Optimization Methods in Energy and Power Systems Lecture
![Optimization Methods in Energy and Power Systems Lecture 7: Nonlinear and Mixed-Integer Nonlinear Programming Optimization Methods in Energy and Power Systems Lecture 7: Nonlinear and Mixed-Integer Nonlinear Programming](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-1.jpg)
![Nonlinear & Mixed-Integer Nonlinear Programming ü Introduction of Nonlinear Programming Problems (NLP) ü Formulating Nonlinear & Mixed-Integer Nonlinear Programming ü Introduction of Nonlinear Programming Problems (NLP) ü Formulating](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-2.jpg)
![Nonlinear Programming (Introduction) ü Variables are continuous ü The nonlinearity can be a part Nonlinear Programming (Introduction) ü Variables are continuous ü The nonlinearity can be a part](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-3.jpg)
![Nonlinear Programming (Introduction) All the representation Forms in LP problems (General, Canonical, and Standard) Nonlinear Programming (Introduction) All the representation Forms in LP problems (General, Canonical, and Standard)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-4.jpg)
![an Obj d/ ec m or F tive ig ht eas Fu be ibl an Obj d/ ec m or F tive ig ht eas Fu be ibl](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-5.jpg)
![Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch Cost Coefficient (€/MW) Unit Output (MW) Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch Cost Coefficient (€/MW) Unit Output (MW)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-6.jpg)
![Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) M. Pourakbari Kasmaei 7 Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) M. Pourakbari Kasmaei 7](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-7.jpg)
![Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) # of Units Generic Formulation Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) # of Units Generic Formulation](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-8.jpg)
![Nonlinear Programming (NLP) Set i /1*2/; Example Variable z; Positive Variables Pg(i); Equations Eq. Nonlinear Programming (NLP) Set i /1*2/; Example Variable z; Positive Variables Pg(i); Equations Eq.](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-9.jpg)
![Nonlinear Programming (NLP) Attention ü For nonlinear problems, finding a suitable solver is important. Nonlinear Programming (NLP) Attention ü For nonlinear problems, finding a suitable solver is important.](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-10.jpg)
![Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-11.jpg)
![Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-12.jpg)
![Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-13.jpg)
![Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-14.jpg)
![Nonlinear Programming (NLP) It seems that there is No Big Difference among Well-known Solvers! Nonlinear Programming (NLP) It seems that there is No Big Difference among Well-known Solvers!](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-15.jpg)
![Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (DNLP) Let us consider a non-smooth Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (DNLP) Let us consider a non-smooth](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-16.jpg)
![Nonlinear Programming (NLP) Ø Discontinuous Nonlinear Program (DNLP) ü Model with Non-smooth Nonlinear terms Nonlinear Programming (NLP) Ø Discontinuous Nonlinear Program (DNLP) ü Model with Non-smooth Nonlinear terms](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-17.jpg)
![Nonlinear Programming (NLP) Set i /1*2/; Variable z; Positive Variables Pg(i); Equations Eq. Obj, Nonlinear Programming (NLP) Set i /1*2/; Variable z; Positive Variables Pg(i); Equations Eq. Obj,](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-18.jpg)
![Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-19.jpg)
![Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-20.jpg)
![Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-21.jpg)
![Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) Failed in Finding the Solution! Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) Failed in Finding the Solution!](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-22.jpg)
![Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) Failed in Finding the Solution! Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) Failed in Finding the Solution!](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-23.jpg)
![Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-24.jpg)
![Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-25.jpg)
![Nonlinear Programming (NLP) Choosing Proper Solver do Matter! M. Pourakbari Kasmaei, 2019 26 Nonlinear Programming (NLP) Choosing Proper Solver do Matter! M. Pourakbari Kasmaei, 2019 26](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-26.jpg)
![Mixed-Integer NLP (MINLP) ü The nature of problem is Nonlinear. ü At least one Mixed-Integer NLP (MINLP) ü The nature of problem is Nonlinear. ü At least one](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-27.jpg)
![an Obj d e m /or ctiv ig F e h t ea F an Obj d e m /or ctiv ig F e h t ea F](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-28.jpg)
