simulation of pi and mpc averaging level control in a water ...control in a water resource recovery...

Simulations and real applications of PI and MPC averaging levelcontrol in a water resource recovery facility

Finn Aakre Haugen1

1University of South-Eastern Norway, Norway, [email protected]

AbstractThe basin level control system in an existing wastewa-ter treatment plant is designed for two different operationmodes: compliant level control for smooth pump flow,and stiff level control. Both a model-based predictivecontroller (MPC) and a standard PI controller are imple-mented, and tested both on a dynamic simulator and onthe real plant. The simulator, the MPC and the PI con-troller tuning are based on a mathematical process modelderived from a material balance of the wastewater in theinlet basin.Keywords: Water resouce recovery facility, equalizationmagazine, level, MPC, Kalman Filter. PI control.

1 IntroductionThis article reports results from a study where alterna-tive controllers of the level of the equalization magazineupstream to the VEAS water resource recovery facility(wrrf) 1 at Slemmestad, Norway has been tested both ona dynamic simulator and on the real plant. This maga-zine comprises the part of the inlet tunnel being closestto the wrrf. VEAS is the largest wrrf in Norway, serv-ing about 700,000 population equivalents (pe), treating inaverage about 3.5 m3/s. The biological treatment at thewrrf will benefit from a smoother hydraulic load than atpresent. One approach to this end is improving the levelcontrol system of the inlet basin, aiming at smoother pumpflow from the basin to the treatment processes (Bolmstedt,2004); (van Overloop et al., 2010).

The implementation of the simulator and the controlsystem used in this study is in LabVIEW (National In-struments) with the MPC algorithm implemented in MAT-LAB code in LabVIEW’s MATLAB Script node. Thesampling time (time-step) of the various discrete-time al-gorithms of simulation, estimation, filtering, and controlis 10 s.

The paper is organized in the following main sections:System description; Controller functions; Results; Con-clusion; Abbreviations; Nomenclature; Acknowledge-ments; References.

1"wrrf" is the terminology recommended by the International WaterAssociation (IWA) and the Water Environment Federation (WEF)

2 System description2.1 Geometrical designFigure 1 depicts the principal geometrical design of theinlet part of the VEAS wrrf.

2.2 Operation modes of the tunnel and basinand requirements to the level control sys-tem

The tunnel and basin are operated in different modes:

• Operation mode #1: Normally low load (tunnelflow) & Compliant level control: The main aim ofthis mode is to obtain smooth pump flow. To this end,compliant level control is implemented: The levelis allowed to vary between the soft limits of 1.5 mand 2.5 m, with 1.8 m as the nominal level setpoint.Lately, these limits have been changed to 1.6 m and2.8 m, respectively, with setpoint 2.3 m. (Both thesesets of specifications are used in this article.)

• Operation mode #2: Normally low load & Stifflevel control: The main aim of this mode is to havethe level close to a relatively low setpoint to en-sure that the pump soaks up solid downfall from thewastewater accumulated in the basin during Opera-tion mode #1. The duration of this operation mode isrelatively short, approximately two hours, each sec-ond day. In this operation mode, the variations of thepump flow will, inevitably, be relatively large as theyare almost the same as the variations of the net inflowto the basin.

In addition to Operation modes #1 and #2, there are oper-ation modes concerning normally high load (at high pre-ciptiation) and to tunnel flushing (building up a volume ofwastewater in a part of the tunnel, and then flushing thetunnel with this volume). This article covers only Opera-tion mode #1.

2.3 Mathematical model of the inlet basinThe basis of both the simulator and the model-based pre-dictive controller is a dynamic model of the liquid level ofthe basin, Eq. 1. The model stems from material balanceassuming the sewage is water.

h(t) =Fin(t)−Fout(t)

A(h)(1)

https://doi.org/10.3384/ecp18153297 297 Proceedings of The 59th Conference on Simulation and Modelling (SIMS 59), 26-28 September 2018,

Oslo Metropolitan University, Norway

Slope, S

-2.4

0

Liquid surface area A [m2]

Tunnel(shaped as inclined

cylinder)

Basin (rectangular)

Swamp

Side view:

3.5 m

Top view:

4.7 m A_swamp[m2]

A_tunnel [m2](varies with liquid

level)

A_basin

Pump inlet

Pumps (eight)

58,4 m

15.2 m

5.2 m

Side channel

3.5 m

A_side_channel [m2]

30 m

Shaft

4.2

(Geometrical proportions not real)

Pump

h [m]

F_pump[L/s]

Controlsignal, u

[L/s]LT

h = 0 m corresponds to-15.0 m absolute level.

