfer functions for whole solar heating systems andapplying them for the dynamic analysis of a real system.

* Tel.: 36 28522000/1414; fax: 36 28410804.E-mail address: [email protected] .

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2014 Elsevier Ltd. All rights reserved.

Renewable Energy 77 (2015) 64 e 78
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Generally speaking, transfer function based modelling is rela-tively new and rare in the analysis of solar heating systems, espe-cially, in domestic case. Besides the latter two references, someexamples are the following: Amer et al. [20] solved a collectormodel with time and one space dimension for the uid tempera-ture using Laplace transformation. Huang and Wang [21] wrote anonlinear two-node collector model into Laplace transformed formto gain transfer functions. Bettayeb et al. [22] used a two-nodemodel to determine collector transfer functions for the uid tem-perature and the absorbed solar energy.

The most prevalent and simple control strategy is the on/off control for solar heating systems in domestic hot water (DHW)

production working with constant ow rate, see e.g. Refs. [7,23] .Several controls using pump ow modulation are used in solar

heating systems: Winn [24] compared on/off, I (integral) and PID(proportional integral differential) controls, Hirsch [25] comparedon/off, P (proportional) and hybrid controls. L of [26] discussed on/off, P, I, PID, adaptive and certain optimal controls. Optimal controlsoften maximize the overall energy gain by ow rate modulation.See Refs. [27 e 29] for the case of no heat exchanger and [30] for thecase of a counter ow heat exchanger.

P and PI (proportional integral) controls for collectors are givenanda PI control is workedout in details fora real collector eldinRef.[19] based on proposed collector transfer functions. Based on studiesin the literature,notmany improvements on controlfor solar heatingsystems used in domestic applications have been established in the

recent few decades. In particular, transfer function based control israther rare. Pasamontes et al. [31] serve with a further example onthe control of the collector eld of a solar cooling system based onthe transfer function for a mathematical model with time delay.

Transfer function based control is more prevalent for industrialprocesses e.g. for solar power or desalination plants [32 e 35] .

In the present study, new, validated transfer functions for solarheating systems used primarily for domestic purposes are pro-posed and used for dynamic analysis and control design. Accordingto a there appointed future research, the present study extends theresults of [19] , where transfer functions, dynamic analysis andcontrols have been proposed for solar collectors. The below workedout transfer functions are unique concerning the linear ODE modelfor solar heating systems in Ref. [4] . If the method for working out

transfer functions for this basic model is presented, corresponding

transfer functions can be derived rather straightforwardly for themore precise extended linear model of [13] based on the samemethod. That is why the linear model of [4] is essential and appliedin this study to work out transfer functions. (Because of limits involume, the transfer functions for the extended model of [13]cannot be derived and detailed here.) More precisely, the trun-cated version of the model of [4] (without modelling the solarstorage) is used in the present study, the validation of which is alsopresented below based on measured data. This means that the hereproposed transfer functions are also validated, since they form analternative representation of the same mathematical model.

The advantages are considerable: Knowing the transfer func-

tions, dynamic analysis of solar heating systems can be carried outand feedback control can be effectively designed based on the well-tried methods of control engineering. A control determined in sucha way is generally much simpler than nonlinear and optimal con-trols and can follow the reference signal much more precisely thanthe most frequently used on/off controls. Perhaps, the simpleapplicability is the main advantage of the linear approach con-cerning transfer functions.

The contributions of the present work in details are thefollowing:

1. Based on a validated and successfully applied ODE model forsolar heating systems [4] , new transfer functions are mathe-matically derived (at the end of Section 3.1) and validated. The

applicability of the transfer functions are interpreted with thedynamic analysis of a real system.2. As a further important application of the transfer functions,

closed-loop (PI) control design is given with stability criteria forsolar heating systems using the methods of control engineering.The ef ciency of the proposed control design is demonstratedbased on simulation results.

The present paper proposes all the concepts of work [19](transfer functions, dynamic analysis, control design) for wholesolar heating systems and not for collectors alone, so the presentcontributions are even more important and more general than theresults of [19] .

The paper is organized as follows: Section 2 describes the model

for solar heating systems, for which the transfer functions are

Nomenclature

Ac collector surface area (m 2) Ah surface area of the heat transfer inside the heat

exchanger (m 2) Ahe surface area of the heat exchanger to the environment

(m 2)c c speci c heat capacity of the collector uid (J/(kg K))c i speci c heat capacity of the uid in the inlet loop (J/

(kg K))c h speci c heat capacity of the heat exchanger material (J/

(kg K))I c (global) solar irradiance on the plane of the collector

(W/m 2)khe heat loss coef cient of the heat exchanger to the

environment (W/(m 2 K))mh mass of the empty heat exchanger (kg)t time (s)

T c collector temperature ( C)T ce collector environment temperature ( C)T hc cold side temperature in the heat exchanger ( C)T he environment temperature of the heat exchanger ( C)T hh hot side temperature in the heat exchanger ( C)T i inlet ( uid) temperature of the system ( C)U L overall heat loss coef cient of the collector (W/(m 2 K))vc (pump) ow rate in the collector loop (m 3/s)vi (pump) ow rate in the inlet loop (m 3/s)V c collector volume (m 3)V h volume of the heat exchanger (m 3) kh heat transfer coef cient inside the heat exchanger (W/

(m 2 K))h 0 collector optical ef ciency (-)F heat exchanger effectiveness (-)r c mass density of the collector uid (kg/m 3)r i mass density of the uid in the inlet loop (kg/m 3)

R. Kicsiny / Renewable Energy 77 (2015) 64 e 78 65


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determined in Section 3 and applied for the dynamic analysis of areal solar heating system. The transfer functions are used fordesigning a stable feedback control in Section 4, where the pro-posed control is applied and evaluated for the same real system.Section 5 contains nal conclusions and future research proposals.

See Ref. [36] for the used concepts of control engineering (Lap-lace transformation, transfer function, step response, P, PI controls,static error, stability, etc.). Maple [37] , with its symbolic solver, isused below for mathematical derivations (e.g. by means of Laplaceand inverse Laplace transformations) mainly to gain the transferfunctions. Matlab [38] is used for validation and to implement andsimulate the PI control for a real solar heating system.

