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Decentralized preview control for multiple disturbance rejection in HVAC systems
Control Eng. Practice,Vol. 2, No. 6, pp. 989-1000, 1994
Pergamon 0967-0661(94)00051-4
Copyright © 1994 Elsevier Science Lid Printed in Great Britain. All rights reserved 0967-0661/94 $7.00 + 0.00
DECENTRALIZED PREVIEW CONTROL FOR MULTIPLE DISTURBANCE REJECTION IN HVAC SYSTEMS 1 M. Zaheer.uddin*, S.A.K. AI.Assadi** and R.V. Patel** *Centre for Building Studies, Concordia University, Montreal, Canada H3G IM8 **Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada H3G 1M8
Abstract: A decentralized preview controller is designed for temperature control of multizone indoor environmental spaces. A two-zone space heating system is considered. The physical system consists of a boiler, heat pumps, distribution network and two environmental zones. By assuming that the outdoor temperature variations are '~previewable",a decentralized preview controller is designed by using a parameter optimization method. The output responses of the resulting decentralized closed-loop bilinear system acted upon by single and multiple disturbances with and without preview action are compared. Also, results showing the robustness property of the controller, and the 24-hour building operation with unoccupied and occupied setpoint tracking using preview control are given. Key words: Multi-Zone Building; HVAC Systems; Temperature Control; Decentralized control ; Preview Control.
1. INTRODUCTION
In a previous study by Zaheer-uddin et al. (1993a), it was shown that because of their modular structure and performance requirements, HVAC systems are good candidates for decentralized control. Since they are also economical as such, they are extensively used in industry e.g. (Haines, 1983). The main findings of the investigation reported in (Zaheeruddin et al., 1993b) have been that the decentralized system response is not significantly different from that of the centralized system, and that the zone temperatures can be regulated to within ~0.2°C of the setpoints using decentralized control. The principal goal of the research reported here is to study whether or not preview control can be used as a means of improving the regulation properties of the decentralized controllers. To this end, the proposal is framed in the context of the specific system (referred to as a multizone space heating (MZSH) system) shown in Fig. 1. The MZSH system (Fig. 1) consists of a boiler
Indoor environmental spaces or zones in large buildings are subjected to multiple disturbances during day-to-day operation. Therefore, good regulation of zone temperatures in the presence of multiple disturbances is a problem of continued interest in the control of indoor environments of buildings. Typical disturbances acting on the zones can be classified into two groups: (i) diurnal variations such as those that occur in outdoor air temperatures, solar radiation fluxes and wind velocities and (ii) internal heat generating sources, viz. lights, occupants and equipment. The diurnal disturbances (referred to here as "external" disturbances) are previewable and can be assumed to be known ahead of time with some degree of accuracy. On the other hand, the intemal heat sources (referred to as "internal" disturbances) are easy to predict since they are related to a building's operating schedule. Coupled with the fact that heating, ventilating and air conditioning (HVAC) systems (which are used to provide conditioned air to zones) have large time constants, and therefore, the effects of disturbances on the plant output are delayed, it seems appropriate to explore the application of preview control concepts e.g. see (Sheridan, 1966; Tomizuka and Rosenthal, 1979; Gunewardana et al., 1979) to improve temperature regulation in buildings.
v2
1 This research w ~ supported by the Natural Sciences and Engineering Rese~ch Council (NSERC) of Canada under grants OGP0036380 and OGP0001345.
~
Yl -Gucpm(j-k2.3) SW- Supl~,w ~ r
E -g.~,,,,~. D- D~am',
HC-HamI~ F.F~
ILW-itm~ ,,mnr
Cl- Cmm~n
(1-I.2....~)
Fig. 1 Schematic diagram of the MZSH system. 989
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M. Zaheer-uddin et al.
which is used to supply water at moderately warm temperatures (between 16-32°C) to the evaporative heat exchangers of the heat pumps. Each environmental zone is installed with its own heat pump. The heat pump which works on the compression refrigeration cycle, receives heat energy from the source water and elevates this energy to a higher temperature and delivers it to the condenser coil of the heat pump. A circulating fan and ductwork arrangement is used to extract the heat from the condenser coil and deliver it to the zone through the diffuser shown in the figure. Thus the zone air is heated to offset the heating load acting on the zone due to cold ambient air temperatures such as those that occur, say, during a cold winter's day. Although only two zones are shown in Fig. l, it is obvious that the same arrangement holds for other zones in buildings with larger numbers of zones. The MZSH system shown in Fig. 1 can be characterized as having three stations (viz. station 1 : boiler, station 2 : zone-l, and station 3 : zone-2). The outputs of each station must be made to track some desired set-points which are known a priori. To achieve this regulation task, use can be made of (i) the station 1 input ul(t) (=v3) to control the output yl(_t) via~'ontroller c3; (ii) the station 2 input u2(t) ( = Iv I v411 ) to control the output Y2(t) via controllers
component of the MZSH system, and by using the energy conservation principle, a bilinear model of the system was developed in a previous study (Zaheer-uddin et al., 1993a). The model equations of the system are c b T b = v 3v 3max ( 1 -- otT b/Tbmax ) - m b CPw ( T b - T l , ) m b cp w (T b -TI2) - a b (T b - T e )
(1)
Cl~T l~ = -vnV4max(Pl-1) + mbcPw(Tb-Tt) - % ( T z - T e)
(2)
ChiT h~ = V4VnmaxP 1 - VlVlmax;(Th, - Tz) - ahl(Thi-Te)
(3)
Cz~ Tz~ = V l V l m a x ~ l ( T h - T z )
(4)
- az(T,-T
p) - a z , 2 ( T z - L 2 )
cI2T 12 = --VSVsmax(P2--1) + mbCPw(Tb-Tl2) - a l 2 ( T l 2 - T e) (5) Ch2T h 2 = V : s m a ~ P 2 - v : 2 ~ , ~ x ; ( T h - T ~ ) - a h ( T h - T Cz2T z2 = v 2 v ~
p)
(6)
~2(Th 2-Tz2) - az2(T z 2-Tp ) + a z,2(Tz -Tz2 ) (7)
where P1-- I + ( P l m a x - 1 ) ( / - ( T h , - T l ) / A T l m a x )
(8)
P2 = 1 + (P2max-l)(1 - (Th2 - Tt2)/AT2max )
(9)
Note that equations (1)-(9) constitute a seventh-order bilinear model of the MZSH system.
clr ~and f4, and (iii) the station 3 input U3(t ) ( = Iv2vsI ) to control the output Y3(t) via the con-
3. THE DECENTRALIZED CONTROL PROBLEM
trollers c 2 and c 5. Thus the control problem is decentralized and there are five local controllers (c l, c2. . . . c5) to achieve the temperature regulation.
The control problem considered is as follows. For the MZSH system, which consists of three interconnected stations (boiler, zones 1 and 2), find a robust decentralized controller so that the output temperatures of zones 1 and 2 and the boiler are asymptotically regulated to desired set-points independent of (fl anv constant unknown disturbances in the outdoor air temperatures acting on the system, and (ii) any non-destabilizing perturbations in the parameters of the MZSH system itself. To formulate the decentralized control problem for the system given by (1)-(9), the dynamics of the MZSH system are first written
From the viewpoint of preview control, the most important external disturbance, namely the outdoor temperature Tp acting on the zones, is considered as the disturbance that is previewable. This external disturbance causes variable heating loads on the zones. It is proposed here to design decentralized preview controllers for the system shown in Fig. 1. The bilinear model equations (Zaheer-uddin et al., 1993a) are presented in Section 2. Then in Section 3, two decentralized controller design problems are defined: the non-linear constrained servomechanism problem (NCSP) (Section 3.1) and the non-linear constrained preview servomechanism problem (NCPSP) (Section 3.2). The designed decentralized controllers, without and with preview action, are implemented on the bilinear system and the resulting closed-loop system responses acted upon by single and multiple disturbances are compared (Section 4). Also included are the results on 24-hour building operation with unoccupied and occupied setpoint tracking using preview control. Conclusions are given in Section 5. 2. ANALYTICAL MODEL By identifying the energy flows to and from each
as: N x(t) = A x(t) + ~_,Bi(x ) ui(t) + E d ( t ) (10a) i=1
Yi(t) = C i x(t),
i = 1,2 ..... N
(10b)
and e i ( t ) = yi(t) - YR(t)
where
l zl ,2 h2 .]T is
(lOc)
s te vec
tor; Ul(t)=v3, u2(t)= Iv 1 va] T and u3(t)= [V2vs]T are the control input vectors; d(t)= [T T ]r is the vector of constant (unmeasurable) disturbances; N=3 is the number of "stations" in the decentralized system; and for station i, ui(t) ~ ]17,m', Yi(t) ~ ~q.P' and yR(t) ~ •P' denote the control inputs, measurable outputs and reference signals (set-points) respectively.
