OPTIMAL CONTROL OF A SOLAR GREENHOUSE. R.J.C. van Ooteghem, J.D. Stigter, L.G. van Willigenburg, G. van Straten

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1 OPTIMAL CONTROL OF A SOLAR GREENHOUSE R.J.C. van Ootegem, J.D. Stigter, L.G. van Willigenburg, G. van Straten Systems and Control Group, Department o Agrotecnology and Food Science, Wageningen University, Mansoltlaan, 678 PA, Wageningen, Te Neterlands Racel.vanOotegem@wur.nl A solar greenouse as been designed tat maximizes solar energy use and minimizes ossil energy consumption. It is based on a conventional greenouse extended wit a eat pump, a eat excanger, an aquier and ventilation wit eat recovery. Te aim is to minimize ossil energy consumption, wile maximizing crop dry weigt and keeping temperature and umidity witin certain limits. Tese requirements are deined in a goal unction, wic is minimized by optimal control. A greenouse wit crop model is used to simulate te process beaviour. It is ound tat open loop optimal control trajectories can be determined. Te boiler use is reduced to a minimum, tus reducing ossil energy use. Compared to a conventional greenouse it is ound tat te energy costs are decreased and te crop dry weigt is increased. Copyrigt 22 IFAC Keywords: optimal control; dynamic model; energy management; gradient metod; model based control. INTRODUCTION Greenouse orticulture is an important branc o industry or Dutc economy. Te intensive crop production owever involves ig input o ossil energy. Te consumption o natural gas in greenouse orticulture is now about 2.5% o te total national consumption. A new greenouse is designed tat maximizes solar energy use and minimizes ossil energy consumption. Tis solar greenouse design is integrated wit climate control to obtain optimal crop growt conditions.. Solar greenouse design In te greenouse design te eat insulation and te transmission o solar radiation are maximized. An aquier is used to store te solar energy. Te aquier consists o a warm- and a cold-water basin in soil. At times o eat demand, te greenouse can be eated wit little energy input wit a eat pump and warm water rom te aquier. At times o eat surplus, te greenouse can be cooled wit a eat excanger and cold water rom te aquier, wile energy is arvested to use at times o eat demand. Te CO 2 supply is independent o boiler operation, tus avoiding te need to use te boiler at times o CO 2 demand. Ventilation wit eat recovery is added to deumidiy te greenouse at times o eat demand..2 Optimal control Te solar greenouse design wit extra control possibilities is a callenge rom te controlengineering point o view. Optimal control as been used to control te greenouse climate in conventional greenouses by Van Henten (994) and Tap (2). Van Henten concluded tat using optimal control could give a signiicant improvement in eiciency o greenouse climate management in teory. Te perormance o te optimal control largely depends on te ability o te control system to deal wit modelling and weater prediction errors. Tap sowed tat only sort-term weater predictions are needed or optimal greenouse climate control. Tey bot state tat good results can be obtained wit optimal control. An indirect gradient metod (Bryson, 999) is used to calculate te optimal control trajectories. Tis metod as proven its eectiveness in conventional greenouse control and many oter ields (Van Willigenburg, 2).

