- Email: [email protected]
Multiple fuzzy model-based temperature predictive control for HVAC systems
Information Sciences 169 (2005) 155–174 www.elsevier.com/locate/ins
Multiple fuzzy model-based temperature predictive control for HVAC systems Ming He a, Wen-Jian Cai a
b
a,*
, Shao-Yuan Li
b
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Institute of Automation, Shanghai JiaoTong University, Shanghai 200030, PR China
Received 4 May 2003; received in revised form 11 February 2004; accepted 11 February 2004
Abstract In this paper, a multiple model predictive control (MMPC) strategy based on Takagi–Sugeno (T–S) fuzzy models for temperature control of air-handling unit (AHU) in heating, ventilating, and air-conditioning (HVAC) systems is presented. The overall control system is constructed by a hierarchical two-level structure. The higher level is a fuzzy partition based on AHU operating range to schedule the fuzzy weights of local models in lower level, while the lower level is composed of a set of T–S models based on the relation of manipulated inputs and system outputs correspond to the higher level. Following this divide-and-conquer strategy, the complex nonlinear AHU system is divided into a set of T–S models through a fuzzy satisfactory clustering (FSC) methodology and the global system is a fuzzy integrated linear varying parameter (LPV) model. A hierarchical MMPC strategy is developed using parallel distribution compensation (PDC) method, in which different predictive controllers are designed for different T–S fuzzy rules and the global controller output is integrated by the local controller outputs through their fuzzy weights. Simulation and real process testing results show that the proposed MMPC approach is effective in HVAC system control applications. 2004 Elsevier Inc. All rights reserved. Abbreviations: AHU: Air-handling unit; CCU: Cooling coil unit; FOPDT: First order plus dead time; FSC: Fuzzy satisfactory clustering; GPC: Generalized predictive
*
Corresponding author. Tel.: +65-6790-6862; fax: +65-6793-3318. E-mail addresses: [email protected] (M. He), [email protected] (W.-J. Cai).
0020-0255/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.02.016
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control; HVAC: Heating, ventilating, and air-conditioning; LPV: Linear varying parameter; MMPC: Multiple model predictive control; MPC: Model predictive control; PDC: Parallel distribution compensation; PID: Proportional–integral–derivative; RMSE: Root mean square error; T–S model: Takagi–Sugeno model; VAV: Variable air volume Keywords: HVAC; AHU; Takagi–Sugeno fuzzy model; Fuzzy satisfactory clustering; Parallel distribution compensation; Multiple model predictive control
1. Introduction In heating, ventilating, and air-conditioning (HVAC) operation, air-handling unit (AHU) plays an essential role for supplying treated air with specified temperature to the conditioned space [1,2]. As there exist severe nonlinearity and time varying characteristic in heat exchange process, it is difficult to find a mathematic model to accurately describe the process over wide operating range. Therefore, maintaining a specified AHU off-coil temperature under different conditions poses a big challenge for practical engineers. So far, classical control techniques, such as on–off switching controller (thermostats) and proportional–integral–derivative (PID) controllers are still widely used in practice as they can be easily implemented, are low cost, and reliable in harsh field conditions. Recently, some complex control strategies based on the classical control concepts have been proposed in attempts to improve the system performance. Among those works, Salsbury [3] described a feedforward control scheme based on a simplified physical model as a supplement of the conventional PI feedback control. Kasahara et al. [4] proposed a robust PID control scheme to deal with the model uncertainty caused by changing characteristics of the plant. Bi and Cai et al. [5] developed an advanced auto-tuning PID controller for both temperature and pressure control. Classic control schemes commonly use the first order or second order plus time delay models to represent process dynamics. The performances of these control schemes are limited when applied to AHU process due to their inherent nonlinearity and time varying nature, especially when significant load disturbances occur. To overcome these drawbacks, applications of intelligent control to HVAC systems have drawn some interests [6–10]. Based on the fuzzy modeling, Sousa et al. [9] proposed a predictive control scheme through iterative numerical optimization. Ghiaus [10] demonstrated that the nonlinearity of the heat exchange process can be well described by a rather simple fuzzy scheme and showed that the fuzzy control resulted in better performance and eliminated the retuning process required by the classical PID controller. However, the variable airflow rates which account for much of the nonlinearity and time varying characteristics in variable air volume (VAV) scheme was not considered in the work.
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One possible way to cope with strongly nonlinearity and time varying characteristic of the process is to use multiple model approaches which have gained popularity in recent years [11–13]. Based on divide-and-conquer strategy, the whole complex system can be partitioned into a set of local subsystems and treated independently, the global control strategy is then determined by integrating local controllers using certain rules. The main benefit is that the linear control strategies such as linear model predictive control (MPC) [14,15] can be implemented to improve the system performance [9,16–19]. Among many modeling methods, fuzzy modeling technique where clustering algorithms are often used to determine the number of fuzzy rules based on the designer’s experience, has certain advantages in forming multiple models since it results smooth behavior across all operating regions and can approximate arbitrary functions [11,16,20–23]. In this paper, a multiple model predictive control (MMPC) is proposed for AHU temperature control. The main contributions of the work are: (1) By using the two-level hierarchical multi-model approach, the complex nonlinear system is divided into a set of local T–S [24] models and the overall system model is constructed by fuzzy integration which resulting a linear varying parameter (LPV) model. In this way, the problem of rule-explosion [25] in fuzzy application is alleviated by dividing single high-dimensional fuzzy set into a collection of low-dimensional fuzzy systems. (2) Fuzzy satisfactory clustering (FSC) algorithm [26] is employed to obtain a proper number of clusters for local T–S models where each cluster is represented by a linear fuzzy rule, which can be easily used to design a linear controller. (3) MPC method is used to design controllers for different models. The local nonlinear controller is the fuzzy weighted integration of linear ones by parallel distribution compensation (PDC) scheme [27]. The global controller output is then aggregated through fuzzy weight scheduler on the higher level. Simulation and testing results show that the proposed MMPC method can achieve good performance in both modeling and temperature control of AHU process.
2. System descriptions and multiple model scheme The schematic diagram of AHU in VAV system is shown as in Fig. 1, which consists of cooling coil, air dampers, fans, chilled water pumps and valves. There are two physical loops in the cooling coil unit (CCU): chilled water loop and air loop as shown in Fig. 2. Chilled water flows from CCU inlet to outlet forced by the chilled water pump and regulating valves with chilled water inlet temperature Tchwi and flow rate m_ chw respectively. Through heat transfer with on-coil air outside the cooling coil pipes, the temperature of the chilled water rises to Tchwo . Supply airflows from the air inlet to outlet of the cooling coil forced by the supply air fan. The dry-bulb temperature, web-bulb
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Exhaust Air
Return Air Return Fan
Control Damper
Filter Cooling Coil Mixed Air
Outside Air
Control Valve
Supply Air
Supply Fan (VSD)
Chilled Water
Fig. 1. Air handling unit.