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HVAC Systems For the system (10), use is made of the general servocompensator given by Davison (1976), and Davison and Solomon (1983)
~ i ( t ) = D i ~i(t)+Oi
ei(t )
f~i* = bl°ck diag [~i ,f~i . . . . . f~i ] Oi* = block diag [Oi , O i . . . . . Oi ] with Oi and f~i completely determined by the disturbances and reference signals acting on the system, e.g., for constant disturbances and reference signals, ~i =0, Oi = 1. Thus, the robust decentralized servomechanism problem for the augmented system described by (10) and (11), is to find N local controllers with inputs Yi(t) ~ IRP'and outputs ui(t) E ~.m,, i = 1. . . . N, SOthat (i) the resulting closed-loop system is asymptotically stable, (ii) asymptotic tracking of the reference signals occurs for all constant disturbances and setpoints, and (iii) for all perturbations in the parameters of the system which do not cause instability in the closed-loop system property (ii) still holds. For the application considered in this paper, m i > Pi, i = I, 2, 3, all the outputs to be regulated are measurable and the disturbances to be regulated and the reference signals to be tracked are constant. In addition, it was found (Zaheer-uddin et al., 1993a) that the linearized models (about the normal operating points) of the MZSH system (1)-(9) with the chosen decentralized structure do not have unstable decentralized fixed modes, or transmission zeros at the origin. The decentralized controller has the structure given by [ei(t)]
command/external input vector for the MZSH system.
robust servomechanism assuming that the externot previewable, the solved is the minimiza-
t!
J(K/);S e(t)r Q e(t)+~,ui(t)rRi ui(t) dt i=1
i = 1. . . . . N
(13a)
x ° i ( t ) - l . 5
j : l ..... 7
(13b)
where x°(t)= [TboTt,oTh,oTz,oTtTh~oT~] r
are the
temperatures at the operating point, vk(t), k=l ..... 5, are the closed-loop control inputs for the MT_SH system, and tf is the duration of the Tp profile (outdoor temperature). The constraint (13a) is required to ensure that the input energy of the system does not exceed its maximum capacity. Note that the vt "s are normalized with respect to their maximum capacities so that they vary between 0 and 1. The constraint (13b) ensures that the system states remain bounded and close to the operating point x°(t). The approach that was used to solve the NCSP involves selecting appropriate Q and R i in (12) and finding the corresponding sets of K~ that minimize J while satisfying the constraints (10a-c), (11), and (13a,b). This problem was solved using the method of successive quadratic programming (Schittkowski, 1986). Note that (Davison and Chang 1986) have also used nonlinear programming to solve such a decentralized control problem. By using the system parameters given in (Zaheer-uddin et. at, 1993b), the optimal decentralized feedback matrix Kd was computed
0.3187
Kd=
0.0
0.0
0.6885 0.0
0.0
0.0
0.4020 0.0
0.0
0.6173 0.0
0.0
0.4998 0.0
0.0
0.9149 0.0
0.0
0.0
0.3702 0.0
0.0
0.7105
0.0
0.0
0.4616 0.0
0.0
0.8406
This gain matrix was obtained with the following values of Q and R~
where ! is the identity matrix.
3.1 The Nonlinear Constrained Servomechanism Problem (NCSP)
0
k = 1, 2. . . . . 5
i= 1. . . . . N
where K i = [Ki O) Ki <2)] , and rio(t) is the operating
To solve the decentralized problem stated above while nal . disturbance TP is . . opttmxzatlon problem to be tion of
0 < vk(t) < 1
(11)
in order to reject the external constant disturbances (step changes in the ambient temperature Tp), and track constant reference signals acting on the system. The matrices
u i ( t ) = V i o ( t ) - K i [~i(/)j,
at the chosen operating point x°(t) (given in section 4) and subject to the constraints (10a-c), (11) and
(12)
3.2 The Nonlinear Constrained Preview Servomechanism Problem (NCPSP) Consider the same bilinear decentralized system defined by (10a-c) and augmented with (11). Also, assume that the outdoor temperature Tpj is previewable and that its values are measured at j=0, 1. . . . . Np. This preview profile can be approximated by a series of step functions. For the NCPSP,
992
M. Zaheer-uddin
the decentralized controller structure consists of two parts: I~d for the decentralized tracking and stabilizing control, and K w for the decentralized previewing control, i.e., lil (t)=
uid(t)+uiP~(t)
subject to the constraints
(10a-c),(11)and
0 < vk(t) < 1 , k = l , 2 . . . . . 5, (15a) 0 x j(t)-l.5 < xj(t) < xOj(t)+ l.5, j = l ..... 7, (15b) where, vk(t), k=l . . . . . 5, are the closed-loop preview control inputs for the MZSH system.
where
[ei(t)] uid(t)=Vio(t)_
et al.
[/~i(1) /~i(2)] ~i (t)]' i = 1 . . . . . N
J
uiP'(t)= ~i'~Kw(i)dl(t) j=0, -
1. . . . ,Np
l=1
Using the same optimization technique (Schittkowski, 1986) as before, the optimal decentralized gain matrix I~ d, and the preview controller gains Kw)i) for one previewing step (y=Np=0), were obtained using the same Q and Ri . These are
J
dj(t)(=To(t))=Cwo+EACwW(t)
,
j=0, 1 . . . . /Vp
l=1
ACw=Cw-Cw, ~ , l=1,2 . . . . . w (t)=the unit step function.
Alp I~ d =
Note that d(t) is computed as the sum of the magnitude of the outdoor temperature at the operating point Cwo and the variations in To from the operating point denoted by ACw, at each preview step. For example, since the external disturbance is assumed to be previewable (known a priori) the amplitudes (step functions) Cwo, .... Cw~ and their time of occurrence are known. In other words, the time-varying outdoor air temperature preview profile is approximated by a series of step functions. Note that the approximation of outdoor temperature by a series of step functions is a common practice in HVAC engineering. The reason is that hourly weather data is currently available from meteorological stations throughout North America. Using such past data, hourly outdoor temperatures are predicted. Between two successive data points either ramp or step function approximation can be used see (Zaheer-uddin et. al, 1993b). For simplicity, step functionapproximation was used to construct the outdoor temperature preview profile. With this knowledge, the corresponding gains gw, for j =0, 1. . . . ,Np were computed. Thus, the decentralized preview control input is updated via uiP~(t). Given this preview structure of TO the decentralized gains are defined such that
Kw=[(blockdiag [[Kwj(i). .i=l' ... l(a= [ {block diag [l(i(J),i=l . . . . .
N ] ] , j = 0 , 1 . . . . ,Np]
N]/,J=I,2 ]
Given the above preview profile of T , j =0, 1. . . . . Np, over the time duration tf , the NCPSP for the augmented decentralized system defined by (10) and (11) is to tl
Minimize f(I(d, Kw)= I 0
e(t) r Q e ( t ) + ~ u i ( t )
Riui(t) dt
i=1
(14)
0.31942
0.0
0.0
0.68863
0.0
0.0
0.0
0.40222
0.0
0.0
0.61737
0.0
0.0
0.49986
0.0
0.0
0.91492
0.0
0.0
0.0
0.37041
0.0
0.0
0.71059
0.0
0.0
0.46174
0.0
0.0
0.84063
[0.00349] Kwo(1)=0"00849 ' Kwo(2)=[0.00187
(3)_ [0.00352] - [0.003091
' Kwo
4. SIMULATION RESULTS 4.1
Open-Loop Response
Consider first the open-loop response of the MZSH system to highlight important physical characteristics. The system operating point refers to a mild-day (winter) operating condition when the outdoor temperature TOo is -2 ° C with Teo = 20 ° C. The choice of the operating point of the system depends on the magnitude of the anticipated load on the system. Usually the operating point can be selected such that it corresponds to 50% load capacity. The following operating point, reflects an average load condition on the MZSH system:
Tbo=27.26 ° C , Ttlo=25.58°C , Th~o=29.84° C , Tz~o=21.96° C Tl2o=25.51° C , Th~o=27.94°C , Tz2o=20.66°C , v lo=0.4, V2o=0.45, V3o=0.5, V4o=0.5, v50=0.5 Under this load condition, the zone temperatures Tz, and T are expected to reach 21.96°C and 20.66 °c respectively, and the boiler water temperature to reach 27.26°C. To examine the system response, a step change in T O is introduced such that the new outdoor temperature is -6°C (resembling a somewhat colder day) and a -1 "C step change in Teo. Furthermore, a -1 ° C step change is introduced in the initial temperatures of Tzl, Tz2 and Tb from the operating point: T~,° = 21.96° C ; Tz2° = 20.66° C ; and Tbo = 27.26° C. How the open-loop system responds to this situation is depicted in Fig. 2. The boiler tem-
HVAC Systems
Set-Point
~o
993
Set-Point
Set-Point
t,F 04
a)
I
t.I
:
U
I
u
I
~.1
u
!