2 .3 Solar greenouse model For application o optimal control an accurate model o te controlled processes is necessary. Van Henten (994) and Tap (2) ound tat parts o te greenouse beaviour were not well described by teir models. Tis aects te perormance o te optimal control. Te model sould be as small as possible wit respect to te number o dierential equations, controls and disturbances or good insigt and ast calculation. It sould also give a good description o te dynamic variables as a unction o te applied controls and disturbances. Te dynamic model used in tis researc consists o models o greenouse climate (Heesen, 997), crop potosyntesis (Farquar, 98) and crop evaporation (Stangellini, 987). Te model as been validated wit greenouse data, and was ound to give an accurate description o te processes. Te model as been extended wit te new solar greenouse elements (eat pump, eat excanger, ventilation wit eat recovery). Te main disturbance is te weater, wic can be orecasted quite accurately or one or two days aead. Te calculation o control trajectories or te new eating system o te solar greenouse will give a better insigt o te proit o te solar greenouse design. 2. MATERIALS AND METHODS For optimal control purposes a model is needed tat gives an adequate description o te controlled processes. Te model needs to be suiciently complex to include all processes in a broad working area, e.g. te crop temperature range sould not be constrained to 3 C. To limit computation time, te number o states as to be small. Preerence is given to a wite model, since te internal variables ave a pysical meaning and can be easily interpreted. 2. Greenouse and crop model Te greenouse model used in tis researc as been developed by Heesen (997) based on te researc by Van Henten (994), De Zwart (996), De Jong (99) and Bot (983). A potosyntesis model (Farquar, 98) and an evaporation model (Stangellini, 987) are used to simulate te crop responses. Tis model (Van Ootegem, 23) is used or all calculations in tis paper. Te greenouse model is written in state space orm ( t, x, u v) x & =, were t is time, x are states, u are control inputs, v are external inputs (disturbances) and is a nonlinear unction. Te contents o tese variables are given in table. Te state equations ave been ormed based on te laws o conservation o entalpy and matter. Te dynamic beaviour o te states is described using irst order dierential equations. states control inputs disturbances Table : States, control inputs and disturbances symbol description T r temperature indoor roo [K] T a temperature indoor air [K] T c temperature crop [K] T s temperature soil (upper layer) [K] T l temperature lower net [K] T u temperature upper net [K] C a_co2 CO 2 concentration indoor air [kg m -3 ] C a_h2o H 2 O concentration indoor air [kg m -3 ] W crop dry weigt [kg m -2 ] lsd window aperture lee-side [..] wsd window aperture windward-side [..] op vr option ventilation eat recovery [/] CO2 valve position CO 2 supply [..] l valve position lower net [..] u valve position upper net [..] e valve position eat excanger [..] p valve position eat pump [..] I o incoming sortwave radiation [W m -2 ] v o wind speed outdoor [m s - ] T o temperature outdoor [K] T sk temperature sky [K] T o_n temperature wet bulb [K] C o_co2 CO 2 concentration outdoor [kg m -3 ] Greenouse model Te greenouse model is based on a Venlo-type greenouse wit a Nort-Sout orientation (Heesen, 997). Te greenouse as no ligting since tis gives energy costs, and no screen since tis would interere wit te maximum solar radiation intake. Te roo as a double glass cover. Te greenouse as two eating nets: a lower and an upper net, wic can be eated wit a boiler. Te lower net can be eated wit a eat pump and cooled by a eat excanger. Te eat pump can only be used at times o low eat demand since it can only eat to about 4 C. It is assumed tat te CO 2 supply is independent o te boiler eat supply. Ventilation wit eat recovery is applied by preeating te outdoor air wit indoor air. For te eat and mass transport te ollowing elements are taken into account: indoor air, crop, eating pipes, roo and soil. Tese elements are modelled as lumped parameter models, were it is assumed tat te compartments are internally omogeneous. Soil and roo are divided into two layers/parts. Crop model Te crop is grown on substrate. Te substrate is placed in a gutter, covered wit wite plastic. It is assumed tat water and nutrient supply is well-controlled and not limiting to crop potosyntesis and evaporation. Crop growt is based on te potosyntesis rate: te production o assimilates rom CO 2 and water wit solar radiation. Assimilates are used or crop growt and maintenance. Te potosyntesis rate is determined by solar radiation, temperature, CO 2 concentration, umidity and lea area index, based on Farquar (98). Te dark respiration rate is determined by temperature and indicates te assimilate use at nigt. Te crop evaporation is modelled according to te researc by Stangelinni (987). Evaporation is necessary to cool te crop leaves and to realise te transport o water and nutrients rom te roots to te upper crop parts.

