Temperature control of two interacting rooms with decoupled PI Control

Transcription

1 Preprints of te 19t World Congress Te International Federation of Automatic Control Cape Town, Sout Africa August 24-29, 2014 Temperature control of two interacting rooms wit decoupled PI Control Meike Stemmann Anders Rantzer Lund University, Sweden ( Lund University, Sweden ( Abstract: Using a simulated model of a ouse consisting of two adjacent rooms, te temperature of te two rooms is controlled wit a PI controller and a decoupled PI controller Tis is compared to MPC control In te simulation eample ere, bot te MPC controller and te decoupled PI controller decreased te interaction between te temperature dynamics of te two rooms, as compared to an independent PI controller in eac room For te eample in tis paper, te control performance of te decoupled PI controller is comparable to tat of MPC, regarding te interactions between te two rooms Keywords: Energy management systems; Building and ome automation; PID control; Predictive control; dynamic decoupling; 1 INTRODUCTION Buildings account for 40 % of total energy consumption in te European Union European Parliament, 2010, in Sweden one tird of te energy used is related to te building sector, and 60% of te energy used in buildings is used for eating and ventilation Persson, 2002 Wit a growing building sector, it is necessary to decrease te energy used by eating and ventilation in buildings, so te total energy used in te buildings sector is not increased Terefore control of eating and ventilation systems in buildings as been an active researc area for many years Many different metods to control temperature and eat in a building ave been proposed, model predictive control (MPC) being one of te most popular Ma et al, 2012, Hazyuk et al, 2012 Model predictive control is cosen many times as te most suitable control metod for buildings, since it can consider predictions of disturbances suc as weater, occupancy or outside temperature On te oter and, MPC requires a model of te building wit sufficiently good accuracy and accurate predictions of te disturbances to give a good control performance Besides estimating a model describing te building dynamics, te tuning of te controller parameters is not necessarily intuitive and not always easy to acieve Anoter control metodology widely used in temperature control and building components control is PI control Salsbury, 2005, Dounis and Caraiscos, 2009 Building components using PI control areis termostats and termostatic valves on radiators Peffer et al, 2011 Te temperature of a room or area is ten controlled locally, witout any knowlege about te temperature control in neigboring rooms Wen te temperature in several rooms in a building is controlled independently for eac room, it can be epected tat te influence of a temperature cange in a neigboring room can degrade te controller performance Te approac in tis paper is to connect te PI controllers for different rooms wit a decoupling network Gagnon et al, 1998 Using suc a network, temperature canges in one room are not affecting te oter room, so tat te local PI controllers are connected to a more centralized control solution For comparison, te same building model is controlled wit an MPC algoritm Te building model used in tis paper for simulation and controller design is presented in Section 2 Section 3 describes te decoupled PI control and te MPC algoritm Net, te simulation results comparing PI controllers witout a decoupling network, PI control wit a decoupling network and an MPC controller are presented in Section 4 Tese results are discussed in Section 5 Finally, Section 6 concludes te paper 2 A MODEL OF TWO NEIGHBORING ROOMS Te building to be controlled is sown in Fig 1 It consists of two rooms wit an interconnecting wall Te ouse is 5 meters long, 25 meters ig and te two rooms ave a widt of 3 meters and 5 meters, respectively It was assumed tat te walls are 02 meters tick and ave a eat conductivity of 004 W/(m C) Eac room as a glass window area of 1 m 2 wit a eat conductivity of 078 W/(m C) and a tickness of 001 meter Te temperature dynamics of te ouse were modeled troug eat transfer, combining te effect of storage and conduction of eat in te building elements and te outside and inside air in a lumped parameter model Felgner et al, 2002, were time delays were neglected At te center of eac room, a temperature node was placed to represent te average temperature in eac room Also, te outside temperature was represented toug a temperature node outside of te two rooms Te resulting model describing te temperature dynamics of te ouse is sown in (1) Copyrigt 2014 IFAC 11722