IJ
4
43
1
~
i
i.I
4
Time (h) (b) Time (h) (e) Time (h ) Figs 2a-c Output responses of the open-loop system to -1 °C step change in the operating point and disturbances (T = 19 °C and Tp = -6 °C). external dhe~bances
perature rises slowly and reaches steady state (as shown in Fig. 2) in about 3.5 hours. The final value of Tb is 26.25° C which is l°C below the operating point. On the other hand, the zone temperatures ( T~, in Fig. 2b and Tz2 in Fig. 2c) are directly influenced by the disturbance, and they reach new steady-state temperatures which are a few degrees below the operating point. For example, Tz, decreases slowly and reaches a steady-state value of 18.9 °C in three hours, which is about 3°C (steady-state error) below the desired set-point. The T~2response in Fig. 2 shows that the input energy to zone-2 cannot sustain the zone-2 temperature at its operating point. The temperature T~: also decreases slowly and reaches a new steady-state value of 17.52°C, which is about 3.14°C below the desired set-point. Some important observations can be drawn from the open-loop responses : (i) the temperature response curves show that the system steady state time is of the order of 3 hours and (ii) the external disturbances influence the zone temperatures much more strongly (almost three times) than the boiler temperature. This latter observation offers evidence that preview of disturbances could be useful in improving the zone temperature regulation.
1
i- 1 / Fig. 3 Block diagram of the decentralized preview control for MZSH system. w(t)
E
IE
J
g
.........
E
......... ®
E
4.2 Closed-Loop Response The closed-loop implementation scheme for the NCPSP is shown in Fig. 3. The cgntroller structure as shown in Fig. 4 consists of two parts: the decentralized output feedback servocompensator control input ud(t), and the preview control input uP(t). It is obvious that for the NCSP, u p (t)=O. When preview action is added, the implementation consists of prior knowledge of the amplitudes of the external disturbance Tp denoted by Cw, and their corresponding computed gains Kw? for j =0, 1. . . . . Np, where j is the index denoting the preview step as discussed in section 3.2. Note that the control structure, the design method, and the implementation technique used in this study differ from the preview control examined by Tomizuka and Rosenthal (1979). To study the effect of adding preview control on the system responses, the system output was simulated
uP(t)
Fig. 4 The structure of the decentralized preview controller for MZSH system.
with (NCPSP) and without (NCSP) preview action by using the same disturbances and initial conditions as those used in the open-loop test (Fig. 2). Figs 5a-c show the output responses. It is apparent that output responses with one-step preview action (dashed lines, Np =0 ) are better in that Tb, Tz, and Tz: reach steady state faster than for the case without preview (solid lines, Ne =-1). Note that the results in Figs 5a-c are for the case when only one preview step (Np =0) of disturbance is considered. Indeed, a further improvement in the system responses can be expected when the external disturbance is approxi-
994
M. Zaheer-uddin et al.
Y ~d
0a
0,4
lt..I
@.l
o.I
I
14
I,li
LI
(a)
21.1
IO
11
III
II
IJI
Time (h )
(b
no preview (N =-1)
Time (h )
..........
)
C
one step preview (Nv =0)
ill
Time (h )
.....................Set-Point
Figs 5a-c Output responses of the closed-loop systems with (NCPSP) and without (NCSP) preview control acted upon by the same magnitude of step changes and disturbances as in Fig. 2. mated by a step function by increasing the preview steps (as shown in Fig. 6) for different classes of disturbance profiles. Two different outdoor temperature profiles (type-a and type-b) are considered in Fig. 6. In each case, they are approximated by four preview intervals (Np =3). The intent is to study the output responses of the system by successively increasing the preview intervals from 0 to 4. Figs 7a-c show the output responses for the disturbance profile type-a shown in Fig. 6. The solid lines in Figs 7a-c are the responses obtained with no preview (N =-1) and the dashed lines are those obtained by considering one preview step ( ~ =0). It is apparent that the addition of one preview step is able to achieve good output responses. The effect of increasing the number of preview steps is illustrated in Figs 7d-f. It is evident that increasing the preview steps improves the system responses. Further evidence of this improvement ill
I
I. . . . . . .
II ,
o:,
-1l
~ - -
~ l l s f = t
.......
Trpe.bpr*~
_ _
1~.a~m,,
1.
-~0
l
~'.s
~
is
~
=:,
Time (h )
Fig. 6 Typical profiles of outdoor air temperatures with step function approximation.
il¢
it.el
li
I
......................
X
.
,
-\
il.r,,
..... I A
lil ll.ll
lira T/.II
II.ll
la.ll
ll.u
os
I
IJ
1
l.s
(a)
I
lli.~I
3.~
ll.llq
Time (It)
(b )
Ne=-I
Time (h)
Ic$
I
IJ
I
li
(e
l
iS
Time (h)
...................... Set-Point
............ Ne=O ll.ll
t.
.k.k!~ l'.
In ,ill \
_f
;7,
,,'-~
.... ....
•v
))_
/2.~
U z~
0
(a
Time (h )
(e ) ..........
Time (h ) N,=I
...................
ii
I
l.i
|
W
..............
I
IJ
Time (h )
(l ) IS=3
Figs 7a-f The output responses of the closed-loop preview control system acted upon by type-a outdoor temperature profile.
HVAC Systems comes from the fact that the magnitude of the performance index decreases as the preview steps is increased. For the results shown in Figs 7a-f, the minimum value obtained for J without preview is J,an-(N~=-i)=l.1271, and the values obtained with preview are:
995
and N =2). This observation suggests that when dealing with a unidirectionally decreasing disturbance profile (type-a), a higher number of preview steps yields better regulation, whereas in the case of a bidirectional disturbance profile (type-b), a smaller number of preview steps would be sufficient to get better zone temperature regulation. This result is encouraging in the sense that over a typical day, the outdoor temperatures increase and decrease about some average temperature. This means that in terms of designing a preview controller for a typical daily operation, a large number of preview steps may not be required.
fmin(N~=0)=0.3397, Jmin(Np= 1)=0.01881, Jmi,(Ne =2)=0.00588 and Jmi~(N~=3)=0.00275 It is obvious that by increasing the preview steps, the system response can be improved. For the system considered in this study, the effect of adding the first preview step is significant. Further increase in the number of preview steps is beneficial in reducing the error from the setpoint although the improvement between successive addition of preview steps is not very significant.
4.3 Robustness Although the advantages of previewing the disturbances is recognized, the fact remains that inaccurate previewing (or forecast) of external disturbances could have an adverse effect on the output responses. On one hand there is the possibility of mismatch in disturbances, while on the other hand, the parameters of the system also change during continuous operation. To test the worst-case scenario, simulation runs were carried out, in which the controller designed by assuming a disturbance profile (type-b) as shown in Fig. 6 was used to reject a type-a disturbance profile. In other words, a complete mismatch of the disturbance profile is assumed. Furthermore, a 25% increase in the system parameters (~, ;t, ~2) was also considered. The results of this simulation are shown in Figs 9a-c. Note that the Tb response (Fig. 9a) with
Further simulations with another class of typical disturbance profile type-b (Fig. 6) were carried out. The corresponding output responses with one preview step (N~=0) and no preview (Ne =-1) are shown in Figs 8a-c. Note that the variations in Tz, and T~: with one preview step are within z~O.05°C as opposed ~-0.16°C for the no preview case. The effect of increasing the preview steps on the output responses of the system is given in Figs 8d-f. It is worth nothing that the improvement in output responses with two or more preview steps (Np = 1 and Np =2) is insignificant (Figs 8d-f), whereas the output responses depicted in Figs 7d-f show a small but finite improvement (for example, compare Figs 7e and 8e for both cases N~ = 1 ~U
all
II.IBI
~7.1|
11111
21J
ILl
I
IJ
!
