Distributed Model Predictive Control Methods For Improving Transient Response Of Automated Irrigation Channels

Transcription

1 Dstrbuted Model Predctve Control Methods For Improvng Transent Response Of Automated Irrgaton Channels Al Khodabandehlou, Alreza Farhad and Al Parsa Abstract Ths paper presents dstrbuted model predctve control methods for mprovng the transent response of feedback-controlled rrgaton channels by managng the water-level reference set ponts across the rrgaton season. The mplemented dstrbuted model predctve control methods explot a two-level and sngle-level archtectures for communcaton. For an automated rrgaton channel, the satsfactory performance of the method wth two-level archtecture for communcaton n mprovng the transent response of automated rrgaton channel s llustrated usng computer smulaton and compared wth the performance of the dstrbuted model predctve control method that explots a sngle-level archtecture for communcaton. It s llustrated that the dstrbuted model predctve control method that explots a two-level archtecture for communcaton has a better performance n mprovng the transent response by better managng communcaton overhead. I. INTRODUCTION An automated rrgaton channel s a system of water transfer for agrcultural purposes that conssts of several pools cascaded by flume gates (see Fg. 1). Each flume gate ncludes an overshot gate, a modem for wreless communcaton, sensors, actuators and a processng devce for makng decson. The adjustment of the flow over upstream gates regulate the downstream water level n each pool under a downstream control. Ths leads to adequate releasng water from reservor n the response to changes n the water levels of pools from the reference set ponts; and hence, near to demand supply that avods water transmsson lost due to oversupply. As the rrgaton pools are very often long, delays for respondng to the changes run the performance of automated rrgaton channel and results n upstream transent error propagaton and amplfcaton phenomenon. Ths results n actuators saturaton and floodng n long automated rrgaton channels. To mtgate these drawbacks, n [1] a supervsory controller s proposed that explots the nformaton avalable from automated rrgaton channel and properly manages the reference set ponts for local A. Khodabandehlou was M.Sc. student n the Department of Electrcal Engneerng, Sharf Unversty of Technology. A. Farhad s Assstant Professor n the Department of Electrcal Engneerng, Sharf Unversty of Technology, Tehran, Iran. Emal: afarhad@sharf.edu. A. Parsa s currently a Ph.D student n the Department of Electrcal Engneerng, Sharf Unversty of Technology. Ths work was supported by the research offce of Sharf Unversty of Technology. controllers of dstrbuted flume gates. To acheve ths goal, the supervsory controller solves a constraned lnear quadratc optmal control problem. But, n long rrgaton channels, the total number of constrants and decson varables are very large. Therefore, to solve ths optmal control problem n real tme usng closed loop control strategy, the recedng horzon dea must be used. However, the computaton overhead (.e., the tme spent for computng the optmal soluton) usng centralzed model predctve control methods, as used n [1], at each recedng horzon wll not be practcal for long rrgaton channels because of large number of constrants and decson varables. For large rrgaton channels, Dstrbuted Model Predctve Control (DMPC) methods [2], [4] must be used that have relatvely much smaller computaton overhead. Towards overcomng ths computatonal scalablty problem, the dstrbuted model predctve control method of [2] can be used. Ths method solves constraned lnear quadratc optmal control problems usng a sngle-level archtecture for communcaton by explotng the computatonal power often avalable at each sub-system n the network. Ths dstrbuted control method conssts of two steps: 1) Intalzaton and 2) Iterated (parallel) computaton and communcaton for exchangng updates of components of the overall decson varable between dstrbuted computng resources. To provde scope for managng the communcaton overheads, motvated by the ntal dea presented n [3], n [4] a dstrbuted model predctve control method wth herarchcal (two-level) archtecture for communcaton and a three-step algorthm ncludng an extra outer terate step are presented for solvng constraned lnear quadratc optmal control problems. Hence, ths method seems to be more sutable to overcome the above computatonal scalablty problem. In Ths method, dstrbuted decson makers are grouped nto q dsjont neghborhoods. Exchange of nformaton between decson makers wthn a neghborhood occurs after each update, whereas the exchange of nformaton between neghborhoods s lmted to be less frequent. Wthn a neghborhood, each decson maker frequently updates ts local component of the overall decson varable by solvng an optmzaton problem of reduced sze. The updated value s then communcated to all other neghborng decson makers. Ths ntra-neghborhood update and communcaton s referred as an nner terate. In addton to nner terates, updates of decson varables from other neghborhoods are receved perodcally.

