OPTIMAL CONTROL OF MICROMIKROALGAE

OPTIMAL CONTROL OF MICROMIKROALGAE GROWTH THROUGH NUTRIENT DILUTION RATES Septia Marga Dartika, Nanda Dewi Oktavianti, Prof. Basuki Widodo, Drs., M.Sc Mathematics Department, Sepuluh Nopember Institute of Technology (ITS) Arief Rahman Hakim Street, Surabaya 60111 Indonesia Abstract Biofuel is one of renewable energy. One of the raw material for manufacture of this alternative biofuel is microalgae. The production of alternative biofuel will increase if the growth of microalgae is greater. So that, need some method to optimize the growth of microalgae. In this study, the inflow of nutrient (𝐼𝑚 ) will be optimal control by applying the Pontryagin maximum principle, the simulation result show that the dried algae production will maximum if the rate minimum weight (0.1) of dilution nutritional value. So, the production of dried algae will increase if the weight of dilution nutritional value is smaller. So that, the production of microalgae will increase in order to get the better and much of alternative biofuel. Keyword : Biofuel, Optimal Control, Pontryagin Maximum Principle.

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Mathematics Department - ITS

INTRODUCTION Fossil fuel is one of unrenewable resource. In this world, fossil fuel is the key to make people still life. The kind of product from fossil fuel are tar, petrochemical, etc. Year by year, the demand of fossil fuel increase. It is possible that fossil fuel can run out so that it is important to keep this resource. The government also make some rule about the using of alternative biofuel. So that, it can reduce the using of fossil fuel. Biofuel is made from organic material, for example are plants and animals. Biofuel is one of renewable resource that can produced by human with specific materials. The materials were grown by human itself. One of the specific material is microalgae. It can grow faster in narrow area. Microalgae contained high quality of oil. It is also be the best resource of oil in the world. Most of oil that we get from the earth is from microalgae that live long years ago. The function of microalgae also like the other plant. For example is produced oxygen by photosynthesis. Cells of microalgae grow in water so that it is more efficient in using of water, carbon dioxide, and nutrient than the other plants. PROBLEM STATEMENT In Thornton (2010) research, mathematical modelling of microalgae growth constructed by using industrial disposal and finished by numerical method. Then, in Hajar (2015) research, it is about carbon dioxide control to produce the maximum microalgae. In this paper, will explain about how to get the optimum control of microalgae growth through nutrient dilution rates in Thornton model with Pontryagin maximum principal in order to get the better and much of alternative biofuel. MATHEMATICAL MODELLING 1) Dynamic Analysis Model of Microalgae Growth This model are from Thornton (2010), 𝑀 𝐴̇ = 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆 − (𝐷𝑟 + ℎ𝑟 )𝐴 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝑀 𝑀̇ = −𝑘2 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆 + 𝐼𝑚 (𝑡) 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝑀 𝑆̇ =∝𝑆 𝐶 − 𝑘3 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆 − (𝐷𝑟 + ℎ𝑟 )𝑆 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝐶̇ = −𝑘1 𝛼𝑆 𝐶 + 𝐼𝐶 (𝑡) where, 𝐴 : concentration of dried microalgae 𝑀 : concentration of nutrient 𝑆 : concentration of glucose 𝐶 : concentration of carbon dioxide 𝑀

𝑝𝑚𝑎𝑥 𝑀+𝑀

: concentration of nutrient in microalgae cells

𝐼𝐶 (𝑡) 𝐼𝑚 (𝑡) 𝐷𝑟 ℎ𝑟 𝛼𝐴 𝛼𝑆

: inflow of carbon diocxide : inflow of nutrient : died level of microalgae : harvest level of microalgae : biomass growth constant : photosynthesis constant

𝑇𝑢𝑟𝑛

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(1) (2) (3) (4)


𝑘1 𝑘2 𝑘3

: change level of 𝐶𝑂2 to (𝐶𝐻2 𝑂)6 : change level of nutrient to dried microalgae : change level of (𝐶𝐻2 𝑂)6 to dried microalgae

And we can get the stationary point from that model in this situation, 𝐴̇ = 𝑀 = 𝑆̇ =̇ 𝐶̇ = 0 so that, the stationary point is: ̅ , 𝑆 ̅ , 𝐶̅ ) 𝑥̅ = (𝐴̅, 𝑀 =(