0.6

FT

Figure 1. Principal design of inlet basin of VEAS wrrf.

whereFout(t) = Fpump(t) (2)

andFin(t) = Finmeas(t)+Finnonmeas(t) (3)

where Finmeas are measured flows and Finnonmeas are unmea-sured flows.

The liquid surface area A in Eq. 1 is calculated fromthe assumed known geometry, and is therefore a functionof the liquid level, h. The ellipsoidal liquid surface in thetunnel is continuously calculated by numerical integration.The dynamics of the pump is taken into account by thelevel controllers. In the real plant, an apparent time-delayof approximately 120 s is observed between the pumpcontrol signal u and the resulting (measured) pump flowFpump:

Fpump(t) = u(t−Tdelaypump) (4)

For conservative controller tuning, the pump dynamics isrepresented by a time-delay of Tdelaypump = 120 s.

2.4 Piping and Instrumentation Diagram(P&I diagram)

Figure 2 shows a P&I diagram of the level control sys-tem. The flow at Vækerø is measured. Vækerø is situ-ated approximately 15 km upstreams the plant, The flowat Vækerø constitutes the main inflow component to theplant, counting for 70-80% of the total tunnel inflow. Thisflow arrives at the plant with a transportation time (time-delay) of approximately 3.0 h, but smoothed, so the trans-portation time is not well-defined. The smoothing is rep-resented with a time constant low pass filter with an esti-mated time constant of 1 h. The Vækerø flow measure-ment is used in simulations. However, it is not used by the

controllers in the real implementations in this study dueto the uncertain information about its contribution to theactual inflow to the basin, as pointed out above.

3 ControllersTwo different controllers are used in this study, namely (a)PI control and (b) MPC.

3.1 PI controllerA standard PI controller (Seborg et al., 2004) is used:

u(t) = Kce(t)+Kc

Ti

∫ t

0e(θ)dθ (5)

The PI controller is tuned with the Skogestad method(Skogestad, 2003) for “integrator plus time-delay” processdynamics, but with a modification where the integral time,Ti, is reduced to obtain faster disturbance compensation(Haugen and Lie, 2013). In general, the following differ-ential equation can represent such dynamics:

y = Kiu(t− τ) (6)

By neglecting Fin, the process model, Eqs. 1 - 4, is on theform of Eq. 6 with

y = h

u = Fpump

Ki =−1A

(7)

The Skogestad PI settings for the model Eq. 6 are:

Kc =1

Ki (Tc + τ)=− A

Tc + τ(8)



LT

1

Tunnel

Basin

LC

1

Pumpinlet

A [m2]

F_tunnel[L/s]

h [m]

u_LC [L/s]= F_pump [L/s]

Levelsetpointh_sp [m]

FT

1

F_Vækerø[L/s]

Time-delay (transportation time),T_delay [d]

F_net_in_internal_meas[L/s]

Vækerø

Pump(=aggregate of

pumps)

F_in_unmeas [L/s]

FT

2

F_pump [L/s]

Note: Geometrical proportions are not real.

Used in feedforward

control

Figure 2. Piping and Instrumentation diagram (P&I D) of level control system of basin.

Ti = 2(Tc + τ) (9)

where Tc is the closed (control) loop time constant speci-fied by the user. In general, the PI controller can be tunedfor satisfactory performance using Tc as follows: By in-creasing Tc, the control systems typically becomes moresluggish: The control signal becomes smoother, and thecontrol error becomes larger. On the other side, by de-creasing the closed loop time-constant, the control sys-tems typically becomes more aggressive (faster): The con-trol signal varies more abruptly, and the control error be-comes smaller.