3 Very ig or low temperatures can cause irreversible damage to te crop. Hig CO 2 concentrations in te indoor air can also cause crop damage, but exact values are not known. In practice a concentration o ppm is used. Hig umidity increases te risk at inection by mould. Te orticulturist can coose te temperature and umidity bounds. For te sortterm te temperature and umidity sould remain witin tese bounds. For te long-term tese bounds may be crossed, wic will be included in te temperature integration at a later stage. 2.2 Optimal control In optimal control, control input trajectories are determined, based on a goal unction. Te control solution consists o actuator trajectories, wic result in temperature, umidity and CO 2 concentration tat optimise a goal unction. Te aim is to minimize ossil energy consumption, wile maximizing crop dry weigt and keeping temperature and relative umidity witin certain limits. In te goal unction costs are deined to penalize ossil energy consumption and to keep temperature and umidity witin bounds. Te dry weigt increment during te control orizon is deined as a negative inal cost. Given a dynamic system wose evolution in time is described by a set o ordinary dierential equations ( t, x, u v) x & =, () n were t is time, x = x R is te state vector, m u = u R is te control input vector, w v = v R is te external input vector and is a non-linear unction. Te goal is to minimize te cosunction ( x, + L( x, dt J ( u) = Φ (2) n+ n+ m+ w+ were Φ : R R and L : R R are dierentiable a suicient number o times wit respect to teir arguments. Te inal time is set to te prediction orizon, wic is equal to one or two days, and tereore will not be subject to optimisation. t Te control inputs are constrained by u i, min i ( i, max K, u u i =, m (3) A control vector u( tat satisies te constraints (3) is called admissible. For te states tere are trajectory constraints, wic are considered sot. Wit tese prerequisites te control problem is to ind * u = arg min J ( u) (4) u given a prediction o v( or t [ t, ], subject to te dierential equations () and te control constraints (3). In oter words, te objective is to ind admissible input trajectories u * ( on te time interval t [ t, ] suc tat te process given by () as states trajectories tat minimize te perormance criterion J. Te resulting control and state trajectories are reerred to as te optimal trajectories. Te unction L [cost s - ] is given by te sum o te penalties or temperature T a, relative umidity RH a and energy consumption Q L( x, = L ( x, + L ( x, L ( x, (5) Ta RHa + Since te penalty unction L as to be integrated in time, te Mayer ormulation (Bryson, 999) is used, wic extends te state vector wit an extra state L. Te penalties or temperature and relative umidity are given by ( x( x ) cx min x( < xmin Lx ( x, xmin x( xmax (6) cx ( xmax x( ) x( > xmax were c x are te cost associated wit exceeding te boundary values x min or x max o state x. Furter penalties are given or energy consumption L ( x, = c Q Q = c Q Q ( Q + Q η Q ) Q boil sum p e (7) were Q sum [W m -2 ] denotes energy used by boiler and eat pump and recovered by te eat excanger. An eiciency actor η =. 5 is introduced or te eat excanger, since only a part o te recovered energy can be reused. Te inal cost Φ are determined by te yield in te orm o crop dry weigt W ( x = c ( t t ) ( W ( t ) W ( )) Φ (8), W t In table 2 te values used in te cosunction are given. Table 2: Cosunction: cost and penalties symbol unit x min x max - cost day - unit T a [ C] 6 24 c T = 5 RH a [%] 85 c RH = 2 CO 2a [ppm] 32 c CO2 = Q sum [W m -2 ] c Q =.677 W [kg m -2 ] c W = 76.8 Tere is no penalty on te CO 2 concentration; te bounds are used or te proportional controller. Te value o c Q corresponds to a cost o.2 per m 3 gas. Te minimization o te cosunction J can be perormed wit many dierent minimization metods. In tis researc an indirect gradient metod (Bryson, 999) is used to calculate te optimal control trajectories.

4 Control orizon and time interval Te control orizon is determined by te computation time, te time interval or te control inputs and te weater orecast time span. Time intervals ranging rom one our to several days are used in researc by Sina and Seginer (989) and Van Henten and Bontsema (99). Tese long time intervals are used because crop growt and development respond slowly to greenouse climate canges. In te solar greenouse te use o solar radiation or eating te greenouse is essential, wic calls or a time interval smaller tan one our. A smaller time interval will result in a longer computation time; tereore te control orizon is limited to a maximum o two days. Te sort-term crop growt is accounted or by dry weigt, wic is a unction o potosyntesis rate. Potosyntesis