![Mixed-Integer NLP (MINLP) ü Like MILP Problems, we have: • Mixed-Binary NLP (MBNLP) Problem Mixed-Integer NLP (MINLP) ü Like MILP Problems, we have: • Mixed-Binary NLP (MBNLP) Problem](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-29.jpg)
![Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) Cannot Be Handled By Commercial Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) Cannot Be Handled By Commercial](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-30.jpg)
![Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) ü Instead of having linear Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) ü Instead of having linear](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-31.jpg)
![Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) ü Assign Binary Variables to Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) ü Assign Binary Variables to](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-32.jpg)
![Mixed-Integer NLP (MINLP) Set i /1*2/; Variable z; Positive Variables Pg(i); Equations Eq. Obj, Mixed-Integer NLP (MINLP) Set i /1*2/; Variable z; Positive Variables Pg(i); Equations Eq. Obj,](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-33.jpg)
![Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) Solution Using KNITRO ---- VAR Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) Solution Using KNITRO ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-34.jpg)
![Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) Non-smooth Terms M. Pourakbari Kasmaei Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) Non-smooth Terms M. Pourakbari Kasmaei](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-35.jpg)
![Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) ü Transform these terms into Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) ü Transform these terms into](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-36.jpg)
![Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) Solution Using KNITRO ---- VAR Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) Solution Using KNITRO ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-37.jpg)
![Some Linearization Techniques Linearization Of Product Terms & SOS 2 Technique M. Pourakbari Kasmaei, Some Linearization Techniques Linearization Of Product Terms & SOS 2 Technique M. Pourakbari Kasmaei,](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-38.jpg)
![Some Linearization Techniques 1) Product of Two Binary Variables M. Pourakbari Kasmaei, 2019 39 Some Linearization Techniques 1) Product of Two Binary Variables M. Pourakbari Kasmaei, 2019 39](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-39.jpg)
![Some Linearization Techniques 2) Product of One Binary and One Continuous M. Pourakbari Kasmaei, Some Linearization Techniques 2) Product of One Binary and One Continuous M. Pourakbari Kasmaei,](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-40.jpg)
![Some Linearization Techniques 3) Product of Two Continuous Variables M. Pourakbari Kasmaei, 2019 41 Some Linearization Techniques 3) Product of Two Continuous Variables M. Pourakbari Kasmaei, 2019 41](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-41.jpg)
![Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2) Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-42.jpg)
![Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2) Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-43.jpg)
![Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2) Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-44.jpg)
![Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2) Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-45.jpg)
![Thanks! M. Pourakbari Kasmaei, 2019 46 Thanks! M. Pourakbari Kasmaei, 2019 46](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-46.jpg)
- Slides: 46
![Optimization Methods in Energy and Power Systems Lecture 7 Nonlinear and MixedInteger Nonlinear Programming Optimization Methods in Energy and Power Systems Lecture 7: Nonlinear and Mixed-Integer Nonlinear Programming](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-1.jpg)
Optimization Methods in Energy and Power Systems Lecture 7: Nonlinear and Mixed-Integer Nonlinear Programming Mahdi Pourakbari Kasmaei, 2019 Thursday, 11 April 2019
![Nonlinear MixedInteger Nonlinear Programming ü Introduction of Nonlinear Programming Problems NLP ü Formulating Nonlinear & Mixed-Integer Nonlinear Programming ü Introduction of Nonlinear Programming Problems (NLP) ü Formulating](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-2.jpg)
Nonlinear & Mixed-Integer Nonlinear Programming ü Introduction of Nonlinear Programming Problems (NLP) ü Formulating NLP Problems ü Introduction of Mixed-Integer NLP (MINLP) ü Formulating MINLP Problems M. Pourakbari Kasmaei, 2019 2
![Nonlinear Programming Introduction ü Variables are continuous ü The nonlinearity can be a part Nonlinear Programming (Introduction) ü Variables are continuous ü The nonlinearity can be a part](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-3.jpg)
Nonlinear Programming (Introduction) ü Variables are continuous ü The nonlinearity can be a part of: • Objective function • The feasible region that is determined by nonlinear constraint(s). M. Pourakbari Kasmaei, 2019 3
![Nonlinear Programming Introduction All the representation Forms in LP problems General Canonical and Standard Nonlinear Programming (Introduction) All the representation Forms in LP problems (General, Canonical, and Standard)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-4.jpg)