3.2 MPC with Kalman FilterA nonlinear MPC is used (Grüne and Pannek, 2011). Theoptimization (minimization) problem of the MPC used inthis study is2

minu

[J =

∫ t0+Tp

t0C1e(t)2 +C2u(t)2 dt

](10)

The optimization or decision variable is the pump flow(control variable). e = hsp− h is control error. u = Fpumpis rate of change of pump flow (control variable). C1 andC2 are cost coefficients. Tp is the prediction horizon. t0is the present point of time. The MPC finds the sequenceof sampled future pump flow values that gives the opti-mal balance or compromise between small control errorand small rate of change of pump flow. To save the com-putional demand, control signal locking is used, i.e. thenumber of allowable values during the prediction horizionis set to Np.

A time-constant filter with time constant T pump wasincluded in the calculation of the applied control signal tothe pump to obtain improved pump flow smoothing.

The MPC uses a Kalman Filter estimate of the total un-measured flows, see below.

2Although the objective function, J, is here on a continuous-timeform, a corresponding discrete-time form is of course used in the com-puter implementation.

The MPC takes into account level constraints and con-trol variable (pump flow) constraints. MATLAB’s fmin-con function is used as optimization function.

Kalman Filter. The estimate of the total of the unmea-sured flows is calculated with an Extended Kalman Filter(EKF) (Simon, 2006) based on the process model, Eqs. 1 –3 with total unmeasured flow, Finnonmeas modelled as a “ran-dom walk” augmentation state variable. The time delay inEq. 4 was neglected as it is assumed to have negligibleeffect on the estimates.

To summarize, the model used in the EKF comprisesthe following two differential equations:

h =Finmeas +Finnonmeas −Fpump

A(h)+w1 (11)

Finnonmeas = 0+w2 (12)

where w1 and w2 are random process disturbances.The process output measurement used by the EKF to

correct the predicted state estimate is the measured level.The process measurement noise covariance was set to

R = [0.1] (13)

and the process disturbance covariance was set to

Q =

[0.1 00 10−6

](14)

In the tuning, R and Q(1,1) were fixed to more or lessrandom values, and then Q(2,2) was used as the ultimatetuning parameter since it directly affects the estimate ofFinnonmeas . Q(2,2) was adjusted by trial-and-error to give asufficiently fast while not too noisy flow estimate. R mighthave been set to the variance of representative time-seriesof the level measurement, but this was not implemented.In practice, it is mainly the ratio between the variances thatdetermines the behaviour of the estimator.



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1.2

1.4

1.6

1.8

2

2.2

[m]

DKS PI control. LC01_sp:green. LT01:blue.

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0

1000

2000

3000

4000

[L/s

]

DKS PI control. u:LC01_PI_out:blue. IPU_FT:green. F_wash:magenta.

142.2 142.3 142.4 142.5 142.6 142.7 142.8 142.9


-100

-50

0

50

100

[(L/

s)/m

in]

DKS PI control. du_dt:blue. du_dt_high_lim and du_dt_low_lim:red.

Figure 3. Results with real PI level control with original PIsettings. (Time range: 21 h.)

4 Results4.1 Real PI level control with original PI set-

tingsFigure 3 shows real responses with the following originalPI controller settings (before the retuning presented in thisarticle):

Kc =−3200 (L/s)/m, Ti = 1000 s (15)

which have been found with trial and error. The level set-point is 1.8 m. The absolute value of the rate of change ofthe pump flow, |u|, becomes substantially larger than thespecified maxium of 20 (L/s)/min. The fraction of timethat |u| is larger than 20 (L/s)/min is 57 %.

4.2 Simulated level control with retuned PIcontroller

The simulator was driven by real flow measurements. ThePI controller was tuned with the Skogestad method withTc = 1000 s adjusted by trial and error on the simulator tosatisfy the specifications. The resulting PI settings are:

Kc =−2000 (L/s)/m, Ti = 2000 s (16)

The simulated responses are shown in Figure 3 (togetherwith responses with MPC). The specification to the rate ofchange of pump flow is met, see the lower plot in Figure4. Also, the level is within the limits.