Fig. 2. Schematic drawing of CCU.
temperature and airflow rate of the on-coil air are Tai , Taiwb and m_ a , respectively. Likewise, the off-coil dry-bulb and wet-bulb air temperatures descend to Tao and Taowb , through heat transfer with chilled water in the cooling coil pipes. The off-coil temperature Tao and chilled water flow rate m_ chw are the process output to be controlled and manipulated variables, respectively. The on-coil temperature of the chilled water is assumed to be constant, and the airflow rate m_ a varies in corresponding to cooling load demand of the conditioned space, these two variables are considered as disturbances to the process. Thus the output Tao can be described as Tao ¼ f ðm_ chw ; m_ a ; Tai ; Tchwi Þ
ð1Þ
where f is a nonlinear time varying function between the system output and the state variables.
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In steady-state, Eq. (1) can be explicitly expressed by [28] Q¼
c1 m_ ea e ðTai Tchwi Þ a 1 þ c2 m_m_chw
ð2Þ
and Q ¼ cchw m_ chw ðTchwo Tchwi Þ ¼ ca m_ a ðTai Tao Þ
ð3Þ
where Q is cooling load, cchw and ca are the specific heats of chilled water and air, c1 and c2 and e are parameters related to the mechanical properties of AHU system. Combine Eqs. (2) and (3), Tao can be described as Tao ¼ Tai
_ echw ðc1 =ca Þm_ e1 a m ðTai Tchwi Þ m_ chw þ c2 m_ ea
ð4Þ
As the transient response of the dynamics for air and chilled water loop are very difficult to model accurately, it can be approximated in a small region respectively, by a first order plus dead time (FOPDT) models [3,4] and given by Tao ðsÞ Kchw eLchw s ¼ m_ chw ðsÞ 1 þ Tchw s
ð5aÞ
Tao ðsÞ Ka eLa s ¼ m_ a ðsÞ 1 þ Ta s
ð5bÞ
where Kchw , Tchw , Lchw , Ka , Ta and La are the process gain, time constant and time delay for chilled water loop and air loop respectively, which varies along with the air and water flow rate changes: if the airflow rate or chilled water flow rate is high, the time constant and time delay will decrease, and vice versa. A hierarchical fuzzy model is effective in the case of the modeling target has many input variables, and the premise input variables have different dynamics. In AHU temperature control system, airflow rate and chilled water flow rate can be considered as the premise variables. Since the control objective is to maintain Tao at a constant temperature by adjusting the m_ chw , and its dynamic behavior is severely affected by m_ a , a hierarchical fuzzy model based on fuzzy partition of m_ a is proposed. The hierarchical two-level structure is shown in Fig. 3, where the higher level is a scheduler based on the fuzzy partition of airflow rate m_ a on its operating range and the lower lever is constructed by a set of T–S models. As manipulated input m_ chw and system output Tao in a particular region is approximated by a local linear T–S model, local controllers can be designed based on respective linear models and the overall control action is obtained by combining local control outputs through their fuzzy weights. In the higher level of this structure, the fuzzy partition of m_ a is based on the knowledge of the AHU process, the membership function based on L fuzzy partitions of m_ a is shown in Fig. 4.
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Fuzzy partition of m⋅ a Higher Level Scheduler
Weight
Lower Level T-S Models
Weight
Sub T-S Model
Sub T-S Model
Sub T-S Model
Fig. 3. Hierarchical multi-model structure.
Fig. 4. Membership function of m_ a in higher level.
3. Multi-model identification using fuzzy satisfactory clustering Corresponding to the higher level partition, there are L local T–S models in the lower level, where the fuzzy weight of each T–S model is determined by the fuzzy partition of m_ a . Consider a data set Z which is composed of the input– output data of AHU system. Following the hierarchical structure, there are L subsets Z1 ; . . . ; ZL with Z ¼ Z1 [ Z2 [ ZL , where each subset data is a data set corresponding to an operation range of the system and can be used for local T– S model identification. For jth subset Zj , j ¼ 1; . . . ; L, define a data pair zk ¼ ½uk ; yk T 2 Rdþ1 , k ¼ 1; . . . ; Nj , where uk is the generalized input vector, which combines system inputs and past outputs, yk is system output. Assume the jth subset Zj is divided into cj clusters, i.e. the jth local T–S model can be correspondingly characterized by cj rules fR1 ; R2 ; . . . ; Rcj g. GK [23] clustering algorithm can be used to finds partition matrix U ¼ ½lij c N and cluster centers V ¼ ½v1 ; . . . ; vc by minimizing the objective function,
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min J ðZ; V ; U Þ ¼
c X N X i¼1
161
lmi;j d 2 ðzj ; vi Þ
j¼1
where dðzj ; vi Þ is the distance of data vector zj from cluster prototype vi . To apply the proper number of clusters, FSC adopts GK algorithm is used to identify T–S model for jth subset described as follows: Algorithm 1 (T–S identification based on FSC). Step 1: Set the initial clusters number cj ¼ 2. Step 2: Using GK algorithm, with initial partition matrix, divide data set Zj into cj parts fA1 ; AP 2 ; . . . ; Acj g, and obtain the partition matrix cj U ¼ ½li;k cj Nj , where i¼1 li;k ¼ 1. Step 3: For each cluster, identify the consequent parameters using stable-state Kalman filter method [24]. For each different cluster, the local model is described as R1 R2
if ðuk ; yk Þ 2 A1
then y1 ¼ p0;1 þ p1;1 uk ð1Þ þ þ pd;1 uk ðdÞ
if ðuk ; yk Þ 2 A2
then y2 ¼ p0;2 þ p1;2 uk ð1Þ þ þ pd;2 uk ðdÞ ð6Þ
.. . Rcj
if ðuk ; yk Þ 2 Acj
then ycj ¼ p0;cj þ p1;cj uk ð1Þ þ þ pd;cj uk ðdÞ
Step 4: Compute the system output ^y corresponding to input zk , , c cj j X X ^y ¼ li;k yi li;k i¼1
ð7Þ
i¼1
~ , use the following equation to calculate l ~i correFor a new input u sponding to ith rule [29] ~i ð~ uÞ ¼ l
Pcj j¼1
1 2=ðm1Þ ~ ; vxi =DAxi u ~ ; vxj DAxi u
ð8Þ
where vxi denotes the projection of the ith cluster center vi onto gen~ ; vxi measures the distance of the new input eralized input space; DAxi u vector from the projection of the cluster center vxi ; m > 1 is a parameter that controls the fuzziness of clusters. Then the predicted output ~y can be calculated by Eq. (7). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi PNj ð^yk yk Þ2 , as the perStep 5: Use root mean square error, RMSE ¼ N1j k¼1 formance index to evaluate the modeling results. If the RMSE is less than the pre-specified number, the current cluster number is satisfied. Otherwise, go to Step 6.
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Step 6: Find a sample from the given data set, which is most different from the current cluster centers v1 vcj and make P it as a new center vcj þ1 . The dissimilarity is calculated by n ¼ arg minn 1 6 i;j 6 c ðlni lnj Þ. i6¼j
Step 7: Forming v1 vcj þ1 as the new initial cluster center, compute the updating initial partition matrix. Let cj ¼ cj þ 1, go to Step 2. The whole system output is then fuzzy aggregated by fuzzy weights determined by PL
j¼1
^y ¼ PL
xj ^yj
j¼1
ð9Þ
xj
where xj , j ¼ 1; . . . ; L is the membership degree of the jth local T–S model. 4. Multi-model predictive control The MPC scheme for individual model and the hierarchical MMPC structure for overall AHU temperature control system are shown in Figs. 5 and 6, respectively. The local generalized predictive control (GPC) algorithm is to determine optimal control strategy based on quadratic cost function in the next N steps through the Nu steps optimal control policies. The fuzzy weighted integration of local controllers’ outputs is calculated as the global manipulated input m_ chw of AHU process.
