Presentation at Modelica Conference 2015

1. 𝑀𝑎𝑔𝑑𝑎𝑙𝑒𝑛𝑎 𝐴𝑥𝑒𝑙𝑠𝑠𝑜𝑛1
, 𝐹𝑟𝑒𝑑𝑟𝑖𝑘 𝑀𝑎𝑔𝑛𝑢𝑠𝑠𝑜𝑛2
, 𝑇𝑜𝑖𝑣𝑜 𝐻𝑒𝑛𝑛𝑖𝑛𝑔𝑠𝑠𝑜𝑛1
A FRAMEWORK FOR NONLINEAR MODEL PREDICTIVE
CONTROL IN JMODELICA.ORG
11. 𝑀𝑜𝑑𝑒𝑙𝑜𝑛, 𝐿𝑢𝑛𝑑, 𝑆𝑤𝑒𝑑𝑒𝑛
2. 𝐷𝑒𝑝𝑎𝑟𝑡𝑚𝑒𝑛𝑡 𝑜𝑓 𝐴𝑢𝑡𝑜𝑚𝑎𝑡𝑖𝑐 𝐶𝑜𝑛𝑡𝑟𝑜𝑙, 𝐿𝑢𝑛𝑑 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦, 𝑆𝑤𝑒𝑑𝑒𝑛

2. • Background
 Model Predictive Control
Optimal control problem
Example
 JModelica.org
Reason for new framework
Optimica
Discretization
• Comparison between frameworks
 Theory
 Benchmark
• Features
• Conclusions and further work
OVERVIEW

3. • Optimization-based control strategy
• Repeated on-line solution of an open-loop Optimal Control Problem (OCP)
• Feedback by measuring the state prior to each optimization
• Main advantages
 Easily extends to multivariable processes
 Instrinscally handles constraints
• Main limitations
 Computationally demanding
 Solution not guaranteed
MODEL PREDICTIVE CONTROL

4. • Objective function min
𝑥,𝑦,𝑢,𝑤 𝑡0
𝑡 𝑓
𝑤𝑠𝑒𝑡 − 𝑤 𝑇
𝑄 𝑤𝑠𝑒𝑡 − 𝑤 𝑑𝑡
𝑤 = (𝑥 𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑒𝑑, 𝑦𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑒𝑑, 𝑢 𝑐𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑒𝑑)
𝑠. 𝑡.
• Model 𝐹 𝑥 𝑡 , 𝑥 𝑡 , 𝑦 𝑡 , 𝑢 𝑡 = 0
• Initial condition 𝑥 𝑡0 = 𝑥0
𝑥 𝐿 ≤ 𝑥 𝑡 ≤ 𝑥 𝑈
• Constraints 𝑔 𝑥 𝑡 , 𝑥 𝑡 , 𝑦 𝑡 , 𝑢 𝑡 ≤ 0
∀𝑡 ∈ [𝑡0, 𝑡𝑓]
OPTIMAL CONTROL PROBLEM

5. MODEL PREDICTIVE CONTROL
Sample period = 10s
At each sample point
• Measure state
• Optimize
• Apply first input

6. Open-source platform for simulation and optimization of Modelica models
Optimization problem formulated in Optimica
 Extension of the Modelica language which enables formulation of
objective functions and constraints
• JModelica.org has a framework for optimization -> Open-Loop framework
• Main goals of MPC framework
 Decrease the total computation time for each optimization
Utilize the fact that the structure of the discretized OCP does not change between
optimizations
 Make JModelica.org easier to use for MPC schemes
Streamline the setup
Provide relevant features
JMODELICA.ORG

7. optimization CSTR_opt(objectiveIntegrand=(c_ref-cstr.c)^2 +
(T_ref-cstr.T)^2 + (Tc_ref-cstr.Tc)^2,
startTime=0.0,
finalTime=150)
CSTR cstr(T(max=350));
input Real u(start = 350,min=230,max=370)=cstr.Tc;
parameter Real c_ref = 500;
parameter Real T_ref = 320;
parameter Real Tc_ref = 300;
constraint
cstr.c <= 1000;
end CSTR_opt;
OPTIMICA CODE

8. • JModelica.org uses direct collocation to transcribe the OCP to a
NonLinear Program (NLP)
 Optimization problem is considered at a number of collocation
points 𝑡 = {𝑡 𝑐1, 𝑡 𝑐2, … , 𝑡 𝑐𝑛}, rather than ∀𝑡 ∈ 𝑡0, 𝑡𝑓
• Underlying NLP solver is IPOPT
 Solution not guaranteed
 Important with good initial guess
DISCRETIZATION

9. OL framework
• Pre-processing
 OCP transcribed to NLP
 Initial guess extracted
 Scaling
 Solver object created
• Solution
 NLP sent to solver
• Post-processing
 Result is processed
 Result object created and
returned
COMPARISON BETWEEN FRAMEWORKS
MPC framework
• Initialization – Only done once
 OCP transcribed to NLP
 Initial guess extracted
 Scaling
 Solver object created
• Pre-processing
 Initial condition updated
 Initial guess updated
 Prediction horizon shifted
• Solution
 NLP sent to solver (warm started)
• Post-processing
 Result is processed
 First input in optimal control sequence
returned

10. • Developed by F. Casella
• Objective
 𝑝𝑠𝑒𝑡 = 8.35 𝑀𝑃𝑎
 𝑙𝑜𝑎𝑑 𝑠𝑒𝑡 = 1
𝑡0
𝑡 𝑓
8.35 − 𝑝 2
+ 1 − 𝑙𝑜𝑎𝑑 2
𝑑𝑡
• Constraints
 𝜎 ≤ 260 𝑀𝑃𝑎
 0 ≤ 𝑢 ≤ 1
 0 ≤ 𝑢 ≤
0.1
60
BENCHMARK MODEL
Combined-Cycle Power Plant (CCPP)

11. BENCHMARK RESULTS
Sample period = 100s

12. • Soften constraints
 OCP may become infeasible if hard constraints are applied
𝑥0 > 𝑥 𝑈, 𝑥 𝑡0 = 𝑥0, 𝑥 𝑡 ≤ 𝑥 𝑈
 Hard constraints need to be softened
• Initial guess
 MPC framework automatically uses the solution to one optimization as
the initial guess for the next
• Unsuccessful optimization
 If an optimization is unsuccessful the MPC framework returns the
second input in the last sucessful optimization
FEATURES
+ 𝜀(𝑡)

13. • We obtain the same solution using the MPC framework as using the
OL framework
• The MPC framework is significantly faster than the OL framework
 CCPP: 𝑇𝑡𝑜𝑡 ≈ −70%
 Mainly due to time-consuming discretization only done once
• The MPC framework handles a lot of things internally, making
scripting easier
CONCLUSIONS

14. • Some optimization options are incompatible with the MPC
framework
• More functionality
 Different ways of softening variable bounds and constraints
 Shifting of dual variables
 Method to obtain the value of the objective function
FURTHER WORK

15. magdalena. axelsson@modelon. com