2. Mathematical model and validation

This section presents the mathematical model for solar heatingsystems from Ref. [4] without the equation corresponding to thesolar storage. The proposed transfer functions will be determinedbased on this truncated model, the validation of which is alsocarried out.

2.1. Mathematical model

Consider the solar heating system in Fig. 1.The mathematical model of the system in Fig. 1 is the following

(see also [4] ):

dT c t dt

Ac h 0r c c c V c

I c t U L Ac r c c c V c

T cet T c t v c V c

T hh t T c t ;

(1a)

dT hh t dt

r c c c v c

c h mh2 r c c c

V h2

T c t T hh t kh Ah

c h mh2 r c c c

V h2

T hc t

T hh t Ahe khe =2c h mh2 r c c c

V h2

T he t T hh t ;

(1b)

dT hc t dt

r ic iv i

c h mh2 r ic i

V h2

T it T hc t kh Ah

c h mh2 r ic i

V h2

T hh t

T hc t Ahe khe =2c h mh

2 r ic iV h2

T he t T hc t :

(1c)

Remark 2.1

1. Here, vc is always maximal (constant) to keep the collectortemperature at an all-time minimal level. This maximizes theef ciency of the collector and the solar heating system in case of any xed value of vi. (Otherwise, vc could be variable in themodels.) vi will be variable in Section 4 , since it will be used as

manipulated variable in the proposed control.2. Basically, the effect of wind speed is not included in the model(1a e c), nevertheless, one may consider it within the coef cientU L (see e.g. Ref. [39] ).

2.2. Validation

Model (1a e c) is applied for a real solar heating system used forDHW production. The system is installed at the Szent Istv an Uni-versity (SZIU) G od oll }o, Hungary [40] . The system (SZIU system) hasthe following parameter values [12] : Ac 33.3 m 2, h 0 0.74,r c 1034 kg/m 3, c c 3623 J/(kg K), V c 0.027 m 3, U L 7 W/(m 2 K),c h 464.76 J/(kg K), m h 37 kg, V h 0.005 m 3, kh 2461.5 W/(m 2 K) Ah 2 m 2, khe 5 W/(m 2 K), Ahe 0.24 m 2, r i 1000 kg/m 3,

c i 4200 J/(kg K), vc 0 or 16.3 l/min ( 0.000272 m 3/s), vi 0 or10.5 l/min ( 0.000175 m 3/s).

The calculations have been done in Matlab (and Matlab Simu-link). Fig. 2 shows the Simulink diagram of model (1a e c).

For validation, the measured values of T i, I c , T ce, T he , vc and vi arefed into the computer model of (1a e c) as inputs as well asthe initialvalues of T c , T hh and T hc then the modelled and measured values of the outlet temperature T hc arecompared. ( T i and T hc aremeasuredonthe inlet and outlet pipe just before and after the storage side of theheat exchanger, respectively.) Fig. 3 shows the comparison of themodelled and measured temperatures for two days: 22ndSeptember 2012 and 2nd November 2012. The simultaneous oper-ating states of the pumps (on/off) are also shown in the gure.

The average of the absolute difference between the modelled

and measured outlet temperatures is 2.6

C on 22nd September2012 and 1.8 C on 2nd November 2012. In proportion to the dif-ference between the maximal and minimal measured temperaturevalues, the time average of the absolute difference (or absoluteerror) is 7.2% on 22nd September 2012 and 6.5% on 2nd November2012 (c.f. [12] ).

It can be concluded that the model tracks the physical processescharacteristically right with acceptable accuracy in view of severalengineering aims (studying and developing solar heating systems).See e.g. Refs. [41] , where the reasonability of such precision isreinforced for similar systems. Thus we can accept and apply themathematical model (1a e c).

Remark 2.2

In fact, the outlet temperature T hc is dif cult to model because of the small volume of the heat exchanger, which causes rather highand fast changes in the modelled temperature mainly when thepump ow rates are changing frequently (see Fig. 3).

3. Transfer functions

In this section, the transfer functions are derived and applied fordynamic analysis. Here, vi is assumed to be constant.

3.1. Transfer function derivation

First, Eqs. (1a e c) should be rewritten from time domain to

Laplace domain with Laplace transformation:Fig. 1. Scheme of the solar heating system.

R. Kicsiny / Renewable Energy 77 (2015) 64 e 7866


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sT c s T c 0 Ac h 0r c c c V c

I c s U L Ac r c c c V c

T ces T c s

v c V c

T hh s T c s ;(2a)

sT hh s T hh 0 r c c c v c

c hm

h2 r c c c V

h2

T c s T hh s

kh Ah

c h mh2 r c c c

V h2

T hc s T hh s

Ahe khe =2c h mh

2 r c c c V h2

T he s T hh s ; (2b)

sT hc s T hc 0 r ic iv i

c h mh2 r ic i

V h2

T is T hc s

kh Ah

c h mh2 r ic i

V h2

T hh s T hc s

Ahe khe =2c h mh

2 r ic i

V h

2

T he s T hc s ; (2c)

where the variables in Laplace domain are marked with overbars, sis the independent complex variable in Laplace domain, T c (0),T hh (0), T hc (0) are the initial values of the state variables T c , T hh , T hc . Itis a great advantage that the system of linear ODEs (1a e c) has beensimpli ed to the system of linear algebraic equations (2a e c).Rearranging equations (2a e c), we get:

T c s W c 0 sT c 0 W c 1 sI c s W c 2 sT hh s

W c 3 sT ces; (3a)

T hh s W hh 0 sT hh 0 W hh 1 sT c s W hh 2 sT hc s

W hh 3 sT he s; (3b)

T hc s W hc 0 sT hc 0 W hc 1 sT is W hc 2 sT hh s

W hc 3 sT he s; (3c)

Fig. 2. Computer model of the solar heating system.

Fig. 3. Validation of the used mathematical model.