𝐼𝑚 (𝑡) 𝑀𝑇𝑢𝑟𝑛 𝑘1 𝐼𝑚 (𝑡)(𝐷𝑟 + ℎ𝑟 ) 𝑘2 𝐼𝑐 (𝑡) − 𝑘1 𝑘3 𝐼𝑚 (𝑡) 𝐼𝑐 (𝑡) , , , ) (5) 𝑘2 (𝐷𝑟 + ℎ𝑟 ) ∝𝐴 𝑝𝑚𝑎𝑥 (𝑘2 𝐼𝑐 (𝑡) − 𝑘1 𝑘3 𝐼𝑚 (𝑡)) − 𝑘1 𝐼𝑚 (𝑡)(𝐷𝑟 + ℎ𝑟 ) 𝑘1 𝑘2 (𝐷𝑟 + ℎ𝑟 ) 𝑘1 ∝𝑠

= (1.6260; 1.7095; 3.9837; 0.1479) Then, we have to analysis the stability and the controllability in that stationary point in order to know the stability and controllability in equations (1) - (4). And we get the result from the analysis are the equations are stable and controlled. 2) Pontryagin Maximum Principle The Pontryagin maximum principle is used to find out the optimal control solution based on the purpose which maximize the objective function that the control 𝑢(𝑡) is limited on (𝑢(𝑡) ∈ 𝒰). This principle said that Hamiltonian equations will be maximum along 𝒰 which is the possible set. The steps are: Let, plant equation: 𝑥̇ = 𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑡) (6) performance index: 𝑡𝑓

𝐽 = 𝑆(𝑥(𝑡𝑓 ), 𝑡𝑓 ) + ∫ 𝑉(𝑥(𝑡), 𝑢(𝑡), 𝑡)𝑑𝑡

(7)

𝑡0

and the boundary conditions, 𝑥(𝑡0 ) = 𝑥0 and 𝑥(𝑡𝑓 ) = 𝑥𝑓 are free. So that the steps: 1. Make Pontryagin function (Hamilton) 𝐻(𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡), 𝑡) = 𝑉(𝑥(𝑡), 𝑢(𝑡), 𝑡) + 𝜆(𝑡)𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑡) (8) 2. Maximum 𝐻 to all control vector 𝑢(𝑡) 𝜕𝐻 ( )=0 (9) 𝜕𝑢 and get, 𝑢∗ (𝑡) = ℎ(𝑥 ∗ (𝑡), 𝜆∗ (𝑡), 𝑡) (10) ∗ 3. Use the result from step 2 into step 1 and find out the optimal 𝐻 𝐻 ∗ (𝑥 ∗ (𝑡), ℎ(𝑥 ∗ (𝑡), 𝜆∗ (𝑡), 𝑡), 𝜆∗ (𝑡), 𝑡) = 𝐻 ∗ (𝑥 ∗ (𝑡), 𝜆∗ (𝑡), 𝑡) (11) 4. Finish the state and costate equations, 𝜕𝐻 𝜕𝐻 𝑥 ∗ (𝑡) = + ( ) ; 𝜆∗ (𝑡) = − ( ) (12) 𝜕𝜆 𝜕𝑥 5. Substitute the solution 𝑥 ∗ (𝑡), 𝜆∗ (𝑡) from step 4 into control optimal expression 𝑢∗ in step 2 to get the optimal control.

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3) Optimal Control Case In this situation, it is denoted by 𝑢(𝑡). In general case, mathematical model of optimal control is, 𝑥̇ = 𝑓(𝑥(𝑡), 𝑢(𝑡), 𝑡) (13) in order to find the control 𝑢(𝑡) of optimal objective function 𝑡

𝐽(𝑢(𝑡)) = Φ(𝑥(𝑡𝑓 ), 𝑡𝑓 ) + ∫𝑡 𝑓 𝐿(𝑥(𝑡), 𝑢(𝑡), 𝑡)

(14)

