Simultaneous Process Scheduling and. Control: A Multiparametric Programming. Based Approach

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Simultaneous Process Scheduling and

Control: A Multiparametric Programming

Based Approach

Baris Burnak,

†,†

Justin Katz,

‡,†

Nikolaos A. Diangelakis,

‡,†

and Efstratios N.

Pistikopoulos

∗,‡,†

†Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station TX, 77845

‡Texas A&M Energy Institute, Texas A&M University, College Station TX, 77845 E-mail: [email protected]

The Supporting Information comprises the discrete state space models of the open loop and closed loop systems, along with their step and impulse responses, respectvely. We also present an example of the simultaneous decisions by the scheduler, surrogate model, and the controller.

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Appendix: State space models of the open loop and

closed loop systems

We provide the state space matrices derived to approximate the open loop high-fidelity CSTR model as follows.

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A =                    9.67 · 10−1 8.00 · 10−4 −2.30 · 10−3 _{−9.15 · 10}−2 _{−2.31 · 10}−3 _{−2.00 · 10}−3 _{5.12 · 10}−4 2.10 · 10−3 9.65 · 10−1 −1.19 · 10−2 _{8.36 · 10}−3 _{−7.93 · 10}−2 _{4.75 · 10}−3 _{−3.89 · 10}−2 7.49 · 10−3 −7.53 · 10−3 _{8.61 · 10}−1 _{2.14 · 10}−2 _{−1.06 · 10}−1 _{3.47 · 10}−2 _{1.36 · 10}−1 6.56 · 10−2 1.72 · 10−2 1.27 · 10−2 9.54 · 10−1 1.04 · 10−2 2.27 · 10−1 −3.09 · 10−2 1.87 · 10−2 1.79 · 10−2 2.19 · 10−2 3.88 · 10−4 9.85 · 10−1 8.79 · 10−2 −3.98 · 10−2 −1.44 · 10−1 _{−1.51 · 10}−2 _{−1.93 · 10}−2 _{−7.41 · 10}−2 _{−1.38 · 10}−2 _{5.65 · 10}−1 _{1.48 · 10}−2 1.49 · 10−2 1.13 · 10−1 −3.65 · 10−3 _{−3.18 · 10}−3 _{−8.20 · 10}−3 _{5.79 · 10}−3 _{6.95 · 10}−1                    B =                    2.35 · 10−3 −1.47 · 10−3 _{−1.82 · 10}−3 _{3.36 · 10}−3 1.46 · 10−3 −1.36 · 10−3 _{3.38 · 10}−3 _{−1.76 · 10}−3 2.69 · 10−3 −5.12 · 10−3 _{7.85 · 10}−3 _{−7.13 · 10}−3 −7.75 · 10−3 _{5.44 · 10}−3 _{1.62 · 10}−2 _{6.43 · 10}−3 −3.56 · 10−3 _{2.36 · 10}−3 _{5.31 · 10}−3 _{1.37 · 10}−3 1.51 · 10−2 −1.36 · 10−2 _{−3.41 · 10}−2 _{−5.63 · 10}−3 −1.28 · 10−2 _{−4.08 · 10}−3 _{−1.10 · 10}−2 _{−3.00 · 10}−3                    C =                    −5.37 · 10−6 5.32 · 10−6 −5.07 · 10−5 −2.50 · 10−5 −8.96 · 10−7 4.86 · 10−5 5.89 · 10−5                    D =       2.73 · 10−2 −7.48 3.03 2.02 · 10−2 −8.16 · 10−2 _{6.13 · 10}−2 _{4.94 · 10}−1 5.82 4.97 9.08 · 10−1 −3.16 · 10−1 _{−3.06 · 10}−1 _{1.61 · 10}−2 _{4.02 · 10}−2 −7.05 7.30 1.34 2.82 · 10−1 −4.33 · 10−1 5.46 · 10−2 4.80 · 10−2      

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Similarly, the closed loop state space models are determined as follows. Surrogate model 1 xt+1 =       0.004 −0.001 0.002 −0.031 −0.010 0.045 −0.118 −0.026 0.118       xt+       −7.2 −4.7 −3.1       10−4Qtotal,t yt=       0.340 −0.037 0.066 0.072 −0.040 0.031 0.048 −0.042 0.041       xt (S1) Surrogate model 2 xt+1=       0.045 0.027 −0.012 0.089 −0.022 −0.035 0.027 0.021 −0.092       xt+       2.4 · 10−4 0.130 9.0 · 10−5 −0.749 2.9 · 10−6 −0.716          Qtotal,t CSP P2,t    yt=       0.105 −0.038 −0.018 0.738 −1.005 −0.381 0 0 0       xt (S2) Surrogate model 3 xt+1 =       −0.011 −0.012 −0.016 −0.067 0.112 0.117 0.134 −0.148 0.220       xt+       2.5 · 10−4 0.171 9.8 · 10−5 −0.620 −5.5 · 10−5 _0.192          Qtotal,t CSP P3,t    yt=       0.014 −0.008 0.004 0 0 0 0.516 −1.081 0.477       xt (S3)

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Appendix: Step and impulse responses of the open loop

and closed loop systems

The step response and the impulse response of the state space model is provided in Figures S1 and S2, respectively. Also, the step and impulse responses of the surrogate models are provided in Figures S3 - S5. C P1 [m ol /L] ×10 -4 -4 -2 0 Qtotal -0.4 -0.2 0 aR2 ∈[0, 0.5) -0.4 -0.2 0 aR2 ∈[0.5, 1] -0.4 -0.2 0 aR3 ∈[0, 0.55) -0.4 -0.2 0 aR3 ∈[0.55, 1] C P2 [m ol /L] ×10 -4 -4 -2 0 -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 -0.2 0 0.2 0.4 Time [min] 0 200 400 C P 3 [m ol /L] ×10 -4 -4 -2 0 Time [min] 0 200 400 -0.4 0 0.4 0.8 Time [min] 0 200 400 -0.4 0 0.4 0.8 Time [min] 0 200 400 -0.4 0 0.4 0.8 Time [min] 0 200 400 -0.4 0 0.4 0.8

Figure S1: Step responses of the identified open loop model with respect to the system inputs

and the scheduling variable Qtotal.