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where

where t c , t hh , t hc are the so-called time constants of the collectorand the hot and cold sides of the heat exchanger, respectively:

t c 1

U L Ac r c c c V c

v c V c

; t hh c hmh r c c c V h

2r c c c v c kh Ah Ahe khe;

t hc c h mh r ic iV h

2r ic iv i kh Ah Ahe khe:

Solving the algebraic equations (3a e c) for T hc s, we get:

T hc s W i1 sT c 0 W i2 sT hh 0 W i3 sT hc 0 W 1 sT is

W 2 sI c s W 3 sT ces W 4 sT he s;(4)

where

where, for simplicity, the independent complex variable s was not

indicated everywhere.In systems engineering approach, the solar heating system hasan output variable T hc and input variables according to Fig. 4.

The transfer functions are the Laplace transformed form of theoutput T hc s divided with the Laplace transformed form of theinputs T is, I c s, T ce s, T he s. When the transfer function corre-sponding to a given input is determined, the other inputs and T c (0),

T hh (0), T hc (0) are assumed to be zero. Based on Eq. (4) , the transferfunctions for each input are the following:

T hc sT is

W 1 s; T hc s

I c s W 2 s;

T hc sT ces

W 3 s; T hc sT he s

W 4 s:

The response of the output temperature to the initial conditionscan be similarly characterized with the following functions:

T hc sT c 0

W i1 s; T hc sT hh 0

W i2 s; T hc s

T hc 0 W i3 s:

According to the superposition principle of linear systems, theresultant effect of the inputs and the initial temperatures on theoutput is the simple sum of the single effects of each input andinitialtemperatures. Eq. (4) expresses this principle in Laplace domain.

Remark 3.1

The above proposed transfer functions are validated, since theyform an alternative representation of the mathematical model(1a e c) validated in Section 2.2 .

3.2. Dynamic analysis

The transfer functions can be used for the dynamic analysis of solar heating systems. The dynamic features of a systemcan be wellcharacterized with its unit step responses. The unit step responsecorresponding to a given input is the response (output) of thesystem to the input in time domain, supposing that the input is theunit step input, while the other inputs and the initial values of the

state variables are zero. The unit step input is the following:Fig. 4. Block diagram of the solar heating system.

W c 0 s t c

t c s 1; W c 1 s

t c

t c s 1$ Ac h 0

r c c c V c ; W c 2 s

t c

t c s 1$v c V c

; W c 3 s t c

t c s 1$ U L Ac

r c c c V c ;

W hh 0 s t hh

t hh s 1; W hh 1 s

t hh

t hh s 1

$ r c c c v c

c h mh2 r c c c V h2

; W hh 2 s t hh

t hh s 1

$ kh Ah

c hmh2 r c c c V h2

;

W hh 3 s t hh

t hh s 1$ Ahe khe =2c h mh

2 r c c c

V h2

; W hc 0 s t hc

t hc s 1; W hc 1 s

t hc t hc s 1

$ r ic iv ic h mh

2 r ic i

V h2

;

W hc 2 s t hc

t hc s 1$ kh Ahc h mh

2 r ic i

V h2

; W hc 3 s t hc

t hc s 1$ Ahe khe =2c hmh

2 r ic i

V h2

;

W i1 s W hc 2 W hh 1 W c 0

1 W hh 1 W c 2 W hc 2 W hh 2; W i2 s

W hc 2 W hh 01 W hh 1 W c 2 W hc 2 W hh 2

; W i3 s W hc 0 1 W hh 1 W c 2 1 W hh 1 W c 2 W hc 2 W hh 2

;

W 1 s W hc 1 1 W hh 1 W c 2 1 W hh 1 W c 2 W hc 2 W hh 2

; W 2 s W hc 2 W hh 1 W c 1

1 W hh 1 W c 2 W hc 2 W hh 2; W 3 s

W hc 2 W hh 1 W c 31 W hh1 W c 2 W hc 2 W hh 2

;

W 4 s W hh 3 W hc 2 W hc 3 W hc 3 W hh 1 W c 2

1 W hh 1 W c 2 W hc 2 W hh 2;



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Input t 0; t < 0 ;

1; t 0 ; ; (5)

the Laplace transformed form of which is the following:

Input s 1s: (6)

The unit step response as output can be given with the followingformula in Laplace domain using the transfer function for the cor-responding input ( W (s)):

Output s W s1s : (7)

The unit step response in time domain can be determined fromOutput s:

Output t L 1 W s1s

; (8)

where L 1 denotes the inverse Laplace transformation.

As a part of the dynamic analysis, the effect of the initial valuesof the state variables on the system can be also investigated in timedomain based on the inverse Laplace transformed form of theproduct of the given initial condition and the correspondingtransfer function ( W i(s)) (here, the inputs and the other initialvalues are zero again):

Output t L 1 W is$Initial condition

Initial condition $L 1 W is : (9)

Apply the described dynamic analysis for the solar heatingsystem. The unit step responses of the system corresponding to theinputs T i, I c , T ce, T he are the following:

T hc t L 1 W 1 s1s; T hc t L 1 W 2 s1s

;

T hc t L 1 W 3 s

1s

and T hc t L 1 W 4 s

1s

:

The responses corresponding to the initial conditions T c (0),T hh (0) and T hc (0) are the following:

T hc t T c 0L 1 W i1 s ; T hc t T hh 0L

1 W i2 s

and T hc t T hc 0L 1 W i3 s :

3.2.1. Application for the SZIU systemThe above dynamic analysis is applied for the SZIU system. The

tap water is led into the storage side of the heat exchanger withtemperature T i. The output uid is the required DHW with tem-perature T hc (see Fig. 1). Here, the pumps are considered switchedon. The response functions to each input are provided below.Some of them are also depicted (but not all because of limits involume).