Fig. 5. Basic idea of predictive control.
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Higher Level
163
m⋅ a
Fuzzy Weight Scheduler
Lower Level Local T-S model/ Controller No.1
Weight1
Set Point
m⋅ chw
Local T-S model/ Controller No.L
Tao AHU Process
+ WeightL
Hierarchical MMPC
Fig. 6. Structure of MMPC for AHU temperature control system.
Sub-TS Model Scheduler
Input
Controller 1
Rule 1
Weight
+ ∆u j Controller cj
Rule cj
Weight
PDC Fig. 7. Local GPC scheme.
Based on PDC scheme shown as in Fig. 7, the jth T–S model composed of cj linear rules of Eq. (7) can be represented by a linear discrete-time CARIMA structure with the form Aðz1 ÞyðkÞ ¼ Bðz1 Þuðk 1Þ þ nðkÞ=D
ð10Þ
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where D ¼ 1 z1 is difference operator, yðkÞ, uðkÞ are output and control sequence of system. nðkÞ is a zero mean white noise. Aðz1 Þ, Bðz1 Þ are polynomials of z1 described by Aðz1 Þ ¼ 1 þ a1 z1 þ a2 z2 þ þ ana zna
ð11Þ
Bðz1 Þ ¼ b0 þ b1 z1 þ b2 z2 þ þ bnb znb multiplying both sides of Eq. (10) by D, Aðz1 ÞDyðkÞ ¼ Bðz1 ÞDuðk 1Þ þ nðkÞ
ð12Þ
Define quadratic cost function as J¼
N2 h Nu h i2 X i X 2 ^y ðk þ jÞ wðk þ jÞ þ kðjÞ Duðk þ j 1Þ j¼N1
ð13Þ
j¼1
where ^y ðk þ jÞ is an j-step ahead predicted output, wðk þ jÞ is the reference output, N1 and N2 are the minimums and maximum predictive horizons, Nu is control horizon, kðjÞ is the weight sequence. The j-step ahead prediction can be obtained as ^y ðk þ jÞ ¼ Gj ðz1 ÞDuðk þ j 1Þ þ Fj ðz1 ÞyðkÞ 1
1
1
1
ð14Þ
1
where Gj ðz Þ ¼ Ej ðz ÞBðz Þ, and Ej ðz Þ, Fj ðz Þ are polynomials of z1 satisfying the following Diophantine equation: 1 ¼ Ej ðz1 ÞDAðz1 Þ þ zj Fj ðz1 Þ with deg Ej ¼ j 1, deg Fj ¼ na . Let Duðk þ j 1Þ ¼ 0, (j > Nu ), the optimized control sequence can be obtained by optimizing the cost function of Eq. (13). The Eq. (14) can be represented as ^y ¼ G~ uþf where 3 3 2 ^y ðk þ 1Þ DuðkÞ 7 6 6 Duðk þ 1Þ 7 6 ^y ðk þ 2Þ 7 7 6 7 6 7 ~¼6 ^y ¼ 6 7 u .. .. 7 6 7 6 5 4 . . 5 4 ^y ðk þ N Þ Duðk þ N 1Þ 2 3 g0 0 0 6 g g0 0 7 6 1 7 G¼6 .. 7 .. .. 6 .. 7 4 . . 5 . . gN 1 gN 2 gN Nu 2
ð15Þ
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165
N
define w ¼ ½ wðk þ 1Þ wðk þ 2Þ wðk þ N Þ , the solution of the future incremental control vector ~ u can be obtained by oJ =o~u ¼ 0 ~ u ¼ ðGT G þ kIÞ1 GT ðw f Þ
ð16Þ
T with the corresponding cost function J ¼ ðG~u þ f wÞ ðG~u þ f wÞ þ k~uT u~. Notice that only DuðkÞ, the first element of vector ~u is applied and the procedure is repeated at the next sampling time. For jth obtained local T–S model, the GPC algorithm can be easily to be designed for each rule to get the incremental control signals Dui , i ¼ 1; 2; . . . ; cj . Based on the obtained cj incremental control signals, the jth local subsystem controller incremental output can be obtained by the aggregation through the PDC structure (Fig. 7) Pcj i¼1 li;k Dui Duj ¼ P ð17Þ cj i¼1 li;k
The whole incremental control signal is then aggregated according to the fuzzy weight determined by the higher level scheduler (Fig. 6) PL j¼1 xj Duj Du ¼ PL ð18Þ j¼1 xj The MMPC design procedure can be summarized as the following Algorithm: Algorithm 2 (The MMPC algorithm for AHU control). Step 1: Develop the data set composed by the input–output data of AHU system. Step 2: Divide the whole system into L T–S models based on the fuzzy partition of varying airflow rate on its operation range. Step 3: Identify T–S models using Algorithm 1. Design GPC controller (16) for each rule. Step 4: Measure the actual output off-coil air temperature and airflow rate; determine fuzzy weights xj , j ¼ 1; . . . ; L of local T–S models. Compute the membership degree li , i ¼ 1; . . . ; cj of each linear model belongs to the local T–S models. Step 5: Compute the control signal Dui of each MISO linear model and aggregate it through the membership degree by Eq. (17). Then calculate the whole incremental control Du signal by Eq. (18). Step 6: Calculate the system control output u ¼ u þ Du. Go back to Step 4 if the system is not terminated.
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5. Verification by simulation Before the real process testing, a simulation study is conducted to verify the proposed MMPC algorithm. Consider a heat exchanger model described by Dougherty and Cooper [13], where the nonlinear dynamic is approximated by three FOPDT models corresponding to its operating range as shown in Table 1. To verify the MMPC algorithm, the switching schedule of the model is organized as below. {10s-210s} Low level
{220s-420s}
10s
{430s-630s}
10s
High level
Middle level
A PI controller based on low level model is designed using Cohen–Coon method for comparison, the parameters Kc , Ti are determined as 1.1 and 0.9, respectively. The MMPC is designed according to the algorithm given in Section 4. Fig. 8 shows the performance of the MMPC and PI controllers under a square wave set point change. It is clear that a better performance is achieved through MMPC, whereas it is difficult to apply one set of PID controller parameters to obtain good performance for the whole operation range. Although tuning the controller at the point of highest process gain to avoid oscillation is possibly one way to overcome this problem, it may result undesired control performance under other operating conditions. Table 1 Heat exchange FOPDT parameters FOPDT model parameters
Low level
Middle level
High level
Kp Ts L
)0.3 0.9 0.8
)0.8 1.1 0.8
)1.6 1.2 0.9
Fig. 8. Controller performance: (a) PID, (b) MMPC. (For color see online version).
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Fig. 9. Pilot plant of centralized HVAC system. (For color see online version).