0

with boundary conditions, 𝑥(𝑡0 ) = 𝑥0 and 𝑥(𝑡𝑓 ) = 𝑥𝑓

THE SOLUTION BY USING OPTIMAL CONTROL In this paper, will explain the purpose from the optimal control is to find out the inflow of nutrient in the model of microalgae growth by minimize the nutrient dilution rates so it will get more microalgae. From equations (1) – (4), will find out the optimal control by using Pontryagin maximum principle. The steps of that method are: a. The equations of microalgae growth model are, 𝑀 𝐴̇ = 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆 − (𝐷𝑟 + ℎ𝑟 )𝐴 (15) 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝑀 𝑀̇ = −𝑘2 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆 + 𝐼𝑚 (𝑡) (16) 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝑀 𝑆̇ = 𝛼𝑆 𝐶 − 𝑘3 ∝𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆 − (𝐷𝑟 + ℎ𝑟 )𝑆 (17) 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝐶̇ = −𝑘1 𝛼𝑆 𝐶 + 𝐼𝐶 (𝑡) (18) b. Objective Function Based on the purpose of this paper, so that the control case in this paper is to get the maximum of dried algae (𝐴) by minimize the nutrient dilution rates (𝐷) from the nutrient inflow (𝐼𝑚 ) in the microalgae growth process. From that case, the objective function is, 𝑡𝑓 𝐷 ) 𝐽(𝐼𝑚 = ∫ (𝐴(𝑇) − 𝐼𝑚 2 (𝑡)) 𝑑𝑡 (19) 2 𝑡0 c. the solution steps: 1) Hamilton Function Model 4

𝐷 𝐻(𝐴, 𝑀, 𝑆, 𝐶, 𝐼𝑚 , 𝜆) = 𝐴(𝑡) − 𝐼𝑚 2 (𝑡) + ∑ 𝜆𝑖 𝑓𝑖 2 𝑖=1

𝐷 𝑀 𝐻 = 𝐴(𝑡) − 𝐼𝑚 2 (𝑡) + 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆𝜆1 − (𝐷𝑟 + ℎ𝑟 )𝐴𝜆1 2 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝑀 − 𝑘2 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆𝜆2 + 𝐼𝑚 (𝑡)𝜆2 + 𝛼𝑆 𝐶𝜆3 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝑀 − 𝑘3 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆𝜆3 − (𝐷𝑟 + ℎ𝑟 )𝑆𝜆3 − 𝐼𝐶 (𝑡)𝜆4 𝑀 + 𝑀𝑇𝑢𝑟𝑛 2) Maximize 𝐻 to control vectors 𝑢(𝑡) = 𝐼𝑚 (𝑡) 𝜕𝐻 =0 𝜕𝑢

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(20)


𝜕𝐻 =0 𝜕𝐼𝑚 −𝐷𝐼𝑚 + 𝜆2 = 0 𝐷𝐼𝑚 = 𝜆2 𝜆2 ̅̅̅ 𝐼𝑚 = 𝐷 We use the minimal value of 𝐼𝑚 = 0.2 it mean the minimal value of nutrient which get in to the microalgae cells and the maximum is 𝐼𝑚 = 4.2. So we can denote, 0.2 ≤ 𝐼𝑚 ≤ 4.2 0.2 , ̅̅̅ 𝐼𝑚 ≤ 0.2 𝐼𝑚 = {̅̅̅ 𝐼𝑚 , 0.2 < ̅̅̅ 𝐼𝑚 < 4.2 4.2 , ̅̅̅ 𝐼𝑚 ≥ 4.2 So that, optimal control 𝐼𝑚 (𝑡)can be write, 𝐼𝑚∗ (𝑡) = min(4.2, 𝑚𝑎𝑥(0.2, ̅̅̅ 𝐼𝑚 ))

(21)

3) Find optimal 𝐻 ∗ Substitute equation (21) into equation (20) so that, 𝐷 𝑀 𝐻 ∗ = 𝐴(𝑡) − 𝐼𝑚 2 (𝑡) + 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆𝜆1 − (𝐷𝑟 + ℎ𝑟 )𝐴𝜆1 2 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝑀 − 𝑘2 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆𝜆2 + ((min(4, max(0, ̅̅̅ 𝐼𝑚 ))))𝜆2 + 𝛼𝑆 𝐶𝜆3 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝑀 − 𝑘3 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆𝜆3 − 𝑘1 𝛼𝑆 𝐶𝜆4 + 𝐼𝐶 (𝑡)𝜆4 𝑀 + 𝑀𝑇𝑢𝑟𝑛 4) Finish the state and costate equations to find out the optimal system. a) State equation 𝜕𝐻 ∗ 𝑀 = 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆 − (𝐷𝑟 + ℎ𝑟 )𝐴 (22) 𝜕𝜆1 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝜕𝐻 ∗ 𝑀 = −𝑘2 𝛼𝐴 (𝑝𝑚𝑎𝑥 ) 𝑆 + (min(4, max(0, ̅̅̅ 𝐼𝑚 ))) (23) 𝜕𝜆2 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝜕𝐻 ∗ 𝑀 = 𝛼𝑆 𝐶 − 𝑘3 𝛼𝐴 (𝑝𝑚𝑎𝑥 )𝑆 (24) 𝜕𝜆3 𝑀 + 𝑀𝑇𝑢𝑟𝑛 𝜕𝐻 ∗ = −𝑘1 𝛼𝑆 𝐶 + 𝐼𝐶 (𝑡) (25) 𝜕𝜆4 b) Costate equation 𝜕𝜆1 𝜕𝐻 ∗ =− = (𝐷𝑟 + ℎ𝑟 )𝜆1 − 1 (26) 𝜕𝑡 𝜕𝐴 𝜕𝜆2 𝜕𝐻 ∗ 𝛼𝐴 𝑝𝑚𝑎𝑥 𝑆 (−𝜆1 + 𝑘2 𝜆2 + 𝑘3 𝜆3 ) =− = 𝜕𝑡 𝜕𝑀 (𝑀 + 𝑀𝑇𝑢𝑟𝑛 ) 𝛼𝐴 𝑝𝑚𝑎𝑥 𝑀𝑆 (−𝜆1 − 𝑘2 𝜆2 + 𝑘3 𝜆3 ) + (27) (𝑀 + 𝑀𝑇𝑢𝑟𝑛 )2 𝜕𝜆3 𝜕𝐻 ∗ =− = −𝛼𝑆 𝜆3 + 𝑘1 𝛼𝑆 𝜆4 (28) 𝜕𝑡 𝜕𝑆 The next step is finish equations (22) – (28) by numerical using MATLAB.