4.3 Simulated level control with MPCThe MPC was tuned with C1 = 1 (fixed) and C2 = 0.05by trial and error. The time step was 120 sec, and theprediction horizon was 30 minutes, corresponding to 15time steps. Control signal blocking with Np = 3 blocks

Figure 4. Results with simulated PI level control (magentacurves) and MPC level control (blue curves). (Time range: 21h.)

of equal size (5 time steps) was implemented. The pumptime-constant filter was set by trial and error to Tpump =700 s.

The simulated responses are shown in Figure 4. Thespecification to the rate of change of pump flow is metexcept for a small time interval around time t = 6 h, seethe lower plot in Figure 4. The level is within the limitsexcept for a small time interval around t = 14 h.

4.4 Real level control with retuned PI con-troller

A retuned PI level controller was applied to the real plant.The retuning was based on the Skogestad method. It wasdecided to increase the normal level setpoint from origi-nally 1.8 m to 2.3 m. This level increase implies that theliquid surface area, A, increases, thereby reducing the pro-cess gain. The reduction of the process gain allows for amore relaxed PI tuning (larger Tc).

The Skogestad tuning was based on the specificationTc = 1500 s. A time-delay of Tdelaypump = 120 s was in-cluded in the process model to account for pump controldynamics. The liquid surface area was fixed to A = 2000m2 in the tuning, which is approximately the area at theoperating point of level h = 1.8 m.

The resulting tuning became

Kc =−1240 (L/s)/m, Ti = 3240 s (17)

Figure 5 shows the results on the real plant. As seen fromthe lower plot, the specifications to the maximum rate ofthe pump flow is satisfied.

The PI settings above are now (as of May 2018) in or-dinary use at the plant.



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t [d] relative to 01-Jan-2017 00:00:00. Start: 29-Sep-2017 12:00:00. Stop: 30-Sep-2017 09:00:00 (winter-time).

1.8

2

2.2

2.4

2.6

2.8

[m]

Compliant PI control. LC01_sp:green. LT01:blue. High_level_lim and Low_level_lim:red.

271.6 271.7 271.8 271.9 272 272.1 272.2 272.3


0

1000

2000

3000

4000

[L/s

]

Compliant PI control. LC01_PI_out:blue. F_wash:magenta.

271.6 271.7 271.8 271.9 272 272.1 272.2 272.3


-100

-50

0

50

100

[(L/

s)/m

in]

Compliant PI control. du_dt:blue. du_dt_high_lim and du_dt_low_lim:red.

Figure 5. Results with PI control on the real plant. (Time range:21 h.)

4.5 Real level control with MPCDue to practical reasons, only initial results of MPC levelcontrol applied to the real plant can be shown in this arti-cle. Both a simulation study similar to the study presentedabove, and real tests were accomplished. Both these testswere accomplished without including the lowpass filterto smooth the pump flow. The MPC pump control sig-nal showed abrupt changes at the points of time whenthe MPC decided to change the control signal, see Figure6. The control signal resembles a piecewise constant sig-nal. This behaviour is actually understandable since u(t)is zero except at the short transitions, making the rate ofchange term, u(t)2, in the MPC objective relatively small.

Several solutions were tried to mitigate the abrupt con-trol signal changes. The best solution turned out to bethe pump lowpass filter, but from these tests, we onlyhave simulation results, cf. Section 4.3. Unfortunately,the equipment was not longer available for a practical testwhich might have confirmed the good simulation resultswith the improved MPC.

100.4 100.5 100.6 100.7 100.8 100.9 101 101.1 101.2

t [d] relative to 01-Jan-2017 00:00:00. Start: 11-Apr-2017 09:30:00. Stop: 12-Apr-2017 06:30:00.

2

2.5

3

[m]

Compliant MPC control. LC01_sp (green). LT01_meas (blue). Max_level_MPC and Min_level_MPC (red)

100.4 100.5 100.6 100.7 100.8 100.9 101 101.1 101.2


0

1000

2000

3000

4000

[L/s

]

Compliant MPC control. LC01_MPC_out (blue). F_in_Kalman_estim (green). F_wash (magenta)

100.4 100.5 100.6 100.7 100.8 100.9 101 101.1 101.2


-100

-50

0

50

100

[(L/

s)/m

in]

Compliant MPC control. du_dt (blue). du_dt max and min (red). Mean(du_dt) (cyan).