2 Cape Town, Sout Africa August 24-29, 2014 T out T 1 T 2 Fig 1 Sketc of te two rooms to be controlled Te average temperature of room 1 is denoted by T 1 and te average temperature of room 2 by T 2 Te outside temperature is denoted by T out M 1 c dt 1 dt M 2 c dt 2 dt = A 1 R 1 (T out T 1 ) + A i R i (T 2 T 1 ) + Q 1 (1) = A 2 R 2 (T out T 2 ) + A i R i (T 1 T 2 ) + Q 2 Here, te resistances R 1 and R 2 represent conduction of eat between te temperature nodes in te center of eac room and te outside temperature troug te room air, te outer walls and te windows Te resistance R i represents te conduction of eat troug te inner wall between te neigboring rooms Te area of te inside wall is denoted by A i and te area of te outside walls of room 1 and room 2 by A 1 and A 2, respectively Te terms on te left-and-side corresponds to storage of eat in te room air, were M 1 and M 2 are te air masses in te two rooms and c is te specific eat capacity of air Te eat flow into te two rooms is denoted by Q 1 and Q 2 Tis model was implemented in MATLAB/Simulink to simulate te temperature dynamics of te ouse sown in Fig 1 It can be seen in (1), tat te interconnecting wall between te two rooms introduces a coupling between te temperature dynamics For te use in PI and MPC control, te temperature dynamics need to be epressed as a transfer function model and a state-space model Disregarding te outside temperature, wic was assumed to be an unknown disturbance, te state-space representation of (1) is sown in (2) and te transfer function representation in (3) d dt ( ) = A T 2 y(t) = ( ( ) + B T 2 ) ( ) T 2 ( ) Q1 Q 2 Te states and te outputs of te state-space system (2) are te temperatures T 1 and T 2 in te two rooms Te inputs are te eat flows Q 1 and Q 2 into room 1 and room 2, respectively (2) T 1 (s) = G 11 (s)q 1 (s) + G 12 (s)q 2 (s) (3) T 2 (s) = G 21 (s)q 1 (s) + G 22 (s)q 2 (s) In te transfer function representation (3), G 12 (s) and G 21 (s) represent te interaction of te temperature dynamics between te two rooms 3 TEMPERATURE CONTROL Te temperature of te ouse in Fig 1 was controlled wit an independent PI controller in eac of te two rooms, PI controllers wit a decoupling network and an MPC controller Compared to te two single PI controllers te decoupled PI controllers take te interaction between te two rooms in te ouse into consideration and are epected to reduce te interaction in te temperature dynamics Since te MPC is a controller wit multiple inputs and outputs, it takes te interaction into account as well 31 Decoupled PI control For eac of te rooms, a PI controller was designed to control te temperature of eac room separately, disregarding te coupling of te temperature dynamics between te rooms Te parameters of te PI controllers were determined troug Ziegler-Nicols step response metod Astrom and Hagglund, 2006 To remove te effect of te coupling, a decoupling network was introduced between te PI controllers and te process Here inverted decoupling was applied, wic as bot a simple realization and a diagonal decoupling matri Gagnon et al, 1998, Garrido et al, 2011 Figure 2 depicts a control system wit inverted decoupling, were te process inputs u 1 (s) and u 2 (s) are a combination of te respective controller output c 1 (s) or c 2 (s) and te oter respective process input Te control signal for inverted decoupling is sown in (4) u 1 (s) = c 1 (s) + u 2 (s) G 12(s) (4) G 11 (s) u 2 (s) = c 2 (s) + u 1 (s) G 21(s) G 22 (s) Wit tis, te transfer function from te controller outputs to te process outputs is a diagonal matri were te effect of te coupling is removed Since only eating of te rooms was considered, te control signals were limited to be positive To cope wit tis input constraint, an anti-windup strategy was added to te decoupled PI controller Because of te structure of inverted decoupling, an anti-windup metod as employed for single PID controllers could be applied directly Gagnon et al, 1998 Te anti-windup strategy used ere is based on