rate is aected by solar radiation, temperature, umidity and CO 2 concentration. To include long-term crop growt and development, temperature integration will be introduced in te goal unction over a range o tree to six days. In tis paper te use o open loop optimal control on a solar greenouse including te eat pump, eat excanger, separate CO 2 supply and ventilation wit eat recovery is described. Te temperature integration will be included in te optimal control at a later stage. In tis paper te control orizon is one day, te time interval or te control inputs is al an our and te maximum integration time step is minute. Tis means tat 48 values are determined by te optimal control or eac control input. Control inputs In conventional greenouse control, te greenouse climate is controlled by euristic rules and setpoints. Te optimal control algoritm uses a simulation o te greenouse climate wit varying control input values along te control orizon to calculate te goal unction value. Te greenouse climate can be modelled wit dierent control inputs. In conventional control o te greenouse temperature, setpoints are deined or te eating and ventilation temperature. I tese setpoints are used as control inputs or te optimal control, an internal (euristic) control as to be included to calculate te desired ealow or eating valve position. Te incorporation o te euristic control rules in te optimal control decreases te optimal control reedom. Tereore te greenouse climate model used in tis researc is based on actuator values as control inputs. Only two optimal control inputs are used in te optimisation. For ventilation te combined window aperture [..2] is used. It is split into te leeside lsd and windward-side wsd window aperture lsd wsd > > For eating/cooling te combined eating valve position [-..2] is used. It is split into te valve positions or eat excanger e, eat pump p, lower net l and upper net u e p l u < 2 < < 2 < 2 < < 2 State dependent control input bounds Based on a priori knowledge o te system, te ollowing bounds are set on te control inputs to pus te optimal control solutions into te correct direction. Te bounds are based on te simulated values o te initial states o te control interval (al an our). From te states, te values o temperature indoor air T a and relative umidity indoor air RH a (based on H 2 O concentration indoor air C a_h2o ) are used to determine te input bounds. Te maximum and minimum values or T a and RH a are equal to te boundary values given in table 2. Control input bound on combined window aperture [..2] < max = Ta < Ta max and RH a RH amax Tis can be interpreted as: Less ventilation i temperature T a and relative umidity RH a are below teir upper bounds. Control input bounds on combined eating valve position [-..2] max max min = = = T T a min a max < T T a < T a a < T < T a max a min Tis can be interpreted as: No eating wit te boiler i temperature T a is above its lower bound T amin. No eating wit te eat pump i temperature T a is above its upper bound T amax. No cooling wit te eat excanger i temperature T a is below its lower bound T amin. Te valve position CO 2 supply CO2 is controlled wit a proportional controller. Te CO 2 setpoint CO 2a-sp [ppm] is determined based on te combined window aperture and te incoming sortwave radiation I o CO 2a-sp CO2a max ( CO2a max CO2a min ) 4 I > o I o = =. CO CO ( ) (9) ( ) CO2( 2a-sp 2a t Tis valve position is constrained to te range [..].

5 I ventilation wit eat recovery is used, 7% o te sensible eat is recovered. Tis is used at times o eat demand, wic is determined by te use o eat pump or boiler ( > or > ). p Calculation Te open loop optimal control trajectories are calculated over te course o one day. Measured weater data is averaged to obtain oneourly weater data. Te results or two days are evaluated, one in summer and one in winter. Te optimal control calculation is started wit constant initial values or bot control input trajectories (, ). Dierent initial values yield dierent optimal control trajectories and dierent values or te cosunction J. To ind a good minimum cosunction a number o constant initial values are tested, and te best combination is used as initial value or te minimisation. l unctions are given or te summer and te winter weater data. cost J Fig. 2 Cosunction in summer (grid o 5*7 values) RESULTS AND DISCUSSION 3. Weater data To test te optimal control, two dierent days are selected, one in summer (998 7 ), and one in winter ( ). On te summer day temperature is between 4 and 9 C; wind speed is between.6 and 4 m s - ; relative umidity is between 48 and 94%; CO 2 concentration is between 29 and 35 ppm and sky temperature is between -4 and 6 C. Te incoming sortwave radiation is given in