6. Real process testing A pilot centralize HVAC is shown as in Fig. 9. The system has three chillers, three zones with three AHUs, three cooling towers and flexible partitions up to twelve rooms. All motors (fans, pumps and compressors) are equipped with VSDs. The system is made very flexible to configure these three units to form different schemes. The cooling coils for the system are two rows with the dimension of 25 · 25 · 8 cm3 as shown in Fig. 10. The measurement signals for the experiment are the water and airflow rates, on-coil air dry-bulb/wet-bulb temperatures, CCU inlet and outlet water temperatures. The experiment is conducted under the following conditions: The chilled water supply temperature is fixed; the cooling load variation is achieved through the air and water flow rates. Fig. 11 shows the fitting result of model Eq. (2) with the parameters c1 ¼ 0:45, c2 ¼ 0:70, e ¼ 0:61. The nonlinear curve of the model surface based on the relation of Tao , m_ chw and m_ a is shown as in Fig. 12. The designed MMPC has the following features: • The higher level is divided into two divisions shown as in Fig. 13 which represents the airflow rate high and low. It is good enough for this application by compromising the control performance and computational burden. • The lower level T–S model has the form of ^y ðkT Þ ¼ f ðyðkT 1Þ; yðkT 2Þ; uðkT 1Þ; uðkT 2ÞÞ • Each local model has three clusters determined by Algorithm 1. Fig. 14
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Fig. 10. An AHU and a CCU of the pilot plant. (For color see online version).
Fig. 11. Fitting result of AHU model. (For color see online version).
Fig. 12. Nonlinear curve of the AHU model surface. (For color see online version).
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Fig. 13. Membership function of m_ a in higher level.
present the membership functions following the high level partition fitted to the projection of the fuzzy partitions onto the antecedent variables. The obtained T–S models are shown below High airflow rate: 1. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A1high Tao ðkT Þ ¼ 2:8982 þ 0:3301Tao ðkT 1Þ þ 0:4889Tao ðkT 2Þ þ 20:0475m_ chw ðkT 1Þ 20:2628m_ chw ðkT 2Þ 2. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A2high Tao ðkT Þ ¼ 1:6189 þ 1:2249Tao ðkT 1Þ 0:1233Tao ðkT 2Þ 5:7551m_ chw ðkT 1Þ þ 5:7448m_ chw ðkT 2Þ 3. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A3high Tao ðkT Þ ¼ 1:8119 þ 0:7492Tao ðkT 1Þ þ 0:1599Tao ðkT 2Þ 29:9554m_ chw ðkT 1Þ þ 28:2628m_ chw ðkT 2Þ Low airflow rate: 4. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A1low Tao ðkT Þ ¼ 4:0015 1:7326Tao ðkT 1Þ þ 2:0112Tao ðkT 2Þ þ 30:8250m_ chw ðkT 1Þ 33:0965m_ chw ðkT 2Þ 5. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A2low Tao ðkT Þ ¼ 3:6281 þ 2:6484Tao ðkT 1Þ 1:7402Tao ðkT 2Þ 14:4422m_ chw ðkT 1Þ þ 14:9606m_ chw ðkT 2Þ 6. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A3low Tao ðkT Þ ¼ 1:3872 þ 0:8126Tao ðkT 1Þ þ 0:11165Tao ðkT 2Þ 33:6744m_ chw ðkT 1Þ þ 32:2856m_ chw ðkT 2Þ
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Fig. 14. Membership functions for the inputs of local T–S models: (a) high airflow rate, (b) low airflow rate. (For color see online version).
Fig. 15 shows that the system nonlinearity can be well captured by the model with RMSE ¼ 0:1621.
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Fig. 15. AHU modeling result by multiple model approach. (For color see online version).
Table 2 GPC parameters for subsystems Controller parameters
N
Nu
k
High airflow rate
Rule 1 Rule 2 Rule 3
5 5 5
2 2 2
0.1 0.1 0.1
Low airflow rate
Rule 1 Rule 2 Rule 3
5 5 5
3 3 3
1 1 1
Corresponding to the lower level T–S models, GPC is designed by Eq. (16) for each rule. The tuning parameters for the controller are given in Table 2. Fig. 16 shows the GPC performance for each rule. Figs. 17 and 18 shows the performance of MMPC controllers with the set point 17 C and airflow rate changes from (0:07 ! 0:10 ! 0:05 ! 0:12) at (time ¼ 0, 300, 600, 900), respectively.
7. Conclusion A MMPC strategy for AHU temperature control of HVAC systems was developed in this paper. Based on the characteristics of AHU process, a twolevel hierarchical modeling and control scheme was proposed by using airflow rate in the higher level as decisive factor to divide the whole complex process into a set of local T–S models. FSC algorithm was employed to find a proper number of fuzzy rules for each T–S model and GPC algorithm was designed for the T–S models using the PDC structure. Compared with the conventional
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Fig. 16. GPC performance for each rule: (a) high airflow rate, (b) low airflow rate. (For color see online version).
single model approaches, the different operating conditions in AHU process can be well described by T–S models which are simple and more suitable for design linear controllers. Simulation and pilot plant testing results demonstrated that the designed MMPC can meet the control performance requirements at different operation points. Since other function blocks in HVAC systems have similar characteristics, the methodology developed in this paper can be easily modified and extended. The MMPC on cooling tower and chiller loops are currently under investigation and the results will be reported later.
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Fig. 17. MMPC control performance. (For color see online version).
Fig. 18. Airflow rate.
References [1] C.W. Carey, J.W. Mitchell, W.A. Beckman, Control of ice storage systems, ASHRAE Trans. 102 (1) (1995) 1345–1452. [2] J.M. House, T.F. Smith, Optimal control of building and HVAC systems, Proc. American Control Conf. 6 (1995) 4326–4330. [3] T.I. Salsbury, A temperature controller for VAV air-handing units based on simplified physical models, HVAC&R Res. 3 (3) (1998) 264–279. [4] M. Kasahara, T. Matsuba, et al., Design and tuning of robust PID controller for HVAC systems, ASHRAE Trans. 105 (2) (1999) 154–166.