2 These are referred to as outer terates. Between outer terates, dstrbuted decson makers contnue to compute and refne the local approxmaton of the optmal soluton, wth fxed values for decson varables from outsde the neghborhood. In ths paper, the dstrbuted model predctve control methods of [2] and [4] are used to solve the constraned lnear quadratc optmal control problem assocated wth the supervsory controller of [1] for automated rrgaton channels. For an automated rrgaton channel, the satsfactory performance of the method wth two-level archtecture for communcaton [4] n mprovng the transent response of automated rrgaton channel and mtgatng the upstream transent error propagaton and amplfcaton phenomenon s llustrated usng computer smulaton and compared wth the performance of the dstrbuted model predctve control method that explots a sngle-level archtecture for communcaton [2]. It s llustrated that the dstrbuted model predctve control method that explots a two-level archtecture for communcaton has a better performance by better managng communcaton overhead. The paper s organzed as follows: Secton II brefly descrbes the dstrbuted model predctve control method of [4] and also the method of [2] (as the method of [2] s a specal case of the method of [4]). Secton III s devoted to modelng automated rrgaton channels, explanng upstream transent error propagaton and amplfcaton phenomenon n automated rrgaton channels and the formulaton of the constraned lnear quadratc optmal control problem to be solved for mtgatng ths phenomenon. Secton IV s devoted to smulaton study and applcatons of dstrbuted model predctve control methods of [4] and [2] n automated rrgaton channels. In Secton V the paper s concluded by summarzng the paper. II. DISTRIBUTED MPC WITH HIERARCHICAL ARCHITECTURE FOR COMMUNICATION The dstrbuted model predctve controller of [4] s concerned wth n nteractng sub-systems: S 1, S 2,..., S n each equpped wth a decson maker wth lmted computatonal power for solvng the followng optmzaton problem n a dstrbuted fashon at each recedng horzon: mn (u 1,...,u n) J(g, u 1,..., u n ), u U, }. (1) Here, g s a collecton of known vectors, J 0 s a fnte-horzon quadratc cost functonal of decson varables wth horzon length N << L, for each = 1, 2,..., n, u R Nm s the decson varable assocated wth sub-system S and U s a closed convex subset of the Eucldean space R Nm that ncludes zero vector. For the smplcty of presentaton, wthout loss of generalty, the dependency of the cost functonal J on g s dropped. Throughout, t s assumed that decson makers have knowledge of known parameters descrbed by g and also the expresson for the cost functonal J n (1) at each recedng horzon. To manage the communcaton overhead, the dstrbuted controller of [4] uses a twolevel archtecture for exchangng nformaton between dstrbuted decson makers. Ths communcaton archtecture nvolves a collecton of dsjont neghborhoods of sub-systems. In each neghborhood at least one decson maker s selected as the neghborhood cluster head such that all the sub-systems of the neghborhood and also all the sub-systems of the nearest neghborng neghborhood are wthn the effectve communcaton range of the neghborhood cluster head so that the communcaton graph between cluster heads s connected. That s, there s a communcaton path between a cluster head to any other cluster heads. Wthout loss of generalty, suppose sub-systems S 1, S 2,...,S n are dstrbuted nto q dsjont neghborhoods, as follows: N 1 = S 1,.., S l1 }, N 2 = S l1+1,..., S l2 },..., N q = S lq 1+1,..., S n }. Then, the dstrbuted model predctve controller of [4] approxmates the soluton of the optmzaton problem (1) for each tme nstant k 0, 1, 2, 3,...