U
(a
3
IJ
2U
Time (h )
0..5
l
IJ
3
~
(b)
Np=-I
1
1J
0.5
Time (h )
........ Np=0
I
U
I
7.5
(c
Time (h )
............... Set-Point Xt!
115 I
I
"~i~\7-" '.2:.- ~t" .....
.1o.~
21,S lq.U
(d)
1,8~
J
JJ
Time (h ) %---1
!1.1 IJ
(• ) ........
Time (h ) %=0
............... % = 1
(f
2
Z$
J
3-5
Time (h ) ...........
%=2
Figs 8a-f The output responses of the closed-loop preview control system acted upon by type-b outdoor temperature profile.
M. Zaheer-uddin et al.
996
(; +=.................... ~'1
]'"""
+....•
J
llll
(a)
Time (h )
05
I
i' )
JJ IJ
~
~1
)l
~
"
Time (h )
(b)
Nominal case
. +.. , o , +
--
Time (h )
(c)
Mismatch in TI, and 25% change in (t,~l,~2 .............
........
-/
Np =O
..........
Np=O
~
2o.? '4.
......
~ ....... ,':.....
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period. To this end, a day with an outdoor temperature profile was considered, as shown in Fig. 10, and the open-loop system response was simulated (for lack of space the figures are omitted). Note that the zone temperatures undergo a maximum variation of 5°C when the MZSH system is operated in openloop. Fig. 10 was assumed to be the preview profile, and a preview controller was designed. The controller so designed was used to simulate the closedloop bilinear system. The corresponding output responses obtained are depicted in Figs lla-c. It is apparent that the output responses with ~ =0 (one preview step) are superior to those without preview. $ 8
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Figs 11a- f Typical daily regulation of the temperatures Tb, Tz, and Tz~ with preview controller (single disturbance case: outdoor temperature profile shown in Fig. 10). The zone temperatures are regulated to within _+0.05°C of the setpoints for most of the day. Also
shown in Figs 1 ld-f are the comparisons of the output responses with one (Np =0) and four preview steps (Np=3). It is apparent that there is no significant advantage in increasing the preview steps (since the decrease in the error from the set-point is very small) so long as there is no mismatch in disturbances. In the event there is a mismatch, the robustness results depicted in Figs 9d-f offer evidence that up to two to three preview steps may be all that is required to achieve good zone temperature regulation under realistic changes in disturbance profiles and/or in the system parameters.
of 7°C compared to 5°C with single disturbance. This suggests that the use of preview controllers would be much more beneficial in multipledisturbance rejection cases. Again decentralized preview controllers with one and several preview steps were designed. The designed preview controller performance was compared with the no preview case as shown in Figs 13a-c. The effect of increasing the previewing steps on output regulation is depicted in Figs 13d-f. A comparison of output responses shown in Figs l la-c (single-disturbance case) and Figs 13a-c (multiple-disturbance case) shows that the boiler, -.one 1 and zone 2 temperatures in the no preview case in Figs 13a-c have somewhat higher overshoot.
4.5 Rejection o f Multiple Disturbances Indoor environmental zones in buildings are not only acted upon by diurnal outdoor temperature profiles (Fig. 10), but are also subject to variable solar radiation fluxes entering through the windows and internal heat gains due to variable occupancy and lighting. To demonstrate the application of preview control to tackle this multiple disturbance rejection problem, two disturbances were considered: namely outdoor temperature and solar gains. The assumed preview profiles for these disturbances are depicted in Figs 10 and 12a respectively. The open-loop system responses to these disturbances are shown in Figs 12b-d. Note that zone temperatures under multiple disturbances undergo a maximum variation
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This is due to the fact that the solar gains entering the zones contribute to the heating of air. Therefore the controller should utilize this free heat energy by reducing the auxiliary energy supplied to the zones. Under these circumstances, the use of the servocompensator alone (no preview case) is not sufficient to achieve good zone temperature regulation. As an example, consider Figs 13b and lib. Note that the use of decentralized output feedback with no preview action results in an overshoot of 0.19°C in Tz, with single disturbance, and 0.25°C with two disturbances. The use of one-step preview control reduces
this maximum overshoot to 0.09" c in the case of single disturbance and about 0.1°C for the case of two disturbances. These variations are further reduced by increasing the number of preview steps (Figs 13d-f). In other words, as the number of disturbances acting on the system increases, the impact of using the preview controllers becomes much more significant compared to the corresponding no-preview cases. Therefore, since building environments are subject to multiple disturbances, preview controllers are good candidates for HVAC systems.
HVAC Systems
999
preview control, consider two examples: in the first example (case-A), consider a single-step preview controller (Ne+=0) for the external disturbance (outdoor temperature is assumed to remain constant), and a four-step preview controller (Ne,=3) for tracking changes in setpoints. In the second example (case-B), consider a two-step preview controller (Np =1) for outdoor temperature (which is assumed to vary as shown in Fig. I0), and a four-step preview controller (Np =3) for setpoint tracking. For case-B, a two step preview controller for outdoor temperature rejection was chosen because previous results showed that two to three preview steps are sufficient to achieve good output regulation.
4.6 Setpoint Tracking for Occupied and Unoccupied Periods Most commercial buildings are operated at different setpoints during the day (occupied periods) and night (unoccupied periods) to save energy. Typically, during unoccupied periods (1800 -0800 h), zone temperature setpoints are decreased to 15°C, and during occupied periods (0800 -1800 h), the zone temperature setpoints are set forward to the normal comfort range (T~ =22°C, Tz=20.7°C). Currently, the rooming heat-up from unoccupied to occupied setpoints, and the evening cool-down from occupied to unoccupied setpoints is achieved by using reset control action. Normally this function is performed on a fixed schedule independent of outdoor and zone temperatures. This is not realistic since outdoor and indoor temperatures change during occupied and as well as unoccupied periods.
Figs 14a-c show the closed-loop system responses for case-A. The actual setpoint strategy shown in the figures requires that the zone temperature be set at 15°C during 1800-0800 h, and T = 22°C and T~2= 20.7°C during occupied hours 0800-1800. To track such ideal settings, a preview setpoint strategy shown by dashed lines in Figs 14a-c was used. This preview strategy consists of two-step preview action which is initiated about two hours ahead of time in the morning, and two hours in the evening, starting at 1800 h. The actual output responses of the system (solid lines) track this preview setpoint strategy so that the zone temperatures reach the day-time setpoints just before the occupants arrive in the building. In the evening, the zone temperatures are decreased as soon as the unoccupied period begins at 1800 h, and they reach the unoccupied setpoint temperature at about 2000 h. The results depicted in Figs 14a-c show that building operation with variable setpoints can be achieved with preview control.
As far as possible, the use of fixed-schedule control even during the unoccupied hours should be avoided, the reason being that the system operation with fixed control while the external disturbances are acting on the system could result in energy waste because of loss of control. On the other hand, output feedback control alone for tracking widely separated setpoints is not likely to give good results. To address this issue the use of preview control is proposed. From the system dynamics, the lead time necessary to heat the space from unoccupied setpoint can be approximately established to occupied setpoint, while ensuring that all the control inputs do not exceed the capacity constraints of the system. For the system considered in this study, this lead time is about two hours. With this knowledge, a four-step preview controller was designed (based on the minimum number of preview steps required to avoid input saturation) to track step changes in reference setpoints. The overall controller now consists of three parts, viz.,
The sets of results shown in Figs 15a-c (open-loop responses) and Figs 15d-f (closed-loop responses) correspond to case-B which is the most general case in which the outdoor temperature was assumed to be time-varying (as in Fig. 10) and the zone temperature setpoints were also varied according to building occupancy (occupied and unoccupied modes). The responses depicted in Figs 15e-f show that the zone temperatures can be regulated to within _+0.2°C throughout the day using decentralized preview controllers. This result is a significant improvement over
6 i(t )= U id (t )+u iP" (t )+u iP"(t ) where ,iP'(t) is the preview input corresponding to the extemal disturbances, and uiP'(t) is for changes in reference setpoints. To illustrate the application of
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the current practice of reset or time schedule control used in buildings. 5. CONCLUSIONS It has been shown that zone temperature regulation in indoor environmental spaces can be significantly improved by augmenting the decentralized controller with preview action. A robust decentralized preview controller has been designed for a multizone space heating system. It has been shown that an increase in preview steps in general improves output regulation. However, the effect of adding more preview steps beyond the first two steps does not improve the output responses significantly. The paper has shown that the preview controller is robust to non-destabilizing variations in the system parameters and any mismatches in the disturbance profile. Results show that the use of preview controllers in the multipledisturbance case significantly improves zone temperature regulation compared to the case with no preview. It has also shown that in 24-hour building operation where the zone temperatures must track setpoints for occupied and unoccupied periods can be achieved in closed-loop mode using preview control. This is a significant improvement over the fixedschedule control strategy currently used in buildings. 6. R E F E R E N C E S
Davison, E.J. (1976), The Robust Decentralized Control of a General Servomechanism Problem, IEEE Trans. Autom. Contr., AC-21, 14-24.