back-calculation Astrom and Hagglund, 2006, were te difference between te saturated and te non-saturated signal is fed back around te integrator in te PI controller wit a time constant T t Wen tere is no saturation, te anti-windup sceme as no effect Oterwise, it will try to drive te output of te integrator, suc tat te control signal is close to its saturation limit 32 MPC control A model predictive control (MPC) algoritm was designed to control te temperature of bot rooms using te statespace model (2) Since te MPC algoritm operates in discrete time, (2) was discretized using zero-order-old discretization wit a sampling time of T s = ours Astrom and Wittenmark 1997 Furtermore, integral action was introduced into te MPC algoritm by introducing a constant disturbance on te input Maciejowski and 11723

3 Cape Town, Sout Africa August 24-29, 2014 r 1 C 1 r 2 C 2 c 1 c 2 G12 G 11 G21 G 22 u 1 u2 G 11 G 12 G 21 G 22 y 2 y 1 Table 1 PI controller parameters for te two rooms and te corresponding damping ξ and natural frequency ω K T i ω ξ Room Room Fig 2 A control system wit inverted decoupling for a system wit two inputs and two outputs Huzmezan, 1997, wic was added as an additional state Te process model used for te MPC algoritm is sown in (5), were A d, B d, C d are te discretized state-space matrices and v(k) is a constant input disturbance ( ) ( ) (k + 1) Ad B = d (k) Bd + u(k) (5) v(k + 1) 0 I v(k) 0 (k) y(k) = (C d 0) v(k) Te state (k) consists of te room temperatures T 1 and T 2 at time k Since te additional state v(k) is not measured, a Kalman filter was used to estimate te statevector of (5) Astrom and Wittenmark, 1997 For an MPC algoritm, predictions ŷ(k) of te process output are needed To determine tese predictions for te prediction orizon H p, te model (5) was used Tese prediction are calculated as sown in (6) Y Hp = S A (k) + S u u(k 1) + S u U Hu (6) were = ŷ 1 (k + 1) ŷ 1 (k + 2) ŷ 1 (k + H p ) T Y Hp U Hu = u(k) u(k + 1) u(k + H u 1) T and u(k) = u(k) u(k 1) Te epressions for S A, S u and S u are presented in te Appendi A Using tese predictions, te optimization problem to be solved for te MPC algoritm is (7) minimize u(k+m) H p m=1 H u ŷ(k + m) r(k + m) 2 Q + u(k + m) 2 R subject to 0 u(k + m) u ma, m = 0,, H u 1 Tis optimization problem was solved using CVX, a package for specifying and solving conve programs Grant and Boyd, 2013, SIMULATION Te model of te two rooms and te control algoritms were implemented in MATLAB/Simulink Te controller parameters for te PI controllers and te corresponding closed-loop damping and natural frequency for bot rooms are sown in Table 4 Te parameters for te MPC controller were prediction orizon H p = 10, control orizon (7) Table 2 Mean square deviation of te room temperaure from its reference value and eat used per our in te case were te reference temperature in room 1 canges T e : mean square error between room temperature and reference temperature Q : eating power used per our Room 1 Room 2 C C 10 3 kw/ 10 3 kw/ PI PI decoupled MPC H u = 5, and te weigting matrices for te cost function R = I and Q = To investigate te effect of interaction in te simulation, te temperature in one of te rooms was canged, wile observing te effect of tis cange of temperature on te oter room Eac room was controlled by eiter an independent PI controller, by PI controllers wit inverse decoupling or by an MPC controller Te simulation results are sown in Fig 3 and Fig 4 In bot figures, te first panel sows te temperature in room 1, te second panel te eat flow used to eat room 1, te tird panel sows te temperature in room 2 and te fourt panel te eat flow needed to eat room 2 Te deviation of te room temperature from its reference temperature and te eat used per our are sown in Table 2 and Table 3 Te deviation of te room temperature from its reference temperature T e and te eat used per our Q are calculated by T e = (1/N) k=t s k=0 (T (k) T ref (k)) and Q = k=t s k=0 Q(k)/t s Here, te lengt of te simulation is denoted by t s, te number of simulated data points by N, te temperature in a room by T, te corresponding reference temperature by T ref, te eat used by Q and te time inde in te simulation by k Te results for te effect of canging te temperature in room 1 on te temperature in room 2 is sown in Fig 3 and Table 2 Te influence on te temperature in room 2 from a cange of reference temperature in room 1 is largest wit te independent PI controllers witout decoupling Bot MPC control and PI control wit inverse decoupling decrease te deviation of te temperature in room 2 from its 20, C reference temperature Te influence of a temperature cange in room 2 on te temperature in room 1 is sown in Fig 4 and Table 3 Te PI controller witout decoupling leads to a larger influence of temperature canges in room 2 on te temperature in room 1 Bot te PI control wit inverted decoupling and te MPC controller decrease te temperature cange in room 1 after a cange of temperature in room