} by takng the followng three steps: Intalzaton: The nformaton exchange between neghborhoods at outer terate t 0, 1, 2,..., t 1} makes t possble for sub-system S to ntalze ts local decson varables as h 0 = ut R Nm, 1,..., n}. For k 1, u 0 = ( u t where u t = ( u t [1] u t [2]... u t [ N 1] 0 ) [0] u t [1]... u t [ N 1] ) R Nm s the approxmated soluton of the optmzaton problem (1) for the tme nstant k 1. For the tme nstant k = 0, u 0 U are chosen arbtrarly. Ths means that the ntalzaton s based on the warm start for t = 0. Inner Iterate: Between every two successve outer terates there are p nner terates. Subsystem S N e (e 1, 2,..., q}) performs p nner terates, as follows: For each nner terate p 0, 1,..., p 1}, subsystem S frst updates ts decson varable va h p+1 = π h + (1 π )h p, (2) where π are chosen subject to π > 0, l 1 j=1 π j = 1,..., n j=l π q 1+1 j = 1 and h =argmn h U J(h 0 1,..., h 0 l e 1, h p l,..., h e 1+1,..., h p l e, h 0 l,..., e+1 h0 n) (note that l 0 = 0, l q = n). Then, t trades ts updated decson varable h p+1 wth all other sub-systems n ts neghborhood N e. Outer Iterate: After p nner terates, there s an outer terate update as follows: u t+1 = λ h p + (1 λ )u t, (3) where t 0, 1, 2,..., t 1} and λ, = 1, 2,..., n, are chosen subject to λ > 0, λ 1 =... = λ l1, λ l1+1 =... = λ l2,..., λ lq 1+1 =... = λ lq (λ lq = λ n ), λ l1 + λ l λ lq = 1. Then, there s an outer terate communcaton,

3 y 1 Flume gate h p Flow Datum pool y h + 1 Flume gate +1 p + 1 (the +1-th gate), p 0 (measured n meter) s the poston of the -th gate, τ (measured n mnutes) s the fxed transport delay, d 0 (measured n meter cube per mnutes - m 3 /mn) s the known off-take flow rate dsturbance taken at the end of pool by user, and γ as well as C n (measured n m 1 2 /mn) and C out (measured n m 1 2 /mn) are constant. The equaton (4) can be wrtten n terms of the storage (ntegrator) equaton (5) and transport (delay) equaton (6), as follows. Fg. 1. An automated rrgaton channel. ẏ (t) = C n z (t) + C out z +1 (t) + C out d (t). (5) n whch the updated decson varables u t+1 are shared between all neghborhoods; and subsequently, between all sub-systems. The above three-step algorthm s repeated t tmes. As shown n [4] when t, then u t converges to the optmal soluton of the optmzaton problem (1). Hence u t = ( u t [0] u t [1]... u t [ N 1] ) R Nm represent the approxmated soluton of the optmzaton problem (1) for the tme nstant k, n whch based on the recedng horzon dea, ts frst component,.e., u t [0] s appled by the sub-system to the system as the control nput of the tme nstant k. III. AUTOMATED IRRIGATION CHANNEL Ths secton s devoted to modelng automated rrgaton channels, explanng upstream transent error propagaton and amplfcaton phenomenon n automated rrgaton channels and the formulaton of the constraned lnear quadratc optmal control problem to be solved by the dstrbuted model predctve controllers of [2] and [4] for mtgatng ths phenomenon. A. Automated Irrgaton Channel Model Throughout, we use the followng model for automated rrgaton channel that has been borrowed from [5]. Consder the automated rrgaton channel shown n Fg. 1. For the -th pool of ths channel we have the followng representaton ẏ (t) = C n z (t τ ) + C out z +1 (t) + C out d (t), = 1, 2,..., n, z =h 3 2, h = y 1 p, (4) z +1 =h , h +1 = y p +1, d = d, d n = d n. γ +1 γ n Here, α > 0 (measured n meter square - m 2 ) s a constant that depends on the pool surface area, y 0 (measured n meter Above Heght Datum - mahd) s the downstream water level at the -th pool, h 0 (measured n meter) s the head over upstream gate (the -th gate), h +1 s the head over downstream gate z (t) = z (t τ ). (6) The storage equaton (5) can be drectly converted to dscrete tme model usng the zero holder technque, whle the transport equaton (6) s converted to dscrete tme model by ntroducng τ T states as follows x,2 (kt ) = z (kt τ T ),..., x, τ T +1 (kt ) = z (kt T ), where the samplng perod T s the bggest common factor of pools transport delays (note that x,1 (kt ) = y (kt ), k 1, 2, 3,...