Davison, E.J. and A. Solomon. (1983), Partial Decentralized Temperature Control of Multi-Zone Buildings, IEEE Conference on Decision and Control., San Antonio, Texas, 10-16. Davison, E.J. and Chang T.N. (1986), Decentralized Controller Design Using Parameter Optimization Methods, Control Theory and Advanced Technology, 2, 131-154. Gunewardana, D.R., M. Tomizuka and D.H Auslander. (1979), Application of Optima 1 Preview Control to Power Plant Cooling Systems, ASME J. Dyn. Sys. Meas. & Contr., 101,162-171. Haines, R.W. (1983), Control Systems for Heating, Ventilating and Air Conditioning, Van Nostrand Reinhold Company, New York. Sheridan,T. B. (1966), Three Models of Preview Control, IEEE Trans. on Human Factors in Electronics, HFE-7, 2, 91-102. Schittkowski, K. (1986), NLPQRL: A Fortran Subroutine for Solving Constrained Non-Linear Programming Problems, Annals of Operations Research, 5,485-500. Tomizuka, M. and D. E. Rosenthal. (1979), On the Optimal Digital State Vector Feedback Controller with Integral and Preview Actions, ASME J. Dyn. Sys. Meas. & Contr., 101, 172-178. Zaheer-uddin M., R.V. Patel and S. AI-Assadi. (1993a), The Design of Decentralized Robust Controllers for Multi-Zone Space Heating Systems, IEEE Trans. on Control Systems Technology, 1,246-261. Zaheer-uddin M., S. AI-Assadi and R.V. Patei. (1993b), Temperature Control of Multi-Zone Space Heating Systems Using Decentralized Robust Controllers, 12th IFAC World Congress, Sydney, Australia, 7, 99-102.
Pergamon 0967-0661(94)00051-4
Copyright © 1994 Elsevier Science Lid Printed in Great Britain. All rights reserved 0967-0661/94 $7.00 + 0.00
DECENTRALIZED PREVIEW CONTROL FOR MULTIPLE DISTURBANCE REJECTION IN HVAC SYSTEMS 1 M. Zaheer.uddin*, S.A.K. AI.Assadi** and R.V. Patel** *Centre for Building Studies, Concordia University, Montreal, Canada H3G IM8 **Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada H3G 1M8
Abstract: A decentralized preview controller is designed for temperature control of multizone indoor environmental spaces. A two-zone space heating system is considered. The physical system consists of a boiler, heat pumps, distribution network and two environmental zones. By assuming that the outdoor temperature variations are '~previewable",a decentralized preview controller is designed by using a parameter optimization method. The output responses of the resulting decentralized closed-loop bilinear system acted upon by single and multiple disturbances with and without preview action are compared. Also, results showing the robustness property of the controller, and the 24-hour building operation with unoccupied and occupied setpoint tracking using preview control are given. Key words: Multi-Zone Building; HVAC Systems; Temperature Control; Decentralized control ; Preview Control.
1. INTRODUCTION
In a previous study by Zaheer-uddin et al. (1993a), it was shown that because of their modular structure and performance requirements, HVAC systems are good candidates for decentralized control. Since they are also economical as such, they are extensively used in industry e.g. (Haines, 1983). The main findings of the investigation reported in (Zaheeruddin et al., 1993b) have been that the decentralized system response is not significantly different from that of the centralized system, and that the zone temperatures can be regulated to within ~0.2°C of the setpoints using decentralized control. The principal goal of the research reported here is to study whether or not preview control can be used as a means of improving the regulation properties of the decentralized controllers. To this end, the proposal is framed in the context of the specific system (referred to as a multizone space heating (MZSH) system) shown in Fig. 1. The MZSH system (Fig. 1) consists of a boiler
Indoor environmental spaces or zones in large buildings are subjected to multiple disturbances during day-to-day operation. Therefore, good regulation of zone temperatures in the presence of multiple disturbances is a problem of continued interest in the control of indoor environments of buildings. Typical disturbances acting on the zones can be classified into two groups: (i) diurnal variations such as those that occur in outdoor air temperatures, solar radiation fluxes and wind velocities and (ii) internal heat generating sources, viz. lights, occupants and equipment. The diurnal disturbances (referred to here as "external" disturbances) are previewable and can be assumed to be known ahead of time with some degree of accuracy. On the other hand, the intemal heat sources (referred to as "internal" disturbances) are easy to predict since they are related to a building's operating schedule. Coupled with the fact that heating, ventilating and air conditioning (HVAC) systems (which are used to provide conditioned air to zones) have large time constants, and therefore, the effects of disturbances on the plant output are delayed, it seems appropriate to explore the application of preview control concepts e.g. see (Sheridan, 1966; Tomizuka and Rosenthal, 1979; Gunewardana et al., 1979) to improve temperature regulation in buildings.
v2
1 This research w ~ supported by the Natural Sciences and Engineering Rese~ch Council (NSERC) of Canada under grants OGP0036380 and OGP0001345.
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which is used to supply water at moderately warm temperatures (between 16-32°C) to the evaporative heat exchangers of the heat pumps. Each environmental zone is installed with its own heat pump. The heat pump which works on the compression refrigeration cycle, receives heat energy from the source water and elevates this energy to a higher temperature and delivers it to the condenser coil of the heat pump. A circulating fan and ductwork arrangement is used to extract the heat from the condenser coil and deliver it to the zone through the diffuser shown in the figure. Thus the zone air is heated to offset the heating load acting on the zone due to cold ambient air temperatures such as those that occur, say, during a cold winter's day. Although only two zones are shown in Fig. l, it is obvious that the same arrangement holds for other zones in buildings with larger numbers of zones. The MZSH system shown in Fig. 1 can be characterized as having three stations (viz. station 1 : boiler, station 2 : zone-l, and station 3 : zone-2). The outputs of each station must be made to track some desired set-points which are known a priori. To achieve this regulation task, use can be made of (i) the station 1 input ul(t) (=v3) to control the output yl(_t) via~'ontroller c3; (ii) the station 2 input u2(t) ( = Iv I v411 ) to control the output Y2(t) via controllers
component of the MZSH system, and by using the energy conservation principle, a bilinear model of the system was developed in a previous study (Zaheer-uddin et al., 1993a). The model equations of the system are c b T b = v 3v 3max ( 1 -- otT b/Tbmax ) - m b CPw ( T b - T l , ) m b cp w (T b -TI2) - a b (T b - T e )
(1)
Cl~T l~ = -vnV4max(Pl-1) + mbcPw(Tb-Tt) - % ( T z - T e)
(2)
ChiT h~ = V4VnmaxP 1 - VlVlmax;(Th, - Tz) - ahl(Thi-Te)
(3)
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(4)
- az(T,-T
p) - a z , 2 ( T z - L 2 )
cI2T 12 = --VSVsmax(P2--1) + mbCPw(Tb-Tl2) - a l 2 ( T l 2 - T e) (5) Ch2T h 2 = V : s m a ~ P 2 - v : 2 ~ , ~ x ; ( T h - T ~ ) - a h ( T h - T Cz2T z2 = v 2 v ~
p)
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~2(Th 2-Tz2) - az2(T z 2-Tp ) + a z,2(Tz -Tz2 ) (7)
where P1-- I + ( P l m a x - 1 ) ( / - ( T h , - T l ) / A T l m a x )
(8)
P2 = 1 + (P2max-l)(1 - (Th2 - Tt2)/AT2max )
(9)
Note that equations (1)-(9) constitute a seventh-order bilinear model of the MZSH system.
clr ~and f4, and (iii) the station 3 input U3(t ) ( = Iv2vsI ) to control the output Y3(t) via the con-
3. THE DECENTRALIZED CONTROL PROBLEM
trollers c 2 and c 5. Thus the control problem is decentralized and there are five local controllers (c l, c2. . . . c5) to achieve the temperature regulation.