4 Cape Town, Sout Africa August 24-29, Temperature room Temperature room Q eat flow from eater room Q eat flow from eater room Temperature room Temperature room 2 T eat flow from eater room T eat flow from eater room Q2 95 Q time ours PI no decoupling PI inverted decoupling MPC time ours PI no decoupling PI inverted decoupling MPC Fig 3 Simlation results for a cange of reference temperature in room 1 for PI control witout decoupling (black solid), PI control wit inverted decoupling (greed dased) and MPC (blue dased) Outside temperature T out = 10 C first panel: Temperature T 1 of room 1 Te red dased curve is te reference temperature During te first 09 ours, te reference is 28 C, afterwards te reference temperature is 19 C second panel: Heat flu Q 1 into room 1 tird panel: Temperature T 2 in room 2 Te red dased curve is te reference temperature fourt panel: Heat flu Q 2 into room 2 Table 3 Mean square deviation of te room temperaure from its reference value and eat used per our in te case were te reference temperature in room 2 canges T e : mean square error between room temperature and reference temperature Q : eating power used per our Room 1 Room 2 C C 10 3 kw/ 10 3 kw/ PI PI decoupled MPC DISCUSSION Canging te temperature in one of two adjacent rooms affects te temperature in te second room as well Having a seperate PI controller for eac room, were eac controller operates independent of te oter one, tis interaction between te temperature dynamics of te two rooms is not taken account of Hence, a decoupling network was added connecting te PI controllers Te results in Fig 3 and Fig 4 sow tat wit tis decoupling network, te deviation of, eg, te temperature in te second room from Fig 4 Simulation results for a cange of reference temperature in room 2 for PI control witout decoupling (black solid), PI control wit inverted decoupling (greed dased) and MPC (blue dased) first panel: Temperature T 1 of room 1 Te red dased curve is te reference temperature second panel: Heat flu Q 1 into room 1 tird panel: Temperature T 2 in room 2 Te red dased curve is te reference temperature During te first 12 ours, te reference is 20 C, afterwards te reference temperature is 29 C fourt panel: Heat flu Q 2 into room 2 its reference temperature, wen te reference temperature in te first room was canged, could be decreased Furtermore, te two adjacent rooms were controlled using a central MPC controller, controlling te temperature of bot rooms at te same time In tis way, interactions of te te temperature dynamics of bot rooms are taken into account as well In te simulation ere (see Fig 3 and Fig 4), te deviation of te room temperature from te reference temperature of te room wit constant reference temperature was less wit te MPC control tan wit two independent PI controllers Often MPC is viewed as te preferable control metod in building contets Hazyuk et al, 2012, Ma et al, 2012, Dounis and Caraiscos, 2009, since it can take into account predictions of disturbances suc as weater conditions or occupant beavior Also, constraints on, eg, control signals can be included in te MPC formulation Te MPC used ere did not ave predictions of te reference signals or oter disturbances available On te oter and, PI controllers are already used in termostats and termostatic valves on radiators to control te te temperature of a single room Peffer et al, 2011 Furtermore, te tuning for a PI controller is more intuitive tan for an MPC controller and simpler to implement 11725