}). Followng the above conversons, the equvalent dscrete tme model descrbng the dynamcs of the -th sub-system/pool s gven by the followng: s [k + 1] = Ǎs [k] + ˇB z [k] + Ďz +1 [k] + ˇF d [k], y [k] = Čs [k] = 1, 2,..., n In automated rrgaton channel, PI controllers z (s) = C (s)e (s), C (s) = KTs+K T F s 2 +T, e s = u y are used to stablze automated rrgaton channel around the pre-defned reference sgnals u s. Now, by fndng the correspondng dscrete tme transfer functon C (z) and then the correspondng state space representaton, we have the followng dscrete tme representaton for the PI controllers q [k + 1] = Āq [k] + B e [k], q [0] = 0, z [k] = C q [k]. Consequently, ( by ) defnng the augmented state varable s [k] x [k] =, the dynamcs of the automated q [k] rrgaton channel s gven by (7) S : x [k + 1] = A x [k] + B u [k] + F d [k] +v [k], y [k] = C x [k], z [k] = D x [k], for = 1, 2,..., n and k 0, 1, 2,..., }. In the above dynamc model v [k] = M x +1 [k] represents the cascade nterconnecton, x R n s the state varable of dmenson n N =1, 2, 3,...}, u R s the reference set pont, y R and z R are varables to be controlled, and d R s a known off-take dsturbance for the -th sub-system. (7)

4 TABLE I. Pool c n( m 1/2 m 1/2 m3 mn ) cout( mn ) τ(mn) γ( mn ) K T F 1 (pool 2 of the East Goulburn) (pool 5 of the East Goulburn) (pool 8 of the East Goulburn) (pool 9 of the East Goulburn) NUMERICAL VALUES FOR PARAMETERS DESCRIBING A FEW POOLS OF THE EAST GOULBURN MAIN IRRIGATION CHANNEL. Fg. 2. The upstream transent error propagaton and amplfcaton phenomenon. B. Upstream Transent Error Propagaton and Amplfcaton Phenomenon For the purpose of llustratng the upstream transent error propagaton and amplfcaton phenomenon n automated rrgaton channels, n ths secton we consder an automated rrgaton channel consstng of pools 2,5,8 and 9 of the East Gourburn man rrgaton channel located n Vctora, Australa. Numercal values for parameters descrbng these automated pools borrowed from [1] are gven n Table I. Fg. 2 llustrates the response of ths automated rrgaton channel to an off-take dsturbance wth the value of d 4 = 8 m3 mn appled to the last pool (pool 9 of the East Goulburn rrgaton channel) for the frst 15 tme steps (note that for smulatons, the desred steady state values for water levels are set to be 1m above datum n Fg. 1 and for smulatons the datum for the water levels are moved to the desred steady state water levels). Fg. 2 llustrates the upstream transent error propagaton and amplfcaton phenomenon due to nterplay between off-take flow dsturbance and transport delay n ths automated rrgaton channel. As clear from Fg. 2 for the last pool (pool 4) the transent devaton of the water level from the desred set pont s relatvely small. However, ths transent error propagates towards upstream pools and as we moved towards upstream pools, the devaton of water levels from desred set ponts ncreases. In large scale automated rrgaton channels wth large number of pools, at ther worst, these undesrable transent characterstcs can result n nstablty and performance degradaton due to actuator lmtatons. C. Reference Manager As shown n [1] one way to mtgate the above effect s to equp automated rrgaton channels wth a supervsory controller whch properly manages reference set ponts u s of local PI controllers by solvng a quadratc constraned optmal control problem. To formulate ths problem note that the dstrbuted dynamc model (7) for automated rrgaton channels has the followng augmented state space representaton x[k + 1] = Ax[k] + Bu[k] + F d[k], y[k] = Cx[k], (8) z[k] = Dx[k], where x[k] = ( x 1[k] x 2[k]... x n[k] ), u[k] = ( u 1 [k] u 2 [k]... u n [k] ), d[k] = ( d 1 [k] d 2 [k]... d n [k] ), y[k] = ( y 1 [k] y 2 [k]... y n [k] ), z[k] = ( z 1 [k] z 2 [k]... z n [k] ). As the supervsory controller can have a larger samplng perod than the samplng perod of local PI controllers, the samplng perod for supervsory controller s set to be ST, S N. Hence, the dynamc model for supervsory controller s obtaned by takng the model re-samplng approach, whch nvolves holdng the nputs to the system constant for the whole new sample perod, and aggregatng the dynamc (8) across the new sample perod, as follows: x[k + 1] = A S x[k] + ( S 1 j=0 AS j 1 B)u[k] +( S 1 j=0 AS j 1 F d[sk + j]), y[k] = Cx[k], z[k] = Dx[k], k = 0, 1, 2, 3,...