The control problem considered is as follows. For the MZSH system, which consists of three interconnected stations (boiler, zones 1 and 2), find a robust decentralized controller so that the output temperatures of zones 1 and 2 and the boiler are asymptotically regulated to desired set-points independent of (fl anv constant unknown disturbances in the outdoor air temperatures acting on the system, and (ii) any non-destabilizing perturbations in the parameters of the MZSH system itself. To formulate the decentralized control problem for the system given by (1)-(9), the dynamics of the MZSH system are first written
From the viewpoint of preview control, the most important external disturbance, namely the outdoor temperature Tp acting on the zones, is considered as the disturbance that is previewable. This external disturbance causes variable heating loads on the zones. It is proposed here to design decentralized preview controllers for the system shown in Fig. 1. The bilinear model equations (Zaheer-uddin et al., 1993a) are presented in Section 2. Then in Section 3, two decentralized controller design problems are defined: the non-linear constrained servomechanism problem (NCSP) (Section 3.1) and the non-linear constrained preview servomechanism problem (NCPSP) (Section 3.2). The designed decentralized controllers, without and with preview action, are implemented on the bilinear system and the resulting closed-loop system responses acted upon by single and multiple disturbances are compared (Section 4). Also included are the results on 24-hour building operation with unoccupied and occupied setpoint tracking using preview control. Conclusions are given in Section 5. 2. ANALYTICAL MODEL By identifying the energy flows to and from each
as: N x(t) = A x(t) + ~_,Bi(x ) ui(t) + E d ( t ) (10a) i=1
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where
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tor; Ul(t)=v3, u2(t)= Iv 1 va] T and u3(t)= [V2vs]T are the control input vectors; d(t)= [T T ]r is the vector of constant (unmeasurable) disturbances; N=3 is the number of "stations" in the decentralized system; and for station i, ui(t) ~ ]17,m', Yi(t) ~ ~q.P' and yR(t) ~ •P' denote the control inputs, measurable outputs and reference signals (set-points) respectively.
991
HVAC Systems For the system (10), use is made of the general servocompensator given by Davison (1976), and Davison and Solomon (1983)
~ i ( t ) = D i ~i(t)+Oi
ei(t )
f~i* = bl°ck diag [~i ,f~i . . . . . f~i ] Oi* = block diag [Oi , O i . . . . . Oi ] with Oi and f~i completely determined by the disturbances and reference signals acting on the system, e.g., for constant disturbances and reference signals, ~i =0, Oi = 1. Thus, the robust decentralized servomechanism problem for the augmented system described by (10) and (11), is to find N local controllers with inputs Yi(t) ~ IRP'and outputs ui(t) E ~.m,, i = 1. . . . N, SOthat (i) the resulting closed-loop system is asymptotically stable, (ii) asymptotic tracking of the reference signals occurs for all constant disturbances and setpoints, and (iii) for all perturbations in the parameters of the system which do not cause instability in the closed-loop system property (ii) still holds. For the application considered in this paper, m i > Pi, i = I, 2, 3, all the outputs to be regulated are measurable and the disturbances to be regulated and the reference signals to be tracked are constant. In addition, it was found (Zaheer-uddin et al., 1993a) that the linearized models (about the normal operating points) of the MZSH system (1)-(9) with the chosen decentralized structure do not have unstable decentralized fixed modes, or transmission zeros at the origin. The decentralized controller has the structure given by [ei(t)]
command/external input vector for the MZSH system.
robust servomechanism assuming that the externot previewable, the solved is the minimiza-
t!
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i = 1. . . . . N
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x ° i ( t ) - l . 5
j : l ..... 7
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where x°(t)= [TboTt,oTh,oTz,oTtTh~oT~] r
are the
temperatures at the operating point, vk(t), k=l ..... 5, are the closed-loop control inputs for the MT_SH system, and tf is the duration of the Tp profile (outdoor temperature). The constraint (13a) is required to ensure that the input energy of the system does not exceed its maximum capacity. Note that the vt "s are normalized with respect to their maximum capacities so that they vary between 0 and 1. The constraint (13b) ensures that the system states remain bounded and close to the operating point x°(t). The approach that was used to solve the NCSP involves selecting appropriate Q and R i in (12) and finding the corresponding sets of K~ that minimize J while satisfying the constraints (10a-c), (11), and (13a,b). This problem was solved using the method of successive quadratic programming (Schittkowski, 1986). Note that (Davison and Chang 1986) have also used nonlinear programming to solve such a decentralized control problem. By using the system parameters given in (Zaheer-uddin et. at, 1993b), the optimal decentralized feedback matrix Kd was computed
0.3187
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0.0
0.0
0.6885 0.0
0.0
0.0
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0.0
0.6173 0.0
0.0
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0.0
0.9149 0.0
0.0
0.0
0.3702 0.0
0.0
0.7105
0.0
0.0
0.4616 0.0
0.0
0.8406
This gain matrix was obtained with the following values of Q and R~
where ! is the identity matrix.
3.1 The Nonlinear Constrained Servomechanism Problem (NCSP)
0
k = 1, 2. . . . . 5
i= 1. . . . . N
where K i = [Ki O) Ki <2)] , and rio(t) is the operating
To solve the decentralized problem stated above while nal . disturbance TP is . . opttmxzatlon problem to be tion of
0 < vk(t) < 1
(11)
in order to reject the external constant disturbances (step changes in the ambient temperature Tp), and track constant reference signals acting on the system. The matrices
u i ( t ) = V i o ( t ) - K i [~i(/)j,
at the chosen operating point x°(t) (given in section 4) and subject to the constraints (10a-c), (11) and
(12)
3.2 The Nonlinear Constrained Preview Servomechanism Problem (NCPSP) Consider the same bilinear decentralized system defined by (10a-c) and augmented with (11). Also, assume that the outdoor temperature Tpj is previewable and that its values are measured at j=0, 1. . . . . Np. This preview profile can be approximated by a series of step functions. For the NCPSP,
992
M. Zaheer-uddin
the decentralized controller structure consists of two parts: I~d for the decentralized tracking and stabilizing control, and K w for the decentralized previewing control, i.e., lil (t)=
uid(t)+uiP~(t)
subject to the constraints
(10a-c),(11)and
0 < vk(t) < 1 , k = l , 2 . . . . . 5, (15a) 0 x j(t)-l.5 < xj(t) < xOj(t)+ l.5, j = l ..... 7, (15b) where, vk(t), k=l . . . . . 5, are the closed-loop preview control inputs for the MZSH system.
where
[ei(t)] uid(t)=Vio(t)_
et al.
[/~i(1) /~i(2)] ~i (t)]' i = 1 . . . . . N
J
uiP'(t)= ~i'~Kw(i)dl(t) j=0, -
1. . . . ,Np
l=1
Using the same optimization technique (Schittkowski, 1986) as before, the optimal decentralized gain matrix I~ d, and the preview controller gains Kw)i) for one previewing step (y=Np=0), were obtained using the same Q and Ri . These are
J
dj(t)(=To(t))=Cwo+EACwW(t)
,
j=0, 1 . . . . /Vp
l=1
ACw=Cw-Cw, ~ , l=1,2 . . . . . w (t)=the unit step function.
Alp I~ d =
Note that d(t) is computed as the sum of the magnitude of the outdoor temperature at the operating point Cwo and the variations in To from the operating point denoted by ACw, at each preview step. For example, since the external disturbance is assumed to be previewable (known a priori) the amplitudes (step functions) Cwo, .... Cw~ and their time of occurrence are known. In other words, the time-varying outdoor air temperature preview profile is approximated by a series of step functions. Note that the approximation of outdoor temperature by a series of step functions is a common practice in HVAC engineering. The reason is that hourly weather data is currently available from meteorological stations throughout North America. Using such past data, hourly outdoor temperatures are predicted. Between two successive data points either ramp or step function approximation can be used see (Zaheer-uddin et. al, 1993b). For simplicity, step functionapproximation was used to construct the outdoor temperature preview profile. With this knowledge, the corresponding gains gw, for j =0, 1. . . . ,Np were computed. Thus, the decentralized preview control input is updated via uiP~(t). Given this preview structure of TO the decentralized gains are defined such that
Kw=[(blockdiag [[Kwj(i). .i=l' ... l(a= [ {block diag [l(i(J),i=l . . . . .