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[5] Q. Bi, W.-J. Cai, Q.-G. Wang, et al., Advanced controller auto-tuning and its application in HVAC systems, Control Eng. Practice 8 (2000) 633–644. [6] J. Teeter, M.Y. Chow, Application of functional link neural network to HVAC thermal dynamic system identification, IEEE Trans. Industrial Electron. (45) (1998) 170–175. [7] Osman et al., Application of general regression neural network (GRNN) in HVAC process identification and control, ASHRAE Trans. 102 (1) (1996) 1147–1156. [8] X.F. Liu, A. Dexter, Fault-tolerant supervisory control of VAV air-conditioning systems, Energy Build. 33 (2001) 379–389. [9] J.M. Sousa, R. Babusla, H.B. Verbruggen, Fuzzy predictive control applied to an airconditioning system, Control Eng. Practice 10 (5) (1997) 1395–1406. [10] C. Ghiaus, Fuzzy model and control of a fan-coil, Energy Build. 33 (2001) 545–551. [11] R. Murray Smith, T.A. Johansen, Multiple Model Approaches to Nonlinear Modeling and Control, Taylor & Francis, 1997. [12] T.A. Johansen, B.A. Foss, Multiple model approaches to modeling and control (Editorial), Int. J. Control. 72 (7–8) (1999) 575. [13] D. Dougherty, D. Cooper, A practical multiple model adaptive strategy for single-loop MPC, Control Eng. Practice 11 (2003) 141–159. [14] D. Clarke, Advances in Model-Based Predictive Control, Oxford University Press, 1994. [15] E.F. Camacho, C. Bordons, Model Predictive Control, Springer, 1999. [16] J.M. Sousa, U. Kaymak, Fuzzy Decision Making in Modeling and Control, World Scientific, 2002. [17] J.M. Sousa, U. Kaymak, Model predictive control using fuzzy decision functions, IEEE Trans. Syst. Man Cybernet.––Part B: Cybernetics 31 (1) (2001) 54–65. [18] Y.L. Huang, H.H. Lou, et al., Fuzzy model predictive control, IEEE Trans. Fuzzy Syst. 8 (6) (2000) 665–678. [19] J.A. Roubos, S. Mollov, et al., Fuzzy model-based predictive control using Takagi–Sugeno models, Int. J. Approx. Reason. 22 (1999) 3–30. [20] K.M. Passino, S. Yurkovich, Fuzzy Control, Addison Wesley, 1995. [21] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, 1981. [22] I. Gath, A.B. Geva, Unsupervised optimal fuzzy clustering, IEEE Trans. Pattern Anal. Machine Intell. 7 (1989) 773–781. [23] D.E. Gustafson, W.C. Kessel, Fuzzy clustering with fuzzy covariance matrix, Proc. IEEE CDC (1979) 761–766. [24] T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modeling and control, IEEE Trans. Syst. Man Cybernet. 15 (1) (1985) 116–132. [25] G.V.S. Raju, J. Zhou, R.A. Kisner, Hierarchical fuzzy control, Int. J. Control 54 (5) (1991) 1201–1216. [26] N. Li, S.-Y. Li, Y.-G. Xi, Multi-model predictive control based on the Takagi–Sugeno fuzzy models: a case study, Inform. Sci., in press. Corrected Proof, Available online 16 December 2003. [27] H.O. Wang, K. Tanaka, M.F. Griffin, An approach to fuzzy control of nonlinear systems: stability and design issues, IEEE Trans. Fuzzy Syst. 4 (1) (1996) 14–23. [28] Y.-w. Wang, W.-J. Cai et al., A simplified modeling of cooling coils for control and optimization of HVAC systems, Energy Convers. Mgmt., in press. [29] B. Babuska, Fuzzy Modeling for Control, Kluwer Academic Publishers, Boston, 1998.
Multiple fuzzy model-based temperature predictive control for HVAC systems Ming He a, Wen-Jian Cai a
b
a,*
, Shao-Yuan Li
b
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Institute of Automation, Shanghai JiaoTong University, Shanghai 200030, PR China
Received 4 May 2003; received in revised form 11 February 2004; accepted 11 February 2004
Abstract In this paper, a multiple model predictive control (MMPC) strategy based on Takagi–Sugeno (T–S) fuzzy models for temperature control of air-handling unit (AHU) in heating, ventilating, and air-conditioning (HVAC) systems is presented. The overall control system is constructed by a hierarchical two-level structure. The higher level is a fuzzy partition based on AHU operating range to schedule the fuzzy weights of local models in lower level, while the lower level is composed of a set of T–S models based on the relation of manipulated inputs and system outputs correspond to the higher level. Following this divide-and-conquer strategy, the complex nonlinear AHU system is divided into a set of T–S models through a fuzzy satisfactory clustering (FSC) methodology and the global system is a fuzzy integrated linear varying parameter (LPV) model. A hierarchical MMPC strategy is developed using parallel distribution compensation (PDC) method, in which different predictive controllers are designed for different T–S fuzzy rules and the global controller output is integrated by the local controller outputs through their fuzzy weights. Simulation and real process testing results show that the proposed MMPC approach is effective in HVAC system control applications. 2004 Elsevier Inc. All rights reserved. Abbreviations: AHU: Air-handling unit; CCU: Cooling coil unit; FOPDT: First order plus dead time; FSC: Fuzzy satisfactory clustering; GPC: Generalized predictive
*
Corresponding author. Tel.: +65-6790-6862; fax: +65-6793-3318. E-mail addresses: [email protected] (M. He), [email protected] (W.-J. Cai).
0020-0255/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.02.016
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control; HVAC: Heating, ventilating, and air-conditioning; LPV: Linear varying parameter; MMPC: Multiple model predictive control; MPC: Model predictive control; PDC: Parallel distribution compensation; PID: Proportional–integral–derivative; RMSE: Root mean square error; T–S model: Takagi–Sugeno model; VAV: Variable air volume Keywords: HVAC; AHU; Takagi–Sugeno fuzzy model; Fuzzy satisfactory clustering; Parallel distribution compensation; Multiple model predictive control
1. Introduction In heating, ventilating, and air-conditioning (HVAC) operation, air-handling unit (AHU) plays an essential role for supplying treated air with specified temperature to the conditioned space [1,2]. As there exist severe nonlinearity and time varying characteristic in heat exchange process, it is difficult to find a mathematic model to accurately describe the process over wide operating range. Therefore, maintaining a specified AHU off-coil temperature under different conditions poses a big challenge for practical engineers. So far, classical control techniques, such as on–off switching controller (thermostats) and proportional–integral–derivative (PID) controllers are still widely used in practice as they can be easily implemented, are low cost, and reliable in harsh field conditions. Recently, some complex control strategies based on the classical control concepts have been proposed in attempts to improve the system performance. Among those works, Salsbury [3] described a feedforward control scheme based on a simplified physical model as a supplement of the conventional PI feedback control. Kasahara et al. [4] proposed a robust PID control scheme to deal with the model uncertainty caused by changing characteristics of the plant. Bi and Cai et al. [5] developed an advanced auto-tuning PID controller for both temperature and pressure control. Classic control schemes commonly use the first order or second order plus time delay models to represent process dynamics. The performances of these control schemes are limited when applied to AHU process due to their inherent nonlinearity and time varying nature, especially when significant load disturbances occur. To overcome these drawbacks, applications of intelligent control to HVAC systems have drawn some interests [6–10]. Based on the fuzzy modeling, Sousa et al. [9] proposed a predictive control scheme through iterative numerical optimization. Ghiaus [10] demonstrated that the nonlinearity of the heat exchange process can be well described by a rather simple fuzzy scheme and showed that the fuzzy control resulted in better performance and eliminated the retuning process required by the classical PID controller. However, the variable airflow rates which account for much of the nonlinearity and time varying characteristics in variable air volume (VAV) scheme was not considered in the work.
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One possible way to cope with strongly nonlinearity and time varying characteristic of the process is to use multiple model approaches which have gained popularity in recent years [11–13]. Based on divide-and-conquer strategy, the whole complex system can be partitioned into a set of local subsystems and treated independently, the global control strategy is then determined by integrating local controllers using certain rules. The main benefit is that the linear control strategies such as linear model predictive control (MPC) [14,15] can be implemented to improve the system performance [9,16–19]. Among many modeling methods, fuzzy modeling technique where clustering algorithms are often used to determine the number of fuzzy rules based on the designer’s experience, has certain advantages in forming multiple models since it results smooth behavior across all operating regions and can approximate arbitrary functions [11,16,20–23]. In this paper, a multiple model predictive control (MMPC) is proposed for AHU temperature control. The main contributions of the work are: (1) By using the two-level hierarchical multi-model approach, the complex nonlinear system is divided into a set of local T–S [24] models and the overall system model is constructed by fuzzy integration which resulting a linear varying parameter (LPV) model. In this way, the problem of rule-explosion [25] in fuzzy application is alleviated by dividing single high-dimensional fuzzy set into a collection of low-dimensional fuzzy systems. (2) Fuzzy satisfactory clustering (FSC) algorithm [26] is employed to obtain a proper number of clusters for local T–S models where each cluster is represented by a linear fuzzy rule, which can be easily used to design a linear controller. (3) MPC method is used to design controllers for different models. The local nonlinear controller is the fuzzy weighted integration of linear ones by parallel distribution compensation (PDC) scheme [27]. The global controller output is then aggregated through fuzzy weight scheduler on the higher level. Simulation and testing results show that the proposed MMPC method can achieve good performance in both modeling and temperature control of AHU process.