}. (9) After obtanng the re-sampled model, the number of states n the re-sampled model (9) s reduced whle mantanng the nput-output behavor usng balanced truncaton. Consequently, the obtaned reduced model for the supervsory controller has the followng representaton ˆx[k + 1] = ˆx[k] + ˆBu[k] + ˆd[k], y[k] = Ĉ ˆx[k], z[k] = ˆDˆx[k], k = 0, 1, 2, 3,...}, (10) where ˆd[k] represents the effect of known off-take dsturbances on the reduced model. Now, the supervsory controller manages reference set ponts u s of local PI controllers by solvng the followng quadratc constraned optmal control problem [1]. mn J L 1 L(ˆx[0], ˆd 0, r, u) u=(u 1,...,u n) subject to (10) and y [k] [L, H ], u [k] [L, H ] z [k] [E, Z ] [1, n], k [0, L 1], } (11)

5 Fg. 3. DMPC wth two-level archtecture for communcaton. Sold lne: wthout computatonal latency, dotted lne: wth latency. where L s the rrgaton season length, the nterval [L, H ] s the admssble regon for water-levels y s and also decson varables u s, the nterval [E, Z ] s the admssble regon for varable z whch s a measure of water flow rate, and J L (ˆx[0], ˆd L 1 0, r, u) = n L 1 y [k] r 2 Q =1 k=0 + u [k] u [k 1] 2 R + z [k] 2 P (u [ 1] = 0). (12) Here. denotes the Eucldean norm (.e., z 2 P =z P z), ˆx[0] s the vector of known ntal states, ˆdL 1 0 = ˆd[k]} k=0,1,...,l 1, where ˆd[k] = ( ˆd 1[k]... ˆd n[k] ) s a collecton of known vectors that represent the effects of off-take dsturbances, r = ( r 1... r n ) s the vector of desred steady state values for y s, and Q, P 0, R > 0 are weghtng matrces. The frst norm n the cost functonal (12) penalzes devaton of water levels from the correspondng desred values, and the second norm penalzes large changes n the nput vector to the local PI controllers; and therefore, t tres to provde a smooth nput trajectory. The last norm tres to mnmze the nput flow rates as z s are measures of nput flow rates; and therefore, t s desrable to make them as small as possble to keep water n reservor as much as possble. Remark 3.1: As the optmal control problem (11) s a constraned problem and the season length L s long, to solve ths optmzaton problem n real tme usng a closed loop control strategy, the recedng horzon dea must be used. In [1] a centralzed model predctve control method s used to solve ths optmal control problem, n whch ths method uses a centralzed optmzaton method at each recedng horzon. As shown n [6], the computaton overhead for solvng the optmzaton problem assocated wth problem (11) at each recedng horzon usng centralzed optmzaton methods grows as O(n 5 ). However, the overhead usng the methods of [2] and [4] grows as O(n). Ths results n large computatonal latency (.e., tme delay between makng measurements and applyng the correspondng control nputs) when we use centralzed model predctve control methods for large scale automated rrgaton channels whch have large number of sub-systems n. Long computatonal latency sgnfcantly reduces the performance of the supervsory controller. Hence, for large scale automated rrgaton channels a practcal way to solve the constraned optmal control problem (11) usng full computatonal capacty of exstng local decson makers s to mplement the dstrbuted model predctve controller wth ether two-level or sngle-level archtecture for communcaton. Usng these controllers, at each tme nstant k by solvng a constraned optmzaton problem wth horzon length N << L, the set ponts u s for the tme nstant k s obtaned and they are appled to automated rrgaton channel by dstrbuted decson makers. IV. SIMULATION STUDY In ths secton, the satsfactory performance of the dstrbuted model predctve controller of [4] n mtgatng the upstream transent error propagaton and amplfcaton phenomenon s llustrated usng computer smulaton and compared wth the performance of the dstrbuted model predctve controller of [2]. To llustrate the satsfactory performance of the dstrbuted model predctve controller wth two-level archtecture for communcaton n mprovng the transent response of automated rrgaton channels, ths controller s appled to the optmzaton problem (11) formulated for the automated rrgaton channel wth four pools wth numercal values as gven n Table I. Here, t s assumed that the above automated rrgaton channel s subject to an off-take dsturbance wth the