igure. On te winter day temperature is between 2 and 6 C; wind speed is between.4 and 3 m s - ; relative umidity is between 78 and 98%; CO 2 concentration is between 322 and 335 ppm and sky temperature is between -25 and -3 C. Te incoming sortwave radiation is given in igure. I o [W m 2 ] I o [W m 2 ] summer 2 winter t [] Fig. Solar radiation summer and winter day 3.2 Initial values control inputs Te initial values o te control inputs are set to trajectories wit a constant value. Te cosen values inluence te optimal control solution ound, since tere is more tan one minimum or te goal unction J. Te best combination o constant initial values or te control inputs is determined by calculating te cosor a number o input combinations. In igures 2 and 3 te resulting cost cost J Fig. 3 Cosunction in winter (grid o 5*7 values) From igures 2 and 3 it can be seen tat te best initial values or te inputs are not te same or summer and winter. Based on a grid o 5*7 values te best input-combination was determined to calculate te optimal control trajectories. Te best input-combinations ound are denoted wit a dot ( ) summer : winter :.5 = 2. and =. and 3.3 Optimal control summer day =. =.5 In te optimal control trajectories o te summer day te boiler is not used. Te window aperture and valve positions o te CO 2 supply, upper and lower net, eat pump and eat excanger are given in igure 4. Te dased lines or te window aperture indicate ventilation wit eat recovery. In igure 5 te optimal state trajectories are given. Te accompanying costs J are determined by te integrals o te penalties L and te inal costs Φ: J (u) = Φ( W ( t ), t ) = 3.47 t t L Ta dt = 4.22; L RHa dt =.3; L Q dt = -2.7 t.5 2

6 lsd wsd CO2 l, u e p t [] Fig. 4 Optimal control input trajectories summer T a [oc] RH a [%] CO 2a [ppm] Q sum [kw/m 2 ] W [kg/m 2 ] t [] 3.4 Optimal control winter day In te optimal control trajectories o te winter day te eat excanger is not used. Te window opening is small. Te greenouse is mainly eated wit te eat pump during daytime. Te boiler is used i te eat demand is iger during nigttime. Te window aperture and valve positions o te CO 2 supply, upper and lower net, eat pump and eat excanger are given in igure 6. Te dased lines or te window aperture indicate ventilation wit eat recovery. In igure 7 te optimal state trajectories are given. Te accompanying costs J are determined by te integrals o te penalties L and te inal costs Φ: J (u) =.6 Φ( W ( t ), t ) = 4.7 t lsd wsd CO2 l, u e p t L Ta dt =.8; L RHa dt =.; L Q dt = 2.83 t t [] Fig. 6 Optimal control input trajectories winter Fig. 5 Optimal state trajectories summer wit bounds (dased) Wile tere is sunligt, te CO 2 supply CO2 is ully opened most o te time. Wit ig solar radiation tis results in a large increase o te dry weigt W. Tis growt implies a ig potosyntesis rate and terewit a ig use o CO 2, resulting in a low CO 2 concentration CO 2a. Wile te temperature T a is witin its bounds, te eat pump valve p is opened to increase temperature. Te temperature increase causes a decrease in relative umidity RH a, keeping te latter below its bound. It also causes a iger potosyntesis rate, increasing dry weigt W. Te eat excanger valve e is opened between te ours 8 and 2 to decrease te temperature T a, keeping it almost witin its bounds. W [kg/m 2 ] Q [kw/m 2 ] CO [ppm] RH [%] T [oc] sum 2a a a t [] Fig. 7 Optimal state trajectories winter wit bounds (dased)

7 Wile tere is sunligt, te proportional CO 2 control deined by (9) is used to control te CO 2 supply CO2. Most o te time te valve is ully opened, except wen te CO 2 concentration is between 9 and ppm. A small decrease in te CO 2 concentration can be seen between te ours 8 and 7, wic correspond to te ours wit solar radiation. Te potosyntesis rate is lower on te winter day compared to te summer day due to less solar radiation. Tis causes less CO 2 consumption and a smaller increase o dry weigt W. Te eat pump valve p is used to increase temperature during daytime, wen te eat demand is low. Te boiler ( l, u ) is used at nigttime, wen te eat demand is iger, keeping te temperature almost witin its bounds. Due to less solar radiation te increase in dry weigt W tat could be acieved by increasing te temperature T a does not countervail against te cost o eating. Te relative umidity RH a stays well below its bound. Tis is due to a lower crop evaporation rate on account o a lower potosyntesis rate and a lower temperature T a. 3.5 Optimal control witout solar greenouse elements I te solar greenouse elements (eat pump, eat excanger, ventilation wit eat recovery) are removed rom te greenouse model, te model describes a conventional greenouse. Te optimal control is tested on tis conventional greenouse or