2. System descriptions and multiple model scheme The schematic diagram of AHU in VAV system is shown as in Fig. 1, which consists of cooling coil, air dampers, fans, chilled water pumps and valves. There are two physical loops in the cooling coil unit (CCU): chilled water loop and air loop as shown in Fig. 2. Chilled water flows from CCU inlet to outlet forced by the chilled water pump and regulating valves with chilled water inlet temperature Tchwi and flow rate m_ chw respectively. Through heat transfer with on-coil air outside the cooling coil pipes, the temperature of the chilled water rises to Tchwo . Supply airflows from the air inlet to outlet of the cooling coil forced by the supply air fan. The dry-bulb temperature, web-bulb
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Exhaust Air
Return Air Return Fan
Control Damper
Filter Cooling Coil Mixed Air
Outside Air
Control Valve
Supply Air
Supply Fan (VSD)
Chilled Water
Fig. 1. Air handling unit.
Fig. 2. Schematic drawing of CCU.
temperature and airflow rate of the on-coil air are Tai , Taiwb and m_ a , respectively. Likewise, the off-coil dry-bulb and wet-bulb air temperatures descend to Tao and Taowb , through heat transfer with chilled water in the cooling coil pipes. The off-coil temperature Tao and chilled water flow rate m_ chw are the process output to be controlled and manipulated variables, respectively. The on-coil temperature of the chilled water is assumed to be constant, and the airflow rate m_ a varies in corresponding to cooling load demand of the conditioned space, these two variables are considered as disturbances to the process. Thus the output Tao can be described as Tao ¼ f ðm_ chw ; m_ a ; Tai ; Tchwi Þ
ð1Þ
where f is a nonlinear time varying function between the system output and the state variables.
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In steady-state, Eq. (1) can be explicitly expressed by [28] Q¼
c1 m_ ea e ðTai Tchwi Þ a 1 þ c2 m_m_chw
ð2Þ
and Q ¼ cchw m_ chw ðTchwo Tchwi Þ ¼ ca m_ a ðTai Tao Þ
ð3Þ
where Q is cooling load, cchw and ca are the specific heats of chilled water and air, c1 and c2 and e are parameters related to the mechanical properties of AHU system. Combine Eqs. (2) and (3), Tao can be described as Tao ¼ Tai
_ echw ðc1 =ca Þm_ e1 a m ðTai Tchwi Þ m_ chw þ c2 m_ ea
ð4Þ
As the transient response of the dynamics for air and chilled water loop are very difficult to model accurately, it can be approximated in a small region respectively, by a first order plus dead time (FOPDT) models [3,4] and given by Tao ðsÞ Kchw eLchw s ¼ m_ chw ðsÞ 1 þ Tchw s
ð5aÞ
Tao ðsÞ Ka eLa s ¼ m_ a ðsÞ 1 þ Ta s
ð5bÞ
where Kchw , Tchw , Lchw , Ka , Ta and La are the process gain, time constant and time delay for chilled water loop and air loop respectively, which varies along with the air and water flow rate changes: if the airflow rate or chilled water flow rate is high, the time constant and time delay will decrease, and vice versa. A hierarchical fuzzy model is effective in the case of the modeling target has many input variables, and the premise input variables have different dynamics. In AHU temperature control system, airflow rate and chilled water flow rate can be considered as the premise variables. Since the control objective is to maintain Tao at a constant temperature by adjusting the m_ chw , and its dynamic behavior is severely affected by m_ a , a hierarchical fuzzy model based on fuzzy partition of m_ a is proposed. The hierarchical two-level structure is shown in Fig. 3, where the higher level is a scheduler based on the fuzzy partition of airflow rate m_ a on its operating range and the lower lever is constructed by a set of T–S models. As manipulated input m_ chw and system output Tao in a particular region is approximated by a local linear T–S model, local controllers can be designed based on respective linear models and the overall control action is obtained by combining local control outputs through their fuzzy weights. In the higher level of this structure, the fuzzy partition of m_ a is based on the knowledge of the AHU process, the membership function based on L fuzzy partitions of m_ a is shown in Fig. 4.
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Fuzzy partition of m⋅ a Higher Level Scheduler
Weight
Lower Level T-S Models
Weight
Sub T-S Model
Sub T-S Model
Sub T-S Model
Fig. 3. Hierarchical multi-model structure.
Fig. 4. Membership function of m_ a in higher level.
3. Multi-model identification using fuzzy satisfactory clustering Corresponding to the higher level partition, there are L local T–S models in the lower level, where the fuzzy weight of each T–S model is determined by the fuzzy partition of m_ a . Consider a data set Z which is composed of the input– output data of AHU system. Following the hierarchical structure, there are L subsets Z1 ; . . . ; ZL with Z ¼ Z1 [ Z2 [ ZL , where each subset data is a data set corresponding to an operation range of the system and can be used for local T– S model identification. For jth subset Zj , j ¼ 1; . . . ; L, define a data pair zk ¼ ½uk ; yk T 2 Rdþ1 , k ¼ 1; . . . ; Nj , where uk is the generalized input vector, which combines system inputs and past outputs, yk is system output. Assume the jth subset Zj is divided into cj clusters, i.e. the jth local T–S model can be correspondingly characterized by cj rules fR1 ; R2 ; . . . ; Rcj g. GK [23] clustering algorithm can be used to finds partition matrix U ¼ ½lij c N and cluster centers V ¼ ½v1 ; . . . ; vc by minimizing the objective function,
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min J ðZ; V ; U Þ ¼
c X N X i¼1
161
lmi;j d 2 ðzj ; vi Þ
j¼1
where dðzj ; vi Þ is the distance of data vector zj from cluster prototype vi . To apply the proper number of clusters, FSC adopts GK algorithm is used to identify T–S model for jth subset described as follows: Algorithm 1 (T–S identification based on FSC). Step 1: Set the initial clusters number cj ¼ 2. Step 2: Using GK algorithm, with initial partition matrix, divide data set Zj into cj parts fA1 ; AP 2 ; . . . ; Acj g, and obtain the partition matrix cj U ¼ ½li;k cj Nj , where i¼1 li;k ¼ 1. Step 3: For each cluster, identify the consequent parameters using stable-state Kalman filter method [24]. For each different cluster, the local model is described as R1 R2
if ðuk ; yk Þ 2 A1
then y1 ¼ p0;1 þ p1;1 uk ð1Þ þ þ pd;1 uk ðdÞ
if ðuk ; yk Þ 2 A2
then y2 ¼ p0;2 þ p1;2 uk ð1Þ þ þ pd;2 uk ðdÞ ð6Þ
.. . Rcj
if ðuk ; yk Þ 2 Acj
then ycj ¼ p0;cj þ p1;cj uk ð1Þ þ þ pd;cj uk ðdÞ
Step 4: Compute the system output ^y corresponding to input zk , , c cj j X X ^y ¼ li;k yi li;k i¼1
ð7Þ
i¼1
~ , use the following equation to calculate l ~i correFor a new input u sponding to ith rule [29] ~i ð~ uÞ ¼ l
Pcj j¼1
1 2=ðm1Þ ~ ; vxi =DAxi u ~ ; vxj DAxi u
ð8Þ
where vxi denotes the projection of the ith cluster center vi onto gen~ ; vxi measures the distance of the new input eralized input space; DAxi u vector from the projection of the cluster center vxi ; m > 1 is a parameter that controls the fuzziness of clusters. Then the predicted output ~y can be calculated by Eq. (7). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi PNj ð^yk yk Þ2 , as the perStep 5: Use root mean square error, RMSE ¼ N1j k¼1 formance index to evaluate the modeling results. If the RMSE is less than the pre-specified number, the current cluster number is satisfied. Otherwise, go to Step 6.