The unit step response corresponding to T i is the following (seealso Fig. 5):

T hc t 0 :8 0 :031 e0 :58 t 0 :34 e 0:054 t 0 :43 e 0:0055 t :

(10)

The unit step response corresponding to I c is the following (see

also Fig. 6):

T hc t 0 :021 0 :00002 e0:58 t 0 :0026 e 0 :054 t 0 :024 e 0:0055 t :

(11)

The unit step response corresponding to T ce is the following:

T hc t 0 :2 0 :00019 e 0:58 t 0 :025 e 0 :054 t 0 :223 e 0 :0055 t :(12)

The unit step response corresponding to T he is the following:

T hc t 0 :0013 0 :0000026 e0:58 t 0 :00054 e 0:054 t

0:00075 e 0 :0055 t : (13)

The response corresponding to T c (0) is the following(T c (0) 1 C) (see also Fig. 7):

T hc t 0 :05 e0 :58 t 0 :577 e 0:054 t 0 :528 e 0:0055 t : (14)

Fig. 5. Output response of the SZIU system to the unit step of T i.

Fig. 6. Output response of the SZIU system to the unit step of I c .

Fig. 7. Output response of the SZIU system to T c (0)

1

C.



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The response corresponding to T hh (0) is the following(T hh (0) 1 C):

T hc t 0 :48 e0:58 t 0 :419 e 0 :054 t 0 :065 e 0 :0055 t : (15)

The response corresponding to T hc (0) is the following(T hc (0) 1 C) (see also Fig. 8):

T hc t 0:466 e0:58 t 0 :473 e 0 :054 t 0 :061 e 0 :0055 t : (16)

If the different inputs and initial conditions effect at the sametime, the resultant output can be determined with a simple sum of functions (10) e (16) according to the superposition principle:

T hc t 1:021 0 :00002 e0 :58 t 0 :0026 e 0 :054 t 0 :024 e 0:0055 t :

(17)

For visibility, the responses in Figs. 5e 8 are presented for

different time periods.

Remark 3.2

1. According to functions (10) e (16) , the largest effect of the in-puts on the outlet temperature T hc is evoked by the unitchange of T i, and the lowest effect is evoked by the unitchange of T he (since khe 5 W/(m 2 K) is very low because of the insulation of the heat exchanger). Among the initial con-ditions, T hc (0) has the largest and T hh (0) has the lowest effect.It seems that each input has rather low effect. For example, if I c increases by 1 W/m 2, T hc increases only by approximately0.02 C (see Fig. 6). It is reasonable considering the rather high

ow rates vc 16.3 l/min, vi 10.5 l/min. Nevertheless, the

output is proportional to the inputs according to the linearityprinciple, so higher inputs involve (proportionally) higheroutput: e.g. if I c increases by 100 W/m 2, T hc increases byapproximately 2 C.

2. The above dynamic analysis is essentially theoretical, since itrequires a perfect unit step as oneof the inputs while theothersare constant, furthermore, the system must be in a perfectequilibrium at the beginning. Of course, such conditions do notoccur in practice. Nevertheless, this dynamic analysis is astandard and widely used way for characterizing dynamicsystems.

4. System control

Stable control can be designed for solar heating systems usingthe transfer functions and the methods of control engineering. Theaim is to change the outlet temperature as controlled variable ac-cording to a given reference function in time by proper modulationof the inlet pump ow rate vi as manipulated variable. Here, vc isconstant, vi is variable (see Note 1 in Remark 2.1). In this approach,functions T i, I c , T ce, T he are disturbances. This control is summarizedin Fig. 9.

Now, vi(t ) is not constant, so not all coef cients are constant insystem (1a e c) even (1a e c) is not linear in the variables T hc (t ), T i(t )and vi(t ), because of the products vi(t )T hc (t ) and vi(t )T i(t ) in (1c).

Therefore, the classical linear methods of control engineeringcannot be directly applied. First of all, the model (1a e c) should belinearized in a convenient operating point.

4.1. Model linearization

Practically, such an equilibrium of model (1a e c) is chosen asoperating point, which represents a sort of average operatingcondition, that is to say, in case of which functions T c (t ), T hh (t ), T hc (t ),T i(t ), I c (t ), T ce(t ), T he (t ) are constants, where each constant is theapproximate average between the upper and lower limits of its realpossible values. Let T 0c , T 0hh , T

0hc , T

0i , I

0c , T 0ce and T 0he denote the cor-

responding constants at such an operating point. The right hand

side of (1ae

c) are zero at the operating point, since it is anequilibrium:

0 Ac h 0r c c c V c

I 0c U L Ac r c c c V c

T 0ce T 0c

v c

V c T 0hh T

0c ; (18a)

0 r c c c

_V c c h mh

2 r c c c V h2

T 0c T 0hh

kh Ahc h mh

2 r c c c V h2

T 0hc T 0hh

Ahe khe =2c h mh

2 r c c c V h2

T 0he T 0hh ;

(18b)

0 r ic iv 0i

c h mh2 r ic i

V h2

T 0i T 0hc

kh Ahc h mh

2 r ic iV h2

T 0hh T 0hc

Ahe khe =2c h mh2 r ic i

V h2

T 0he T 0hc ; (18c)

The inlet ow rate at the operating point from Eq. (18c) is

v 0i kh Ah T 0hc T

0hh Ahe khe T

0hc T

0he . 2

T 0i T 0hc r ic i

: (19)

Eq. (1c) has the following form:

dT hc t dt

f T hh t ; T hc t ; T it ; T he t ; v it ; (20)

based on which the linearized version of ( 1c) at the operating point

is the following:

Fig. 8. Output response of the SZIU system to T hc (0) 1 C.

Fig. 9. Block diagram of the solar heating system in view of control.