5 Cape Town, Sout Africa August 24-29, 2014 Te fact tat ere te MPC controller performs worse tan te decoupled PI controller concerning reducing te influence of te temperature of one room on te oter could be due to tuning of te parameters or tat te MPC did not take into account any disturbances Neverteless, te perspective tat PI control wit a decoupling network can ave a performance comparable to tat of an MPC controller regarding tis coupling gives te cance to build on already used tecnology to improve temperature control Increasing te amount of rooms adjacent to eac oter, te structure of te decoupled PI controller will also get more comple It remains to be investigated ow tis will affect te controller performance Moreover, te a question is ow te performance of a decoupled PI controller compares to tat of an MPC in case of a more comple representation of a building, wit disturbances from weater conditions or solar radiation, and wen te control signals reac te saturation limits 6 CONCLUSION Te connection of two adjacent rooms troug a common wall introduces and interaction between te temperature dynamics of tese rooms A PI control strategy wit and witout a decoupling network and an MPC controller were used to control te temperature in te two rooms It was observed ow te cange of temperature in one of te rooms affects te cange of temperature in te oter room Te tree control strategies were compared wit respect to tis interaction It was found tat bot MPC control and PI control wit a decoupling network reduce te effect of a temperature cange in te first room on te temperature of te second room In tis simulation PI control wit a decoupling network lead to a smaller effect on te temperature of te second room tan MPC control E Gagnon, A Pomerleau, and A Desbiens Simplified, ideal or inverted decoupling? ISA Transactions, 37(4): , 1998 J Garrido, F Vázquez, F Morilla, and T Hägglund Practical advantages of inverted decoupling Proceedings of te Institution of Mecanical Engineers, Part I: Journal of Systems and Control Engineering, 225(7): , 2011 M Grant and S Boyd Grap implementations for nonsmoot conve programs In V Blondel, S Boyd, and H Kimura, editors, Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, pages Springer-Verlag Limited, 2008 ttp://stanfordedu/~boyd/grap_dcptml M Grant and S Boyd CVX: Matlab software for disciplined conve programming, version 20 beta ttp:// cvrcom/cv, September 2013 I Hazyuk, C Giaus, and D Penouet Optimal temperature control of intermittently eated buildings using model predictive control: Part ii control algoritm Building and Environment, 51: , 2012 Y Ma, A Kelman, A Daly, and F Borrelli Predictive control for energy efficient buildings wit termal storage: Modeling, stimulation, and eperiments Control Systems, IEEE, 32(1):44 64, 2012 J M Maciejowski and M Huzmezan Predictive control Springer, 1997 T Peffer, M Pritoni, A Meier, C Aragon, and D Perry How people use termostats in omes: A review Building and Environment, 46(12): , 2011 A Persson Energianvändning i bebyggelsen Kungliga ingenjörsvetenskapsakademien, 2002 T I Salsbury A survey of control tecnologies in te building automation industry In 16t IFAC World Congress, 2005 ACKNOWLEDGEMENTS Tis work was supported by te Swedis Researc Council troug te LCCC Linnaeus Center Te autors are members of te LCCC Linnaeus Center and te elliit Ecellence Center at Lund University REFERENCES K J Astrom and T Hagglund Advanced PID control Isa, 2006 K J Astrom and B Wittenmark Computer-Controlled systems: teory and design Prentice all, 1997 A I Dounis and C Caraiscos Advanced control systems engineering for energy and comfort management in a building environment - a review Renewable and Sustainable Energy Reviews, 13(6): , 2009 Council European Parliament Directive 2010/31/eu of te european parliament and of te council of 19 may 2010 on te energy performance of buildings Official Journal of te European Union, June 2010 F Felgner, S Agustina, R C Boigas, R Merz, and L Litz Simulation of termal building beaviour in modelica In Proceedings of te 2nd International Modelica Conference, volume 18, page 19 Oberfpaffenofen,

6 Cape Town, Sout Africa August 24-29, 2014 S u = Appendi A PREDICTION MATRICES S A = C I A I C I A 2 I C I A Hu I C I A Hp I C I B I C I A I + I B I Hu 1 C I B I S u = Hu C I B I H p 1 C I B I C I B I 0 0 C I A I + I B I C I B uc 0 0 Hu 1 C I Hu C I Hp 1 C I T 0 B I C I Ā + I B I C I B I B I Hu 1 C I Hp 2 BI CI B I C I 2 B I C I A I + I B I Hp Hu B I C I B I Appendi B SOME LATIN VOCABULARY 11727