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Step 6: Find a sample from the given data set, which is most different from the current cluster centers v1 vcj and make P it as a new center vcj þ1 . The dissimilarity is calculated by n ¼ arg minn 1 6 i;j 6 c ðlni lnj Þ. i6¼j
Step 7: Forming v1 vcj þ1 as the new initial cluster center, compute the updating initial partition matrix. Let cj ¼ cj þ 1, go to Step 2. The whole system output is then fuzzy aggregated by fuzzy weights determined by PL
j¼1
^y ¼ PL
xj ^yj
j¼1
ð9Þ
xj
where xj , j ¼ 1; . . . ; L is the membership degree of the jth local T–S model. 4. Multi-model predictive control The MPC scheme for individual model and the hierarchical MMPC structure for overall AHU temperature control system are shown in Figs. 5 and 6, respectively. The local generalized predictive control (GPC) algorithm is to determine optimal control strategy based on quadratic cost function in the next N steps through the Nu steps optimal control policies. The fuzzy weighted integration of local controllers’ outputs is calculated as the global manipulated input m_ chw of AHU process.
Fig. 5. Basic idea of predictive control.
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Higher Level
163
m⋅ a
Fuzzy Weight Scheduler
Lower Level Local T-S model/ Controller No.1
Weight1
Set Point
m⋅ chw
Local T-S model/ Controller No.L
Tao AHU Process
+ WeightL
Hierarchical MMPC
Fig. 6. Structure of MMPC for AHU temperature control system.
Sub-TS Model Scheduler
Input
Controller 1
Rule 1
Weight
+ ∆u j Controller cj
Rule cj
Weight
PDC Fig. 7. Local GPC scheme.
Based on PDC scheme shown as in Fig. 7, the jth T–S model composed of cj linear rules of Eq. (7) can be represented by a linear discrete-time CARIMA structure with the form Aðz1 ÞyðkÞ ¼ Bðz1 Þuðk 1Þ þ nðkÞ=D
ð10Þ
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where D ¼ 1 z1 is difference operator, yðkÞ, uðkÞ are output and control sequence of system. nðkÞ is a zero mean white noise. Aðz1 Þ, Bðz1 Þ are polynomials of z1 described by Aðz1 Þ ¼ 1 þ a1 z1 þ a2 z2 þ þ ana zna
ð11Þ
Bðz1 Þ ¼ b0 þ b1 z1 þ b2 z2 þ þ bnb znb multiplying both sides of Eq. (10) by D, Aðz1 ÞDyðkÞ ¼ Bðz1 ÞDuðk 1Þ þ nðkÞ
ð12Þ
Define quadratic cost function as J¼
N2 h Nu h i2 X i X 2 ^y ðk þ jÞ wðk þ jÞ þ kðjÞ Duðk þ j 1Þ j¼N1
ð13Þ
j¼1
where ^y ðk þ jÞ is an j-step ahead predicted output, wðk þ jÞ is the reference output, N1 and N2 are the minimums and maximum predictive horizons, Nu is control horizon, kðjÞ is the weight sequence. The j-step ahead prediction can be obtained as ^y ðk þ jÞ ¼ Gj ðz1 ÞDuðk þ j 1Þ þ Fj ðz1 ÞyðkÞ 1
1
1
1
ð14Þ
1
where Gj ðz Þ ¼ Ej ðz ÞBðz Þ, and Ej ðz Þ, Fj ðz Þ are polynomials of z1 satisfying the following Diophantine equation: 1 ¼ Ej ðz1 ÞDAðz1 Þ þ zj Fj ðz1 Þ with deg Ej ¼ j 1, deg Fj ¼ na . Let Duðk þ j 1Þ ¼ 0, (j > Nu ), the optimized control sequence can be obtained by optimizing the cost function of Eq. (13). The Eq. (14) can be represented as ^y ¼ G~ uþf where 3 3 2 ^y ðk þ 1Þ DuðkÞ 7 6 6 Duðk þ 1Þ 7 6 ^y ðk þ 2Þ 7 7 6 7 6 7 ~¼6 ^y ¼ 6 7 u .. .. 7 6 7 6 5 4 . . 5 4 ^y ðk þ N Þ Duðk þ N 1Þ 2 3 g0 0 0 6 g g0 0 7 6 1 7 G¼6 .. 7 .. .. 6 .. 7 4 . . 5 . . gN 1 gN 2 gN Nu 2
ð15Þ
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165
N
define w ¼ ½ wðk þ 1Þ wðk þ 2Þ wðk þ N Þ , the solution of the future incremental control vector ~ u can be obtained by oJ =o~u ¼ 0 ~ u ¼ ðGT G þ kIÞ1 GT ðw f Þ
ð16Þ
T with the corresponding cost function J ¼ ðG~u þ f wÞ ðG~u þ f wÞ þ k~uT u~. Notice that only DuðkÞ, the first element of vector ~u is applied and the procedure is repeated at the next sampling time. For jth obtained local T–S model, the GPC algorithm can be easily to be designed for each rule to get the incremental control signals Dui , i ¼ 1; 2; . . . ; cj . Based on the obtained cj incremental control signals, the jth local subsystem controller incremental output can be obtained by the aggregation through the PDC structure (Fig. 7) Pcj i¼1 li;k Dui Duj ¼ P ð17Þ cj i¼1 li;k
The whole incremental control signal is then aggregated according to the fuzzy weight determined by the higher level scheduler (Fig. 6) PL j¼1 xj Duj Du ¼ PL ð18Þ j¼1 xj The MMPC design procedure can be summarized as the following Algorithm: Algorithm 2 (The MMPC algorithm for AHU control). Step 1: Develop the data set composed by the input–output data of AHU system. Step 2: Divide the whole system into L T–S models based on the fuzzy partition of varying airflow rate on its operation range. Step 3: Identify T–S models using Algorithm 1. Design GPC controller (16) for each rule. Step 4: Measure the actual output off-coil air temperature and airflow rate; determine fuzzy weights xj , j ¼ 1; . . . ; L of local T–S models. Compute the membership degree li , i ¼ 1; . . . ; cj of each linear model belongs to the local T–S models. Step 5: Compute the control signal Dui of each MISO linear model and aggregate it through the membership degree by Eq. (17). Then calculate the whole incremental control Du signal by Eq. (18). Step 6: Calculate the system control output u ¼ u þ Du. Go back to Step 4 if the system is not terminated.