N ] ] , j = 0 , 1 . . . . ,Np]
N]/,J=I,2 ]
Given the above preview profile of T , j =0, 1. . . . . Np, over the time duration tf , the NCPSP for the augmented decentralized system defined by (10) and (11) is to tl
Minimize f(I(d, Kw)= I 0
e(t) r Q e ( t ) + ~ u i ( t )
Riui(t) dt
i=1
(14)
0.31942
0.0
0.0
0.68863
0.0
0.0
0.0
0.40222
0.0
0.0
0.61737
0.0
0.0
0.49986
0.0
0.0
0.91492
0.0
0.0
0.0
0.37041
0.0
0.0
0.71059
0.0
0.0
0.46174
0.0
0.0
0.84063
[0.00349] Kwo(1)=0"00849 ' Kwo(2)=[0.00187
(3)_ [0.00352] - [0.003091
' Kwo
4. SIMULATION RESULTS 4.1
Open-Loop Response
Consider first the open-loop response of the MZSH system to highlight important physical characteristics. The system operating point refers to a mild-day (winter) operating condition when the outdoor temperature TOo is -2 ° C with Teo = 20 ° C. The choice of the operating point of the system depends on the magnitude of the anticipated load on the system. Usually the operating point can be selected such that it corresponds to 50% load capacity. The following operating point, reflects an average load condition on the MZSH system:
Tbo=27.26 ° C , Ttlo=25.58°C , Th~o=29.84° C , Tz~o=21.96° C Tl2o=25.51° C , Th~o=27.94°C , Tz2o=20.66°C , v lo=0.4, V2o=0.45, V3o=0.5, V4o=0.5, v50=0.5 Under this load condition, the zone temperatures Tz, and T are expected to reach 21.96°C and 20.66 °c respectively, and the boiler water temperature to reach 27.26°C. To examine the system response, a step change in T O is introduced such that the new outdoor temperature is -6°C (resembling a somewhat colder day) and a -1 "C step change in Teo. Furthermore, a -1 ° C step change is introduced in the initial temperatures of Tzl, Tz2 and Tb from the operating point: T~,° = 21.96° C ; Tz2° = 20.66° C ; and Tbo = 27.26° C. How the open-loop system responds to this situation is depicted in Fig. 2. The boiler tem-
HVAC Systems
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perature rises slowly and reaches steady state (as shown in Fig. 2) in about 3.5 hours. The final value of Tb is 26.25° C which is l°C below the operating point. On the other hand, the zone temperatures ( T~, in Fig. 2b and Tz2 in Fig. 2c) are directly influenced by the disturbance, and they reach new steady-state temperatures which are a few degrees below the operating point. For example, Tz, decreases slowly and reaches a steady-state value of 18.9 °C in three hours, which is about 3°C (steady-state error) below the desired set-point. The T~2response in Fig. 2 shows that the input energy to zone-2 cannot sustain the zone-2 temperature at its operating point. The temperature T~: also decreases slowly and reaches a new steady-state value of 17.52°C, which is about 3.14°C below the desired set-point. Some important observations can be drawn from the open-loop responses : (i) the temperature response curves show that the system steady state time is of the order of 3 hours and (ii) the external disturbances influence the zone temperatures much more strongly (almost three times) than the boiler temperature. This latter observation offers evidence that preview of disturbances could be useful in improving the zone temperature regulation.
1
i- 1 / Fig. 3 Block diagram of the decentralized preview control for MZSH system. w(t)
E
IE
J
g
.........
E
......... ®
E
4.2 Closed-Loop Response The closed-loop implementation scheme for the NCPSP is shown in Fig. 3. The cgntroller structure as shown in Fig. 4 consists of two parts: the decentralized output feedback servocompensator control input ud(t), and the preview control input uP(t). It is obvious that for the NCSP, u p (t)=O. When preview action is added, the implementation consists of prior knowledge of the amplitudes of the external disturbance Tp denoted by Cw, and their corresponding computed gains Kw? for j =0, 1. . . . . Np, where j is the index denoting the preview step as discussed in section 3.2. Note that the control structure, the design method, and the implementation technique used in this study differ from the preview control examined by Tomizuka and Rosenthal (1979). To study the effect of adding preview control on the system responses, the system output was simulated
uP(t)
Fig. 4 The structure of the decentralized preview controller for MZSH system.
with (NCPSP) and without (NCSP) preview action by using the same disturbances and initial conditions as those used in the open-loop test (Fig. 2). Figs 5a-c show the output responses. It is apparent that output responses with one-step preview action (dashed lines, Np =0 ) are better in that Tb, Tz, and Tz: reach steady state faster than for the case without preview (solid lines, Ne =-1). Note that the results in Figs 5a-c are for the case when only one preview step (Np =0) of disturbance is considered. Indeed, a further improvement in the system responses can be expected when the external disturbance is approxi-
994
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Figs 5a-c Output responses of the closed-loop systems with (NCPSP) and without (NCSP) preview control acted upon by the same magnitude of step changes and disturbances as in Fig. 2. mated by a step function by increasing the preview steps (as shown in Fig. 6) for different classes of disturbance profiles. Two different outdoor temperature profiles (type-a and type-b) are considered in Fig. 6. In each case, they are approximated by four preview intervals (Np =3). The intent is to study the output responses of the system by successively increasing the preview intervals from 0 to 4. Figs 7a-c show the output responses for the disturbance profile type-a shown in Fig. 6. The solid lines in Figs 7a-c are the responses obtained with no preview (N =-1) and the dashed lines are those obtained by considering one preview step ( ~ =0). It is apparent that the addition of one preview step is able to achieve good output responses. The effect of increasing the number of preview steps is illustrated in Figs 7d-f. It is evident that increasing the preview steps improves the system responses. Further evidence of this improvement ill
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HVAC Systems comes from the fact that the magnitude of the performance index decreases as the preview steps is increased. For the results shown in Figs 7a-f, the minimum value obtained for J without preview is J,an-(N~=-i)=l.1271, and the values obtained with preview are:
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and N =2). This observation suggests that when dealing with a unidirectionally decreasing disturbance profile (type-a), a higher number of preview steps yields better regulation, whereas in the case of a bidirectional disturbance profile (type-b), a smaller number of preview steps would be sufficient to get better zone temperature regulation. This result is encouraging in the sense that over a typical day, the outdoor temperatures increase and decrease about some average temperature. This means that in terms of designing a preview controller for a typical daily operation, a large number of preview steps may not be required.
fmin(N~=0)=0.3397, Jmin(Np= 1)=0.01881, Jmi,(Ne =2)=0.00588 and Jmi~(N~=3)=0.00275 It is obvious that by increasing the preview steps, the system response can be improved. For the system considered in this study, the effect of adding the first preview step is significant. Further increase in the number of preview steps is beneficial in reducing the error from the setpoint although the improvement between successive addition of preview steps is not very significant.
4.3 Robustness Although the advantages of previewing the disturbances is recognized, the fact remains that inaccurate previewing (or forecast) of external disturbances could have an adverse effect on the output responses. On one hand there is the possibility of mismatch in disturbances, while on the other hand, the parameters of the system also change during continuous operation. To test the worst-case scenario, simulation runs were carried out, in which the controller designed by assuming a disturbance profile (type-b) as shown in Fig. 6 was used to reject a type-a disturbance profile. In other words, a complete mismatch of the disturbance profile is assumed. Furthermore, a 25% increase in the system parameters (~, ;t, ~2) was also considered. The results of this simulation are shown in Figs 9a-c. Note that the Tb response (Fig. 9a) with
Further simulations with another class of typical disturbance profile type-b (Fig. 6) were carried out. The corresponding output responses with one preview step (N~=0) and no preview (Ne =-1) are shown in Figs 8a-c. Note that the variations in Tz, and T~: with one preview step are within z~O.05°C as opposed ~-0.16°C for the no preview case. The effect of increasing the preview steps on the output responses of the system is given in Figs 8d-f. It is worth nothing that the improvement in output responses with two or more preview steps (Np = 1 and Np =2) is insignificant (Figs 8d-f), whereas the output responses depicted in Figs 7d-f show a small but finite improvement (for example, compare Figs 7e and 8e for both cases N~ = 1 ~U
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M. Zaheer-uddin et al.