8/15

Eq. (1a,b) are linear with respect to all time-dependent func-tions, so the coef cients in these equations remain the same in the

linearized model below ( 22ae

c).Let ~T c t T c t T 0c , ~T hh t T hh t T 0hh ,

~T hc t T hc t T 0hc ,~T it T it T 0i ,

~I c t I c t I 0c , ~T cet T cet T 0ce ,

~T he t T he t T 0he , ~v it v it v0i , with which the linearized

model is

d ~T c t d t

Ac h 0r c c c V c

~I c t U L Ac r c c c V c

~T ce t ~T c t v c V c

~T hh t ~T c t ;

(22a)d ~T hh t

dt r c c c v c

c h mh2 r c c c

V h2

~T c t ~T hh t

kh Ah

c h mh2 r c c c

V h2

~T hc t ~T hh t

A

hek

he=2

c h mh2 r c c c

V h2

~T he t ~T hh t ;

(22b)

d ~T hc t dt

r ic iv 0i

c h mh2 r ic i

V h2

~T it ~T hc t

kh Ah

c h mh2 r ic i

V h2

~T hh t ~T hc t

Ahe khe =2c h mh

2 r ic iV h2

~T he t ~T hc t

r ic i

c h mh2 r ic i

V h2

T 0i T 0hc

~v it :

(22c)

Rewrite ( 22a e c) (with ~T c 0 ~T hh 0 ~T hc 0 0 C) into

Laplace domain:

s~T c s Ac h 0r c c c V c

~I c s U L Ac r c c c V c

~T ces ~T c s v c V c

~T hh s~T c s ;

(23a)s~T hh s

r c c c v c c h mh

2 r c c c V h2

~T c s ~T hh s

kh Ah

c h mh2 r c c c

V h2

~T hc s ~T hh s

Ahe khe =2c h mh

2 r c c c V h2

~T he s ~T hh s ;

(23b)

s~T hc s r ic iv 0i

c h mh2 r ic i

V h2

~T is ~T hc s

kh Ah

c h mh2 r ic i

V h2

~T hh s ~T hc s

Ahe khe =2c h mh

2 r ic iV h2

~

T he s ~

T hc s

r ic i

c h mh2 r ic i

V h2

T 0i T 0hc

~v t ;

(23c)

from which the resultant effect of the inputs is the sumof the single effects according to the linear superpositionprinciple:

~T hc s ~W 1 s

~T is ~W 2 s

~I c s ~W 3 s~T ces ~W 4 s

~T he s

~W 5 s~v is;(24)where

dT hc t d t

f T 0hh ; T 0hc ; T

0i ; T

0he ; v

0i

v f v T hh

T 0hh ; T 0hc ; T

0i ; T

0he ; v

0i

$ T hh t T 0hh v f v T hc

T 0hh ; T 0hc ; T

0i ; T

0he ; v

0i

$

T hc t T 0hc

v f v T i

T 0hh ; T 0hc ; T

0i ; T

0he ; v

0i

$ T it T 0i v f v T he

T 0hh ; T 0hc ; T

0i ; T

0he ; v

0i

$ T he t T 0he v f v v i

$

T 0hh ; T 0hc ; T

0i ; T

0he ; v

0i

$ v it v 0i 0 kh Ah

c h mh2

r ic iV h2

T hh t T 0hh

r ic iv0i kh Ah khe Ahe. 2c h mh

2 r ic i

V h2

$

T hc t T 0hc r ic iv 0i

c h mh2 r i

c iV h2

T it T 0i khe Ahe =2

c h mh2 r i

c iV h2

T he t T 0he r ic i T 0i T

0hc

c h mh2 r i

c iV h2

v it v 0i

(21)

~W 1 s ~W hc 1 s 1

~W hh 1 s ~W c 2 s

1 ~W hh 1 s ~W c 2 s

~W hc 2 s ~W hh2 s

; ~W 2 s ~W hc 2 s

~W hh 1 s ~W c 1 s

1 ~W hh 1 s ~W c 2 s

~W hc 2 ~W hh 2 s

;

~W 3 s ~W hc 2 s

~W hh 1 s ~W c 3 s

1 ~W hh 1 s ~W c 2 s

~W hc 2 s ~W hh2 s

; ~W 4 s ~W hh 3 s

~W hc 2 s ~W hc 3 s

~W hc 3 s ~W hh 1 s

~W c 2 s1 ~W hh 1 s

~W c 2 s ~W hc 2 s

~W hh 2 s;

~W 5 s ~W hc 4 s 1

~W hh 1 s ~W c 2 s

1 ~W hh 1 s ~W c 2 s

~W hc 2 s ~W hh2 s

;



9/15

where

according to the notation in Section 3.1, and

~W hc 1 s ~t hc

~t hc s 1$ r ic iv

0i

c h mh2 r ic i

V h2

;

W hc 2 s ~t hc

~t hc s 1$ kh Ah

c h mh2 r ic i

V h2

;

W hc 3 s ~t hc

~t hc s 1$ Ahe khe =2

c h mh2 r ic i

V h2

;

W hc 4 s ~t hc

~t hc s 1$

r ic i T 0i T 0hc

c h mh

2 r ic i

V h

2

;

where

~t hc c hm h r ic iV h

2 r ic iv 0i kh Ah Ahe khe: (25)

4.2. Control design

The aim is to realize a stable closed-loop control of the solarheating system ( 1a e c) according to Figs. 9 and 10 such that theoutlet temperature T hc (t ) follows a predetermined reference inputT hc ,r (t ) in time precisely enough. This requirement corresponds tothat ~T hc t follows

~T hc ;r t , where ~T hc ;r t T hc ;r t T 0hc .

In effect, the problem is the determination of ~W c such that thecontrol is stable with conveniently small static errors correspond-ing to the inputs ~T hc ;r ,

~T i, ~I c , ~T ce , ~T he . The transfer functions of the

control in Fig. 10 with respect to the reference input and the dis-turbances ~T i,

~I c , ~T ce , ~T he are the following:

~W ~T hc ;~T hc ;r s ~W c s ~W 5 s

1 ~W c s ~W 5 s; (26)

~W ~T hc ;~T i s ~W 1 s

1 ~W c s ~W 5 s; (27)

~

W ~T hc ;~I c s

~W 2 s1 ~W c s ~W 5 s; (28)

~W ~T hc ;~T ce s ~W 3 s

1 ~W c s ~W 5 s; (29)

~W ~T hc ;~T he s ~W 4 s

1 ~W c s ~W 5 s: (30)

The so-called loop gain of the controlled system (the gain ingoing around the feedback loop) is ~W c s ~W 5 s. Let us write itbelow (in (34) and (35) ) in the general transfer function formc c si

~W 0 s:

~

W c s ~

W 5 s c c si

~

W 0 s; (31)

where i and c c are constant and ~W 0 0 1.Consider the possibility of P and PI controls:

P : ~W c s AP ; (32)

PI : ~W c s AP 1 1sT I

AP sT I

1 sT I ; (33)

where AP and T I are constant. Based on Section 4.1, it can be derivedthat the product ~W c s ~W 5 sis in accordance with the general formof Eq. (31) in case of each control type:

P : ~W c s ~W 5 s c c ;P

s0 ~W 0 s; (34)

PI : ~W c s ~W 5 s c c ;PI

s1~W 0 s: (35)

Let us take reference inputs of the following sort (when thedisturbances are zero):

~T hc ;r t c r t j; (36)

where j and c r are constant. If j 0, ~T hc ;r t is step function, if j 1, itis ramp function (only t 0 is of importance in our case).