C P1 [m ol /L] × 10-5 -20 -10 0 Qtotal -0.02 0 aR2∈[0, 0.5) -0.02 0 aR2∈[0.5, 1] -0.02 0 aR3∈[0, 0.55) -0.02 0 aR3∈[0.55, 1] C P2 [m ol /L] × 10-5 -20 -10 0 0 0.02 0.04 0 0.02 0.04 0 0.02 0.04 0 0.02 0.04 Time [min] 0 200 400 C P 3 [m ol /L] × 10-5 -20 -10 0 Time [min] 0 200 400 -0.05 0 0.05 Time [min] 0 200 400 -0.05 0 0.05 Time [min] 0 200 400 -0.05 0 0.05 Time [min] 0 200 400 -0.05 0 0.05

Figure S2: Impulse responses of the identified open loop model with respect to the system

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C P1 [m ol /L] × 10-4 -4 -2 0 Qtotal C P2 [m ol /L] × 10-5 -4 -2 0 Time [min] 0 5 10 15 20 C P3 [m ol /L] × 10-5 -4 -2 0

(a) Step response

C P1 [m ol /L] × 10-4 -1 0 1 Qtotal C P2 [m ol /L] × 10-5 -5 0 5 Time [min] 0 5 10 15 20 C P3 [m ol /L] × 10-5 -5 0 5 (b) Impulse response

Figure S3: Step and impulse responses of Surrogate model 1.

C P1 [m ol /L] × 10-5 -2 0 2 Qtotal _×₁₀-5 -2 0 2 aR2 C P 2 [m ol /L] × 10-4 -1 0 1 ×10 -4 -1 0 1 Time [min] 0 10 20 30 40 C P3 [m ol /L] × 10-5 -1 0 1 Time [min] 0 10 20 30 40 ×10-5 -1 0 1

(a) Step response

C P1 [m ol /L] × 10-5 -1 0 1 Qtotal _×₁₀-5 -1 0 1 aR2 C P 2 [m ol /L] × 10-4 -1 0 1 ×10 -4 -1 0 1 Time [min] 0 10 20 30 40 C P3 [m ol /L] × 10-5 -1 0 1 Time [min] 0 10 20 30 40 ×10-5 -1 0 1 (b) Impulse response Figure S4: Step and impulse responses of Surrogate model 2.

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C P 1 [m ol /L] ×10 -5 -1 0 1 Qtotal _×₁₀-5 -1 0 1 aR3 C P 2 [m ol /L] × 10-5 -1 0 1 ×10 -5 -1 0 1 Time [min] 0 10 20 30 40 C P 3 [m ol /L] ×10-4 -1 0 1 Time [min] 0 10 20 30 40 ×10-4 -1 0 1

(a) Step response

C P1 [m ol /L] × 10-6 -1 0 1 2 Qtotal _×₁₀-6 -1 0 1 2 $a_{R_3$ C P2 [m ol /L] × 10-6 -1 0 1 ×10 -6 -1 0 1 Time [min] 0 10 20 30 40 C P 3 [m ol /L] × 10-5 -5 0 5 Time [min] 0 10 20 30 40 ×10-5 -5 0 5 (b) Impulse response Figure S5: Step and impulse responses of Surrogate model 3.

Appendix: A snapshot of the simultaneous decisions

We present an example of the simultaneous decisions by the scheduler, surrogate model, and

the controller at the end of the 1st _{hour of the operation depicted in Figure 6. Following are}

the states and the remaining parameters of the system at t = 1h.

W =       1.26 0.24 0.65       , CP =       0.24 1.5 × 10−4 0       , aR,tc=−1 =       1 0 0       DRts=0 =       5.81 6.06 5.89       , DRts=1 =       5.79 6.09 5.89       , DRts=2 =       5.82 6.06 5.97       (S4)

Locating the system for the given parameters in Equation S4, the optimal decisions can be evaluated from the corresponding affine functions, as presented in Table S1.

Due to the transition from CP1 to CP2, the surrogate and control actions are mostly

saturated. To highlight the affine expressions, a snapshot from the production period (t = 105 min) is also presented in Table S2. Note that the product concentrations at t = 105 min

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Table S1: Optimal decisions for the system parameters given in Equation S4.

Decision variable Affine expression

Ftotal,ts=0 = −16.7W2+ DRts=0,2+ DRts=1,2 Ftotal,ts=1 = −16.7W3+ DRts=0,3+ DRts=1,3+ DRts=2,3 Ftotal,ts=2 = DRts=2,2 Qtotal = 500 C_PSP₁ = 0 CSP P2 = 0.91CP1 − 0.01CP2 + 0.02 CSP P3 = 0 aR1 = 0 aR2 = 0.55 aR3 = 0

Table S2: Optimal decisions at t = 105 min. Decision

variable Affine expression

Qtotal = Ftotal,ts=0/CP2 + 1.59 × 10 2 CSP P1 = 0 CSP P2 = 0.12 C_PSP₃ = 0 aR1 = 1 − aR2 aR2 = 0.06CP1 + 0.64CP2 − 0.01CP3 − 4.0 × 10 −5_Q total− 0.08CPSP1 − 0.67C SP P2 + 0.98aR,tc=−1− 0.04 aR3 = 0