Figure 6. Results with MPC on the real plant. (Time range: 21h.)

5 Conclusions

Successful averaging level control was obtained with a PIcontroller tuned with the Skogestad tuning method, bothon the simulated plant and on the real plant.

The first implementation of MPC for averaging levelcontrol applied to the real plant was not successful asthe pump flow pattern generated by the controller showedabrupt changes between the time intervals of piecewiseconstant flow. By including a lowpass filter on the controlsignal in the MPC model, successful control was eventu-ally obtained with MPC – on the simulated plant. Un-fortunately, due to practical reasons, this successful MPCimplementation could not be tested on the real plant.

This study has demonstrated, to the author and to thestaff at the plant, the power of using a dynamic simulatorfor tuning and testing control systems.

In the hindsight, one lesson to learn from this study isthat enough time should be allocated to, hopefully, obtainsatisfying simulation-based tests before testing the con-troller on the real plant.

Acknowledgements

The author is thankful for the support provided by the staffat VEAS wrrf, Slemmestad, Norway.



NomenclatureSymbol [Unit] Description

A [m2] Surface area of liquidC1, C2 - Cost coefficients of MPC

e - Control errorFin [L/s] Total inflow to the basin

Finmeas [L/s] Total measured inflows to the basinFinnonmeas [L/s] Total unmeasured inflows to the basin

Fout [L/s] Total outflow of the basinFpump [L/s] Total basin pump flow

h [m] Water level of basinhsp [m] Setpoint of water level of basinKc [(L/s)/m] Gain of PI controllerKi - Process integral gainNp [1] Number of MPC control signal blocksTp [s] Prediction horizon of MPCt0 [s] Present point of time of MPCTc [s] Closed-loop time-constant (Skogestad)

Tdelaypump [s] Approximate time-delay of basin pumpTi [s] Integral time of PI controlleru [L/s] Control signal to the pump|u| [(L/s)/min] Rate of change of pump control signal

AbbreviationsAbbr. Full nameEKF Extended Kalman FilterFT Flow Transmitter (sensor)LC Level ControllerLT Level Transmitter (sensor)MPC Model-Predictive ControlPID Proportional + Integral + Derivativewrrf water resource recovery facility

ReferencesJ. Bolmstedt. Controlling the Influent Load to Wastewater Treat-

ment Plants. Lic. thesis, Lund University, Sweden, 2004.

L. Grüne and J. Pannek. Nonlinear Model Predictive Control.Springer-Verlag, London, 2011. doi:10.1007/978-0-85729-501-9.

F. Haugen and B. Lie. Relaxed Ziegler-Nichols Closed LoopTuning of PI Controllers. Modeling, Identification and Con-trol, 34(2):83–97, 2013. doi:10.4173/mic.2013.2.4.

D. E. Seborg, Th. F. Edgar, and D. A. Mellichamp. ProcessDynamics and Control. John Wiley and Sons, 2004.

D. Simon. Optimal State Estimation. Wiley, 2006.doi:10.1002/0470045345.

S. Skogestad. Simple analytic rules for model reduction andPID controller tuning. Journal of Process Control, 14, 2003.doi:10.1016/S0959-1524(02)00062-8.

P. J. van Overloop, R. R. Negenborn, B. De Schutter B, and N. C.van de Giesen. Predictive control for national water flow opti-mization in the netherlands. Intelligent Systems, Control andAutomation, Springer, 42, 2010.



http://dx.doi.org/10.1007/978-0-85729-501-9

http://dx.doi.org/10.1007/978-0-85729-501-9

http://dx.doi.org/10.4173/mic.2013.2.4

http://dx.doi.org/10.1002/0470045345

http://dx.doi.org/10.1016/S0959-1524(02)00062-8

simulation of pi and mpc averaging level control in a water ...control in a water resource recovery...

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