If i > j holds for i and j in (31) and (36) , the static error of thecontrol corresponding to ~T hc ;r is zero. The control is required to

follow the step inputs ( j 0) precisely, that is with zero static error,so the PI control is chosen, since i 1 > j 0 (see (35) ) is ful lled inthis case. Thus c c c c ,PI , where c c ,PI can be found in the Appendix asEq. (A1) (which can be derived based on Section 4.1).

The static error of the PI control in case of ramp function asreference input ( j 1) is the following:

er ;s limt / ~T hc ;r t

~T hc t c r c c

: (37)

Consider the disturbance ~T it in the following form (when~T hc ;r t and the other disturbances are zero):

~T it c 1 t k; (38)

~W c 1 s W c 1 s; ~W c 2 s W c 2 s;

~W c 3 s W c 3 s; ~W hh 1 s W hh 1 s;

~W hh 2 s W hh 2 s; ~W hh 3 s W hh 3 s;

Fig. 10. Closed-loop control of the solar heating system.



10/15

where k and c 1 are constant. If k 0, ~T it is step function, if k 1 itis ramp function.

The transfer function ~W 1 s corresponding to ~T i should beconsidered in the following form:

~W 1 s c ~T isl

~W ~T i0 s; (39)

where c ~T i is constant and ~W ~T

i0 s 1.If i > k l is ful lled for i, k and l in (31), (38) and (39) , the static

error of the control corresponding to ~T i is zero.It can be derived similarly that ~W 1 s is in accordance with (39) ,

where l 0 and c ~T i is according to (A2) and (A3) in the Appendix .Thus the static error of the control is zero if ~T it is step function

(k 0). If ~T it is ramp function ( k 1), the static error corre-sponding to ~T it is the following:

e1 ;s limt / ~T hc ;r t

~T hc t c ~T ic c

c 1 : (40)

Consider ~I c t , ~T ce t and ~T he t similarly as in (38) :

~I c t c 2 t m ; ~T cet c 3 t n ;

~T he t c 4 t q ;

where m, n , q, c 2, c 3 and c 4 are constant.It can be shown like above that the static error corresponding to

~I c (e2,s), ~T ce (e3,s) or ~T he (e4,s) are zero if m 0, n 0, q 0,respectively. If ~I c t , ~T cet or ~T he t is ramp function ( m 1, n 1 orq 1, respectively), the corresponding static errors are thefollowing, respectively:

e2 ;s c ~I c c c

c 2 ; (41)e3 ;s

c ~T cec c

c 3 ; (42)

e4 ;s c ~T hec c

c 4 ; (43)

where c ~I c , c ~T ce and c ~T he are according to (A3)e (A6) in the Appendix .

The control parameters AP and T I should be such that the ab-solute values of the above static errors are not higher than a pre-

xed positive limit E , based on the constants c r , c ~T i , c ~I c , c ~T ce , c ~T he , c 1,c 2, c 3, c 4 and that c c c c ,PI .

Furthermore, AP and T I should be such that the control is stable.Consider the transfer function with respect to ~T hc ;r that is

~W ~T hc ;~T hc ;r (see (26) ). The controlled system is stable corresponding to ~T hc ;r if the real parts of the zeros of the denominator of ~W ~T hc ;~T hc ;r arenegative. It can be derived that the denominator 1 ~W c s ~W 5 shas the following form:

dr 4 s4 dr 3 s3 dr 2 s2 dr 1 s1 dr 0 : (44)

The constants dr 4, dr 3, dr 2, dr 1, dr 0 can be found in the Appendix((A7) e (A11)) .

The following system of conditions is suf cient for the stabilityaccording to the Routh-Hurwitz criterion:

dr 4 > 0 ; dr 3 > 0 ; dr 2 > 0 ; dr 1 > 0 ; dr 0 > 0;

dr 3 dr 1dr 4 dr 2

> 0;dr 3 dr 1 0dr 4 dr 2 dr 00 dr 3 dr 1

> 0 ;

dr 3 dr 1 0 0dr 4 dr 2 dr 0 00 dr 3 dr 1 00 dr 4 dr 2 dr 0

> 0:

(SC)

The denominators in (27) e (30) are also 1 ~W c s ~W 5 s, so (SC)is suf cient for the stability of the controlled system corresponding

to ~

T i,~

I c , ~

T ce and ~

T he as well.

In sum, the mathematical task of designing a PI control is thefollowing: determine the free control parameters AP and T I suchthat er ;s E , e1;s E , e2;s E , e3;s E , e4;s E hold, andconditions (SC) hold.

Remark 4.1

The Routh-Hurwitz criterion assures the stability of not only thelinearized but the corresponding nonlinear controlled system(where vi(t ) is not constant), since the latter is also stable if the realparts of the zeros of the denominator of ~W ~T hc ;~T hc ;r are negative ac-cording to Lyapunov.

4.2.1. Application for the SZIU systemLet us give an appropriate PI control for the SZIU system.

Consider an end-of-spring period in May. Let T 0hc 55 C, which is

high enough for general domestic purposes. Let T 0hh 56 C (a bit

higher than T 0hc ), T 0i 15

C (approximate average tap water tem-perature), T 0ce 20 C (approximate average daytime environmenttemperature), T 0he 25 C (approximate average temperature in themaintenance house). From these data, the remaining equilibriumvalues T 0c , I 0c and v 0i can be determined according to ( 18a e c):T 0c 60.85 C, I 0c 586.9 W/m

2 (which is in accordance with theapproximate average daytime solar irradiance of a clear day in Mayin Hungary [42] ), v 0i 0.000032 m

3/s ( 1.92 l/min), the maximalvalue of vi is 10.5 l/min (see Section 2.2 ). As further limitation, it isassumed that vi(t ) can vary between zero and the maximal value in3 s that is the absolute value of the speed of ow rate changing (theabsolute value of the acceleration of the ow) is 0.000058 m 3/s 2.