comparison wit te solar greenouse. Values costs J, integrals o te penalties L and te inal costs Φ in summer witout solar greenouse elements: J (u) = 5.93 Φ( W ( t ), t ) = t t L Ta dt = 4.6; L RHa dt =.48; L Q dt = 2.8 t Values costs J, integrals o te penalties L and te inal costs Φ in winter witout solar greenouse elements: J (u) = 7.22 Φ( W ( t ), t ) = 3.3 t t L Ta dt = 2.83; L RHa dt =.7; L Q dt = 7.63 t Te trajectories o te variables T a, RH a and CO 2a (not sown) are comparable to tose sown in igures 5 and 7 wit solar greenouse elements. In table 3 te total amount o energy used Q sum and te increase o dry weigt W are sown. From te results it can be seen tat te energy use in te conventional greenouse is iger and te increase in crop dry weigt is lower tan in te solar greenouse. Table 3: Results energy and dry weigt symbol unit summer winter Q solar sum [W m 2 day ] conv solar.4.8 W [kg m 2 day ] conv CONCLUSIONS From te open loop optimal control results ound, we can conclude tat Optimal control o te solar greenouse is easible. Altoug te model is non-linear and complex, rational optimal control solutions can be ound. Te control and state trajectories can be interpreted easily, since te internal variables ave pysical meaning. Te use o a pre-calculation o te constant initial optimal control values can be used to obtain control trajectories tat are more likely to go to a good minimum o te cosunction. Te results o te optimal control strongly depend on te weater conditions; tereore reliable orecasts are needed. Te boiler, eat pump and eat excanger are used only i it yields a proit in te optimal control goal unction. Tis causes temperature and relative umidity close to teir bounded values. Te use o te solar greenouse elements (eat pump, eat excanger and ventilation wit eat recovery) results in lower energy costs and a iger dry weigt increase. In urter researc, te open loop optimal control will be extended to a receding orizon optimal control to control te greenouse processes. Tap (2) and Van Henten (994) used tis on a conventional greenouse. Tey bot ound tat optimal control in greenouse is easible. For te receding orizon optimal control, initial state values are needed at eac new sampling time to calculate new optimal control trajectories. A state estimation is needed to calculate tese initial states. An extended Kalman ilter could be introduced to estimate tese states. Adaptive control can be used to improve te model by adapting model parameters tat are not or not well known. Tis can also be included in te extended Kalman ilter. ACKNOWLEDGEMENTS Tis researc is unded by EET, te Dutc institute or Economy, Ecology and Tecnology. REFERENCES Bot, G.P.A. (983). Greenouse climate: rom pysical processes to a dynamic model, P.D. tesis, Wageningen Agricultural University, Wageningen, Te Neterlands. Bryson Jr., A.E. (999). Dynamic optimization, Menlo-Park, Caliornia: Addison-Wesley. de Jong, T. (99). Natural ventilation o large multi-span greenouses, P.D. tesis, Wageningen Agricultural University, Wageningen, Te Neterlands. de Zwart, H.F. (996). Analyzing energy-saving options in greenouse cultivation using a

8 simulation model, P.D. tesis, Wageningen Agricultural University, Wageningen, Te Neterlands. Farquar G.D., S. von Caemmerer and J.A. Berry (98). A biocemical model o potosyntetic CO 2 assimilation in leaves o C3 species, Planta, 49, Heesen, L. (997). Deinitie, gevoeligeidsanalyse en evaluatie van een dynamisc model van et kas-gewasproductieproces, M.Sc. tesis, Wageningen Agricultural University, Wageningen, Te Neterlands. Sina, G. and I. Seginer (989). Optimal management o tomato growt in greenouses, Acta Horticulturae, 248, Stangellini, C. (987). Transpiration o greenouse crops an aid to climate management, P.D. tesis, Wageningen Agricultural University, Wageningen, Te Neterlands. Tap, R.F. (2). Economics-based optimal control o greenouse tomato crop production, P.D. tesis, Wageningen Agricultural University, Wageningen, Te Neterlands. van Henten, E.J. and J. Bontsema (99). Optimal control o greenouse climate, In: Matematical and control applications in agriculture and orticulture (Hasimoto, Y. and W. Day (eds.)), IFAC Worksop Series,, van Henten, E.J. (994). Greenouse climate management: an optimal control approac, P.D. tesis, Wageningen Agricultural University, Wageningen, Te Neterlands. van Ootegem (23). Te solar greenouse model; Extension o te greenouse wit crop model, Internal report, Wageningen University, Wageningen, Te Neterlands. van Willigenburg, E.J. van Henten and W.T.M. van Meurs (2). Tree time-scale receding orizon optimal control in a greenouse wit a eat storage tank, In: Proceedings o te Agricontrol 2 conerence, 2-4 July 2, Wageningen, Te Neterlands.