6 Fg. 4. DMPC wth sngle-level archtecture for communcaton. Sold lne: wthout computatonal latency, dotted lne: wth latency. value of d 4 = 8 m3 mn for the frst 135 mn. It s also assumed that x[0] = 0, r = 0, S = 9, L = 240, N = 10, [L, H ] = [ 0.2m, 0.2m] and [E, Z ] = [0, /2 ]. Note that by applyng balanced truncaton, the reduced model has only 22 states nstead of 92 states. Also, a smlar method as used n [1] s used to guarantee the recursve feasblty of DMPCs. The communcaton load for each nner terate communcaton s assumed to be 5sec., whle for each outer terate communcaton the load s assumed to be 50 sec. p s set to be 10 and t = 1. Hence, at each recedng horzon the communcaton overhead of the Dstrbuted Model Predctve Control (DMPC) wth two-level archtecture for communcaton s p 5 + t 50 = = 100 sec., and the overhead of the DMPC wth sngle-level archtecture for communcaton s p 50 = 500 sec. Fg. 3 llustrates the response of the DMPC wth two-level archtecture for communcaton wthout and wth consderng the computatonal latency. As the computaton overhead n average s 13sec., ts computatonal latency n average s ( =)113sec., whch s almost 2/9th of the samplng perod of 9mn. As clear from Fg. 3, the response wth the computatonal latency s smlar to the response wthout latency. Ths result s expected because the computatonal latency here s almost 4 tmes smaller than the samplng perod. As clear from Fg. 3, the magntude of transent errors between water levels and the desred values decreases as we move towards upstream pools. Ths ndcates that the DMPC wth two-level archtecture for communcaton mtgates the upstream transent error propagaton and amplfcaton phenomenon. Fg. 4 llustrates the response of the DMPC wth sngle-level archtecture for communcaton wthout and wth consderng the computatonal latency. Here, the average computaton overhead s 15sec., and hence the computatonal latency n average s 515sec. Fg. 4 clearly llustrates that the performance of the case wth the computatonal latency n dsturbance rejecton s worst than the performance of the case wthout the computatonal latency. Ths results s expected because of large computatonal latency here. Obvously, ths reducton n performance s more sgnfcant n larger channels that have larger communcaton overhead. Fg. 3 and Fg. 4 llustrate that the DMPC wth two-level archtecture has a performance better than the performance of DMPC wth sngle-level archtecture by better managng communcaton overhead. V. CONCLUSION In ths paper, t was llustrated that the dstrbuted model predctve control method wth two-level archtecture for communcaton s a sutable method for mprovng the transent response of automated rrgaton channels. Hence, ths method can be partcularly used n large scale automated rrgaton channels to mtgate the upstream transent error propagaton and amplfcaton phenomenon. REFERENCES [1] M. Kearney and M. Canton, MPC-based reference management for automated rrgaton channels, n the Proceedngs of 2012 Australan Control Conference, Sydney, Australa, pp , [2] B. T. Stewart, A. N. Venkat, J. B. Rawlngs, S. J. Wrght and G. Pannoccha, Cooperatve dstrbuted model predctve control, Systems and Control Letters, 59, pp , [3] B. T. Stewart, A. N. Venkat, J. B. Rawlngs and S. J. Wrght, Herarchcal cooperatve dstrbuted model predctve control, n the Proceedngs of Amercan Control Conference, Marrott Waterfront, Baltmore, MD, USA, pp , [4] A. Khodabandehlo, Dstrbuted model predctve control and ts applcaton to automated rrgaton channels, MSc. Thess, Sharf Unversty of Technology, [5] M. Canton, E. Weyer, Y. L, S. K. Oo, I. Mareels and M. Ryan, Control of large-scale rrgaton networks, Proceedngs of the IEEE, 95(1), pp , [6] A. Farhad, P. M. Dower and M. Canton, Computaton tme analyss of a centralzed and dstrbuted optmzaton algorthms appled to automated rrgaton networks, n the Proceedngs of 2013 Australan Control Conference, Perth, Australa, pp , November 2013.