Let the absolute values of the static errors (37), (40) e (43) be lessor equal than 0.1 C, which is convenient for a DHW heatingapplication. Let the controlled system be stable that is conditions(SC) are satis ed.

Suppose that such fast changing disturbances (with high am-plitudes) act on the controlled system simultaneously, which are

rare even separately in real circumstances, thus they are even moreimprobable in the same time. Let us check in such a case whetherthe controlled system is still able to follow precisely enough arelatively fast changing reference input (with high amplitude). If the controlled system can follow well such an extreme referenceinput under such extreme disturbances (which have small proba-bilities), it will work even more satisfactorily under real circum-stances. Thus the practical applicability of the control will bethoroughly supported.

Sinusoidal inputs are given as the mentioned reference inputand disturbances (see Figs. 11 and 12 ) with initial values corre-sponding to the above operating point of the SZIU system:

T hc ;r 0 ~T hc ;r 0 T 0hc 55

C, amplitude 5 C, initial

gradient 0.5

C/min;

Fig. 11. Reference input to the controlled solar heating system.



11/15


12/15

Fig. 15 shows the control error T hc ,r (t ) T hc (t ).The results show that the absolute value of the control error

decreases permanently below 5% of the initial error (below1.75 C) within 25.0 min, the absolute value of the error decreasespermanently below the prescribed limit 0.1 C within 26.4 min(settling time). Based on these values, the worked out PI control isconsidered satisfactorily fast and accurate in view of the controlpurpose (even in case of extreme reference input anddisturbances).

Remark 4.2

1. On an average clear day in May, at least 440 W/m 2 solar irra-diance is still expected for 5 e 5.5 h after the above settling time26.4 min [41] . 436.9 W/m 2 is the minimal irradiance in thesimulation (see Fig. 12), which proved to be still enough tomaintain the outlet temperature at the desired level (55 C)according to simulation experiences. Thus the controlled outlet

temperature T hc (t ) should be able to follow the reference input(precisely enough) for at least 5 e 5.5 h.

2. Based on simulation data, 591.5 L DHW, at an average tem-perature of 55 C, has been produced in 5.5 h after the abovesettling time 26.4 min (on an average clear day in May),which covers 30% of the daily demand (1990 l [6] ) of theconsumer of the SZIU system (a kindergarten). The producedDHW can be stored in a water storage, which can be dis-charged during the day according to the all-time hot waterdemand.

3. The gained simulation results, where the control error con-verges to zero, underlie the statement of Remark 4.1 corre-sponding to the stability of the nonlinear controlled system(where vi(t ) is not constant), since the nonlinear system modelwas controlled above (see Fig. 13 ).

4. The control error converges to zero with oscillation (see Fig. 15),which is normal in case of PI controllers, since the manipulatedvariable vi(t ) is partially proportional to the time integral of theerror (not only directly to the error). This causes recurringoverruns/oscillation. In fact, vi(t ) cannot be proportional to (theintegral of) the error every time because of its lower (0 l/min)and upper (10.5 l/min) limits (see Fig.14 ). That is why the graphof the control error is not smooth but has breaking points (seeFig. 15 ).

5. Conclusion

Generally speaking, transfer function based modelling is rela-tively new and rare in the analysis of solar heating systems, espe-cially, in domestic case. In addition, not many improvements oncontrol for solar heating systems used in domestic applicationshave been established in the recent few decades. In particular,transfer function based control design is rather rare despite of thesimple applicability, which is the main advantage of the linearapproach concerning transfer functions. Controls based on transferfunctions are generally much simpler than nonlinear and optimalcontrols and can follow the reference signal much more preciselythan the most frequently used on/off controls. This paper intended

to contribute to ful l these research gaps by proposing new,

Fig. 14. T hc,r (t ), T hc (t ) (controlled variable) and vi(t ) (manipulated variable).

Fig. 15. Control error T hc ,r (t )

T hc (t ).



13/15

validated transfer functions based on a validated system model anddesigning stable controls (in particular, a closed-loop PI control) bymeans of the transfer functions.

The transfer functions have been also applied for the dynamicanalysis of a real solar heating system, for which the proposedstable PI control has been applied to make the outlet temperaturefollow a reference input.

According to a there appointed future research, this study hasextended the results of [19] , where transfer functions, dynamicanalysis and controls have been proposed for solar collectors. Sincethe present paper proposes all of these concepts for whole solarheating systems, its contributions are even more important andmore general than the results of [19] .

As the main ndings of the present paper, it can be stated thatthe proposedtransfer functions are successful and relatively easy toapply (see the applications in the paper) for dynamic analysis andstable control design. Furthermore, the proposed PI control isappropriate for the corresponding control purpose because of itsprecision and rapidity even in case of fast changing reference inputand disturbances with high amplitudes.

In essence, the introduced dynamic analysis can be easilyadapted for any solar heating system with an external heatexchanger. The proposed control design can be also comprehen-sively applied for solar heating applications, where the outputtemperature should follow some reference signal in time (e.g. solar

power and desalination plants).The manipulated variable was the ow rate in the inlet loop. The

inlet temperature may be also used as manipulated variable,although, this solution is very rare. On the contrary, variable owrate pumps are comprehensively applied in the practice of solarheating, and more generally, in building systems.

Further research work may deal with the derivation of transferfunctions corresponding to the more precise extended linear modelof [13] . Such a workcan followin essence the steps of the derivationfor the essential basic model of [4] presented above. Furthermore,so-called describing functions may be determined correspondingto nonlinear mathematical models of solar heating systems.Describing functions can be gained by means of harmonic lineari-zation, which is a linearization method (differing from the oneapplied in this paper) corresponding to convenient operating

points, and can be also used for dynamic analysis and controldesign.