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period. To this end, a day with an outdoor temperature profile was considered, as shown in Fig. 10, and the open-loop system response was simulated (for lack of space the figures are omitted). Note that the zone temperatures undergo a maximum variation of 5°C when the MZSH system is operated in openloop. Fig. 10 was assumed to be the preview profile, and a preview controller was designed. The controller so designed was used to simulate the closedloop bilinear system. The corresponding output responses obtained are depicted in Figs lla-c. It is apparent that the output responses with ~ =0 (one preview step) are superior to those without preview. $ 8
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Figs 11a- f Typical daily regulation of the temperatures Tb, Tz, and Tz~ with preview controller (single disturbance case: outdoor temperature profile shown in Fig. 10). The zone temperatures are regulated to within _+0.05°C of the setpoints for most of the day. Also
shown in Figs 1 ld-f are the comparisons of the output responses with one (Np =0) and four preview steps (Np=3). It is apparent that there is no significant advantage in increasing the preview steps (since the decrease in the error from the set-point is very small) so long as there is no mismatch in disturbances. In the event there is a mismatch, the robustness results depicted in Figs 9d-f offer evidence that up to two to three preview steps may be all that is required to achieve good zone temperature regulation under realistic changes in disturbance profiles and/or in the system parameters.
of 7°C compared to 5°C with single disturbance. This suggests that the use of preview controllers would be much more beneficial in multipledisturbance rejection cases. Again decentralized preview controllers with one and several preview steps were designed. The designed preview controller performance was compared with the no preview case as shown in Figs 13a-c. The effect of increasing the previewing steps on output regulation is depicted in Figs 13d-f. A comparison of output responses shown in Figs l la-c (single-disturbance case) and Figs 13a-c (multiple-disturbance case) shows that the boiler, -.one 1 and zone 2 temperatures in the no preview case in Figs 13a-c have somewhat higher overshoot.
4.5 Rejection o f Multiple Disturbances Indoor environmental zones in buildings are not only acted upon by diurnal outdoor temperature profiles (Fig. 10), but are also subject to variable solar radiation fluxes entering through the windows and internal heat gains due to variable occupancy and lighting. To demonstrate the application of preview control to tackle this multiple disturbance rejection problem, two disturbances were considered: namely outdoor temperature and solar gains. The assumed preview profiles for these disturbances are depicted in Figs 10 and 12a respectively. The open-loop system responses to these disturbances are shown in Figs 12b-d. Note that zone temperatures under multiple disturbances undergo a maximum variation
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998
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This is due to the fact that the solar gains entering the zones contribute to the heating of air. Therefore the controller should utilize this free heat energy by reducing the auxiliary energy supplied to the zones. Under these circumstances, the use of the servocompensator alone (no preview case) is not sufficient to achieve good zone temperature regulation. As an example, consider Figs 13b and lib. Note that the use of decentralized output feedback with no preview action results in an overshoot of 0.19°C in Tz, with single disturbance, and 0.25°C with two disturbances. The use of one-step preview control reduces
this maximum overshoot to 0.09" c in the case of single disturbance and about 0.1°C for the case of two disturbances. These variations are further reduced by increasing the number of preview steps (Figs 13d-f). In other words, as the number of disturbances acting on the system increases, the impact of using the preview controllers becomes much more significant compared to the corresponding no-preview cases. Therefore, since building environments are subject to multiple disturbances, preview controllers are good candidates for HVAC systems.
HVAC Systems
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preview control, consider two examples: in the first example (case-A), consider a single-step preview controller (Ne+=0) for the external disturbance (outdoor temperature is assumed to remain constant), and a four-step preview controller (Ne,=3) for tracking changes in setpoints. In the second example (case-B), consider a two-step preview controller (Np =1) for outdoor temperature (which is assumed to vary as shown in Fig. I0), and a four-step preview controller (Np =3) for setpoint tracking. For case-B, a two step preview controller for outdoor temperature rejection was chosen because previous results showed that two to three preview steps are sufficient to achieve good output regulation.
4.6 Setpoint Tracking for Occupied and Unoccupied Periods Most commercial buildings are operated at different setpoints during the day (occupied periods) and night (unoccupied periods) to save energy. Typically, during unoccupied periods (1800 -0800 h), zone temperature setpoints are decreased to 15°C, and during occupied periods (0800 -1800 h), the zone temperature setpoints are set forward to the normal comfort range (T~ =22°C, Tz=20.7°C). Currently, the rooming heat-up from unoccupied to occupied setpoints, and the evening cool-down from occupied to unoccupied setpoints is achieved by using reset control action. Normally this function is performed on a fixed schedule independent of outdoor and zone temperatures. This is not realistic since outdoor and indoor temperatures change during occupied and as well as unoccupied periods.
Figs 14a-c show the closed-loop system responses for case-A. The actual setpoint strategy shown in the figures requires that the zone temperature be set at 15°C during 1800-0800 h, and T = 22°C and T~2= 20.7°C during occupied hours 0800-1800. To track such ideal settings, a preview setpoint strategy shown by dashed lines in Figs 14a-c was used. This preview strategy consists of two-step preview action which is initiated about two hours ahead of time in the morning, and two hours in the evening, starting at 1800 h. The actual output responses of the system (solid lines) track this preview setpoint strategy so that the zone temperatures reach the day-time setpoints just before the occupants arrive in the building. In the evening, the zone temperatures are decreased as soon as the unoccupied period begins at 1800 h, and they reach the unoccupied setpoint temperature at about 2000 h. The results depicted in Figs 14a-c show that building operation with variable setpoints can be achieved with preview control.
As far as possible, the use of fixed-schedule control even during the unoccupied hours should be avoided, the reason being that the system operation with fixed control while the external disturbances are acting on the system could result in energy waste because of loss of control. On the other hand, output feedback control alone for tracking widely separated setpoints is not likely to give good results. To address this issue the use of preview control is proposed. From the system dynamics, the lead time necessary to heat the space from unoccupied setpoint can be approximately established to occupied setpoint, while ensuring that all the control inputs do not exceed the capacity constraints of the system. For the system considered in this study, this lead time is about two hours. With this knowledge, a four-step preview controller was designed (based on the minimum number of preview steps required to avoid input saturation) to track step changes in reference setpoints. The overall controller now consists of three parts, viz.,
The sets of results shown in Figs 15a-c (open-loop responses) and Figs 15d-f (closed-loop responses) correspond to case-B which is the most general case in which the outdoor temperature was assumed to be time-varying (as in Fig. 10) and the zone temperature setpoints were also varied according to building occupancy (occupied and unoccupied modes). The responses depicted in Figs 15e-f show that the zone temperatures can be regulated to within _+0.2°C throughout the day using decentralized preview controllers. This result is a significant improvement over
6 i(t )= U id (t )+u iP" (t )+u iP"(t ) where ,iP'(t) is the preview input corresponding to the extemal disturbances, and uiP'(t) is for changes in reference setpoints. To illustrate the application of
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Figs 15a-f 24-hour operation (case-B) of the MZSH system. (a-c open-loop responses; d-f closed-loop with preview control)
the current practice of reset or time schedule control used in buildings. 5. CONCLUSIONS It has been shown that zone temperature regulation in indoor environmental spaces can be significantly improved by augmenting the decentralized controller with preview action. A robust decentralized preview controller has been designed for a multizone space heating system. It has been shown that an increase in preview steps in general improves output regulation. However, the effect of adding more preview steps beyond the first two steps does not improve the output responses significantly. The paper has shown that the preview controller is robust to non-destabilizing variations in the system parameters and any mismatches in the disturbance profile. Results show that the use of preview controllers in the multipledisturbance case significantly improves zone temperature regulation compared to the case with no preview. It has also shown that in 24-hour building operation where the zone temperatures must track setpoints for occupied and unoccupied periods can be achieved in closed-loop mode using preview control. This is a significant improvement over the fixedschedule control strategy currently used in buildings. 6. R E F E R E N C E S
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