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SIMULATION USING MATLAB Table 1. value of parameters in simulation Parameters value Constanta of biomass growth 10.2 𝛼𝐴 Concentration of nutrient in saturation cells 0.4 𝑝𝑚𝑎𝑥 Level of microalgae death 0.46 𝐷𝑟 Level of microalgae harvest 2 ℎ𝑟 Concentration of half saturation nutrient that get 4 𝑀𝑇𝑢𝑟𝑛 when outside nutrient Inflow of nutrient 0.2 𝐼𝑚 Constanta of photosynthesis 67.6 𝛼𝑆 0.4 Change level of 𝐶𝑂2 to (𝐶𝐻2 𝑂)6 𝑘1 Change level of nutrient to dried microalgae 0.05 𝑘2 0.05 Change level of (𝐶𝐻2 𝑂)6 to dried microalgae 𝑘3 Inflow of carbon dioxide 4 𝐼𝐶 0 0 0 0 Let the initial conditions: (𝐴 , 𝑀 , 𝑆 , 𝐶 ) is (3, 0.2, 5, 2.5), (3, 0.4, 10, 5), and (3, 0.6, 15, 7.5).

 Initial condition (𝐴0 , 𝑀0 , 𝑆 0 , 𝐶 0 ) = (3, 0.6, 15, 7.5)

Figure 1. Initial condition given control with initial condition (𝐴0 , 𝑀0 , 𝑆 0 , 𝐶 0 ) = (3, 0.6, 15, 7.5) CONCLUSION Based on the analysis in the step before, the conclusion are: 1. The dynamic system in Thornton microalgae growth is stable and controlled. The stationary point is 𝐸 = (𝐴, 𝑀, 𝑆, 𝐶) = (1.6260; 1.7095; 3.9837; 0.1479) 2. The result from the flow of nutrient (𝐼𝑚 (𝑡)) as optimal control by using Pontryagin Maximum Principle is: 0.2 , ̅̅̅ 𝐼𝑚 ≤ 0.2 ̅̅̅ 𝐼𝑚 = {𝐼𝑚 , 0.2 < ̅̅̅ 𝐼𝑚 < 4.2 ̅̅̅ 4.2 , 𝐼𝑚 ≥ 4.2 or it can write as: 𝐼𝑚∗ (𝑡) = min(4.2, 𝑚𝑎𝑥(0.2, ̅̅̅ 𝐼𝑚 )) 6


3. The best control for microalgae growth is: (𝐴0 , 𝑀0 , 𝑆 0 , 𝐶 0 ) = (3, 0.6, 15, 7.5) 4. The result of objective function with value of nutrient dilution rate 𝐷 = 0.1 and initial condition (𝐴0 , 𝑀0 , 𝑆 0 , 𝐶 0 ) = (3, 0.6, 15, 7.5) and the result of objective function 67.7631 𝑔[𝐴]𝑚−3. REFERENCESS [1] Nanda. 2016. Tugas Akhir “Kendali Optimal Pertumbuhan Mikroalga Melalui Tingkat Pengenceran Nutrisi”. Jurusan Matematika FMIPA. Surabaya: ITS. [2] Widodo, Basuki. 2012. “Pemodelan Matematika”. ITS Press: Surabaya.

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