Acknowledgement

The author thanks the Editor for the encouraging help in thesubmission process, the anonymous Referees for their valuable

comments and Prof. Istv an Farkas and the Department of Physicsand Process Control (SZIU) for the measured data. The authoralso thanks his colleagues in the Department of Mathematics inthe Faculty of Mechanical Engineering (SZIU) for theircontribution.

Appendix

The equations below have been derived based on the relationsin the main text (with the symbolic solver of Maple) and arereferred to in the text, where needed. For example, c c ,PI has beenderived based on the expressions of ~W 5 sand ~W c sin Sections 4.1and 4.2 .

c ~T i 2 r ic iv0i Ahe khe 2 kh Ah 2U L Ac c c r c v c

Ahe khe 2 kh Ah U L Ac =den ; (A2)

where

c ~I c 4 v c kh Ah h 0 Ac r c c c =den ; (A4)

c ~T ce 4 v c kh Ah U L Ac r c c c =den ; (A5)

c ~T he Ahe khe 4 kh Ah 2U L Ac c c r c v c

Ahe khe 4 kh Ah U L Ac Ahe khe =den :(A6)

dr 4 T I V c c c c h mh r ic iV h r c c h mh r c c c V h; (A7)

c c ;PI 2 AP r ic i T 0i T 0hc Ahe khe c c r c v c 2 kh Ah U L Ac 2 kh Ah r c c c v c 2c c r c v c U L Ac Ahe khe U L Ac .

T I 4 r c c c v c U L Ac kh Ah 2 r c c c v c U L Ac khe Ahe 4 r c c c v c U L Ac c i r iv0i 4 kh Ah r c c c v c khe Ahe 2 r ic i$

U L Ac khe Ahe v0i A

2he k

2he U L Ac 2khe Ahe r c c c v c r ic iv

0i A

2he k

2he r c c c v c 4 kh Ah r c c c v c r ic iv

0i 4 kh$

U L Ac khe Ahe 4 kh Ah U L Ac r ic iv0i :

(A1)

den 2 Ahe khe v0i 4U L Ac v

0i 4 kh Ah v

0i c c r c v c 2 Ahe khe v

0i 4 kh Ah v

0i U L Ac c ir i 4 kh Ah

2 Ahe khe Ac U L A2he k2he 4 kh Ah Ahe khe c c r c v c A

2he k

2he 4 kh Ah Ahe khe Ac U L:

(A3)



14/15

dr 0 2 AP r ic i T 0i T 0hc Ahe khe c c r c v c 2 kh Ah U L Ac

2 kh Ah r c c c v c 2c c r c v c U L Ac Ahe khe U L Ac :

(A11)

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dr 3 2c h mh r c c c V c 2 r 2c c 2c V h V c T

0i 2T

0hc c h mh r c c c V c 2T

0hc r

2c c

2c V hV c c i r i AP 2V c v

0i v c V h $

r c c c U L Ac V h c h mh 2 v c 2 v0i V hV c V

2h v c c

2c r

2c Ahe khe 2 kh AhV h V c V

2h U L Ac r c c c c i$

r i r c c c v c U L Ac c 2h m2h v c V h 2 v c V c c

2c r

2c 2 Ahe khe 4 kh Ah V c U L Ac V h r c c c c h mh

Ahe khe 2 kh Ah V hV c c 2c r 2c T I ;

(A8)

dr 2 2U L Ac 2 r c c c v c c h mh 2 v c V h 4 v c V c c 2c r

2c 2 Ahe khe 4 kh Ah V c 2U L Ac V h r c c c $

T 0i 2 r c c c v c T 0hc 2U L Ac T

0hc mh c h 4T

0hc v c V c 2T

0hc v c V h c

2c r

2c 4 kh Ah 2 Ahe khe T

0hc V c

2T 0hc V h U L Ac r c c c r ic i AP 2 r c c c v c v0i 2U L Ac v

0i c h mh 2V h v c v

0i 4 v c V c v

0i c

2c r

2c 4 kh Ah v

0i

2 Ahe khe v0i V c 2 v c 2 v

0i Ac U L 2 kh Ah v c Ahe khe v c V h c c r c Ahe khe 2 kh Ah Ac U LV h c i r i

4 kh Ah v c 2 Ahe khe v c 2 v c U L Ac c c r c 4 kh Ah 2 Ahe khe Ac U L mh c h 4 kh Ah v c 2 Ahe khe v c V c

2 kh Ah v c 2 Ahe khe v c V h c 2c r2c A

2he k

2he 4 kh Ah Ahe khe V c Ahe khe 2 kh Ah Ac U LV h c c r c T I

2c h mh r c c c V c 2c 2c r

2c V h V c T

0i 2T

0hc c h mh r c c c V c 2T

0hc c

2c r

2c V h V c c i r i AP ;

(A9)

dr 1 2 Ahe khe 4 kh Ah 4U L Ac v c c c r c 2 Ahe khe 4 kh Ah U L Ac T 0i 4U L Ac 2 Ahe khe

4 kh Ah T 0hc v c c c r c 4 kh Ah 2 Ahe khe Ac U LT

0hc c ir i AP 4 Ac U Lv

0i 4 kh Ah v

0i 2 Ahe khe v

0i

$

v c c c r c 2 Ahe khe v0i 4 kh Ah v

0i Ac U L c i r i 2 Ahe khe 4 kh Ah Ac U L A

2he k

2he 4 kh Ah Ahe khe v c $

c c r c A2he k2he 4 kh Ah Ahe khe Ac U L T I 2 r c c c v c 2 Ac U L mh c h 4V c 2V h v c c 2c r 2c 2 Ahe $

khe 4 kh Ah V c 2U L Ac V h r c c c T 0i 2T 0hc U L Ac 2T

0hc r c c c v c mh c h 2T

0hc V h 4T

0hc V c v c c

2c r

2c $

4 kh Ah 2 Ahe khe T 0hc V c 2T

0hc V hU L Ac r c c c c i r i AP ;

(A10)

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