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5. Verification by simulation Before the real process testing, a simulation study is conducted to verify the proposed MMPC algorithm. Consider a heat exchanger model described by Dougherty and Cooper [13], where the nonlinear dynamic is approximated by three FOPDT models corresponding to its operating range as shown in Table 1. To verify the MMPC algorithm, the switching schedule of the model is organized as below. {10s-210s} Low level
{220s-420s}
10s
{430s-630s}
10s
High level
Middle level
A PI controller based on low level model is designed using Cohen–Coon method for comparison, the parameters Kc , Ti are determined as 1.1 and 0.9, respectively. The MMPC is designed according to the algorithm given in Section 4. Fig. 8 shows the performance of the MMPC and PI controllers under a square wave set point change. It is clear that a better performance is achieved through MMPC, whereas it is difficult to apply one set of PID controller parameters to obtain good performance for the whole operation range. Although tuning the controller at the point of highest process gain to avoid oscillation is possibly one way to overcome this problem, it may result undesired control performance under other operating conditions. Table 1 Heat exchange FOPDT parameters FOPDT model parameters
Low level
Middle level
High level
Kp Ts L
)0.3 0.9 0.8
)0.8 1.1 0.8
)1.6 1.2 0.9
Fig. 8. Controller performance: (a) PID, (b) MMPC. (For color see online version).
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Fig. 9. Pilot plant of centralized HVAC system. (For color see online version).
6. Real process testing A pilot centralize HVAC is shown as in Fig. 9. The system has three chillers, three zones with three AHUs, three cooling towers and flexible partitions up to twelve rooms. All motors (fans, pumps and compressors) are equipped with VSDs. The system is made very flexible to configure these three units to form different schemes. The cooling coils for the system are two rows with the dimension of 25 · 25 · 8 cm3 as shown in Fig. 10. The measurement signals for the experiment are the water and airflow rates, on-coil air dry-bulb/wet-bulb temperatures, CCU inlet and outlet water temperatures. The experiment is conducted under the following conditions: The chilled water supply temperature is fixed; the cooling load variation is achieved through the air and water flow rates. Fig. 11 shows the fitting result of model Eq. (2) with the parameters c1 ¼ 0:45, c2 ¼ 0:70, e ¼ 0:61. The nonlinear curve of the model surface based on the relation of Tao , m_ chw and m_ a is shown as in Fig. 12. The designed MMPC has the following features: • The higher level is divided into two divisions shown as in Fig. 13 which represents the airflow rate high and low. It is good enough for this application by compromising the control performance and computational burden. • The lower level T–S model has the form of ^y ðkT Þ ¼ f ðyðkT 1Þ; yðkT 2Þ; uðkT 1Þ; uðkT 2ÞÞ • Each local model has three clusters determined by Algorithm 1. Fig. 14
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Fig. 10. An AHU and a CCU of the pilot plant. (For color see online version).
Fig. 11. Fitting result of AHU model. (For color see online version).
Fig. 12. Nonlinear curve of the AHU model surface. (For color see online version).
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Fig. 13. Membership function of m_ a in higher level.
present the membership functions following the high level partition fitted to the projection of the fuzzy partitions onto the antecedent variables. The obtained T–S models are shown below High airflow rate: 1. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A1high Tao ðkT Þ ¼ 2:8982 þ 0:3301Tao ðkT 1Þ þ 0:4889Tao ðkT 2Þ þ 20:0475m_ chw ðkT 1Þ 20:2628m_ chw ðkT 2Þ 2. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A2high Tao ðkT Þ ¼ 1:6189 þ 1:2249Tao ðkT 1Þ 0:1233Tao ðkT 2Þ 5:7551m_ chw ðkT 1Þ þ 5:7448m_ chw ðkT 2Þ 3. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A3high Tao ðkT Þ ¼ 1:8119 þ 0:7492Tao ðkT 1Þ þ 0:1599Tao ðkT 2Þ 29:9554m_ chw ðkT 1Þ þ 28:2628m_ chw ðkT 2Þ Low airflow rate: 4. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A1low Tao ðkT Þ ¼ 4:0015 1:7326Tao ðkT 1Þ þ 2:0112Tao ðkT 2Þ þ 30:8250m_ chw ðkT 1Þ 33:0965m_ chw ðkT 2Þ 5. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A2low Tao ðkT Þ ¼ 3:6281 þ 2:6484Tao ðkT 1Þ 1:7402Tao ðkT 2Þ 14:4422m_ chw ðkT 1Þ þ 14:9606m_ chw ðkT 2Þ 6. If ðTao ðkT 1Þ; Tao ðkT 2Þ; m_ chw ðkT 1Þ; m_ chw ðkT 2ÞÞ 2 A3low Tao ðkT Þ ¼ 1:3872 þ 0:8126Tao ðkT 1Þ þ 0:11165Tao ðkT 2Þ 33:6744m_ chw ðkT 1Þ þ 32:2856m_ chw ðkT 2Þ
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Fig. 14. Membership functions for the inputs of local T–S models: (a) high airflow rate, (b) low airflow rate. (For color see online version).
Fig. 15 shows that the system nonlinearity can be well captured by the model with RMSE ¼ 0:1621.
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Fig. 15. AHU modeling result by multiple model approach. (For color see online version).
Table 2 GPC parameters for subsystems Controller parameters
N
Nu
k
High airflow rate
Rule 1 Rule 2 Rule 3
5 5 5
2 2 2
0.1 0.1 0.1
Low airflow rate
Rule 1 Rule 2 Rule 3
5 5 5
3 3 3
1 1 1
Corresponding to the lower level T–S models, GPC is designed by Eq. (16) for each rule. The tuning parameters for the controller are given in Table 2. Fig. 16 shows the GPC performance for each rule. Figs. 17 and 18 shows the performance of MMPC controllers with the set point 17 C and airflow rate changes from (0:07 ! 0:10 ! 0:05 ! 0:12) at (time ¼ 0, 300, 600, 900), respectively.
7. Conclusion A MMPC strategy for AHU temperature control of HVAC systems was developed in this paper. Based on the characteristics of AHU process, a twolevel hierarchical modeling and control scheme was proposed by using airflow rate in the higher level as decisive factor to divide the whole complex process into a set of local T–S models. FSC algorithm was employed to find a proper number of fuzzy rules for each T–S model and GPC algorithm was designed for the T–S models using the PDC structure. Compared with the conventional
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Fig. 16. GPC performance for each rule: (a) high airflow rate, (b) low airflow rate. (For color see online version).
single model approaches, the different operating conditions in AHU process can be well described by T–S models which are simple and more suitable for design linear controllers. Simulation and pilot plant testing results demonstrated that the designed MMPC can meet the control performance requirements at different operation points. Since other function blocks in HVAC systems have similar characteristics, the methodology developed in this paper can be easily modified and extended. The MMPC on cooling tower and chiller loops are currently under investigation and the results will be reported later.
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Fig. 17. MMPC control performance. (For color see online version).
Fig. 18. Airflow rate.
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