Nonlinear Programming (Introduction) All the representation Forms in LP problems (General, Canonical, and Standard) are also available in NLP problems. M. Pourakbari Kasmaei, 2019 4
![an Obj d ec m or F tive ig ht eas Fu be ibl an Obj d/ ec m or F tive ig ht eas Fu be ibl](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-5.jpg)
an Obj d/ ec m or F tive ig ht eas Fu be ibl nc No e R tion nl eg in io ea n r Formulating NLP Problems (General Form) M. Pourakbari Kasmaei, 2019 5
![Nonlinear Programming NLP Example 7 1 Economic Dispatch Cost Coefficient MW Unit Output MW Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch Cost Coefficient (€/MW) Unit Output (MW)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-6.jpg)
Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch Cost Coefficient (€/MW) Unit Output (MW) ü In Practice, the cost function is of order 5 or higher. We use Quadratic Cost Function. M. Pourakbari Kasmaei 6
![Nonlinear Programming NLP Example 7 1 Economic Dispatch NLP M Pourakbari Kasmaei 7 Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) M. Pourakbari Kasmaei 7](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-7.jpg)
Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) M. Pourakbari Kasmaei 7
![Nonlinear Programming NLP Example 7 1 Economic Dispatch NLP of Units Generic Formulation Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) # of Units Generic Formulation](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-8.jpg)
Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) # of Units Generic Formulation of ED M. Pourakbari Kasmaei 8
![Nonlinear Programming NLP Set i 12 Example Variable z Positive Variables Pgi Equations Eq Nonlinear Programming (NLP) Set i /1*2/; Example Variable z; Positive Variables Pg(i); Equations Eq.](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-9.jpg)
Nonlinear Programming (NLP) Set i /1*2/; Example Variable z; Positive Variables Pg(i); Equations Eq. Obj, Eq 1, Eq 2, Eq 3, Eq 4, Eq 5; Table gendata(i, *) a b c upplim lowlim 1 0. 0003 10. 1 671 50 0 2 0. 0013 8. 8 374 40 0; 7. 1: Economic Dispatch (NLP) ü ü Set Table Eq. Obj. . z =e= sum(i, gendata(i, 'a')*Power(Pg(i), 2) + gendata(i, 'b')*Pg(i) + gendata(i, 'c')); Eq 1. . sum(i, Pg(i)) =e= 62; Eq 2. . (2/3)*Pg('1') + Pg('2') =l= 68; Eq 3. . 2*Pg('1') + 0. 3*Pg('2') =g= 50; ü NLP Solver Eq 4(i). . Pg(i) =l= gendata(i, 'upplim'); Eq 5(i). . Pg(i) =g= gendata(i, 'lowlim'); Model NLP_ED /ALL/; Solve NLP_ED USING NLP MINIMIZING Z; M. Pourakbari Kasmaei 9
![Nonlinear Programming NLP Attention ü For nonlinear problems finding a suitable solver is important Nonlinear Programming (NLP) Attention ü For nonlinear problems, finding a suitable solver is important.](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-10.jpg)
Nonlinear Programming (NLP) Attention ü For nonlinear problems, finding a suitable solver is important. • Although the nonlinearity makes some obstacles in finding the global solution, the main obstacles are related to non-convexity. • Non-convexity results in multiple local optimal solutions. M. Pourakbari Kasmaei 10
![Nonlinear Programming NLP Example 7 1 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-11.jpg)
Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1621. 425 +INF 22. 000 40. 000 +INF Solution Using CONOPT MARGINAL. . . ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 LOWER 1045. 000 62. 000 -INF 50. 000 LEVEL 1045. 000 62. 000 54. 667 56. 000 UPPER MARGINAL 1045. 000 1. 000 62. 000 10. 113 68. 000. +INF. -INF 22. 000 40. 000 50. 000 40. 000 . -1. 209 . . 22. 000 40. 000 +INF . . M. Pourakbari Kasmaei 11
![Nonlinear Programming NLP Example 7 1 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-12.jpg)
Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1621. 425 +INF MARGINAL. 22. 000 40. 000 9. 2833 E-8 EPS +INF Solution Using KNITRO ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 M. Pourakbari Kasmaei LOWER 1045. 000 62. 000 -INF 50. 000 LEVEL 1045. 000 62. 000 54. 667 56. 000 UPPER MARGINAL 1045. 000 1. 000 62. 000 10. 113 68. 000 -2. 003 E-7 +INF 8. 0213 E-6 -INF 22. 000 40. 000 50. 000 40. 000 EPS -1. 209 . . 22. 000 40. 000 +INF EPS 12
![Nonlinear Programming NLP Example 7 1 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-13.jpg)
Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1621. 425 +INF 22. 000 40. 000 +INF Solution Using LINDO MARGINAL. . . ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 M. Pourakbari Kasmaei LOWER 1045. 000 62. 000 -INF 50. 000 LEVEL 1045. 000 62. 000 54. 667 56. 000 UPPER MARGINAL 1045. 000 1. 000 62. 000 10. 113 68. 000. +INF. -INF 22. 000 40. 000 50. 000 40. 000 . . 22. 000 40. 000 +INF 2. 1244 E-8 -1. 209 EPS 13
![Nonlinear Programming NLP Example 7 1 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-14.jpg)
Nonlinear Programming (NLP) Example 7. 1: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1621. 425 +INF 22. 000 40. 000 +INF Solution Using LOCALSOLVER Stopped due to time limit (1000 sec) ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 M. Pourakbari Kasmaei LOWER 1045. 000 62. 000 -INF 50. 000 LEVEL 1045. 000 62. 000 54. 667 56. 000 UPPER 1045. 000 62. 000 68. 000 +INF -INF 22. 000 40. 000 50. 000 40. 000 . . 22. 000 40. 000 +INF 14
![Nonlinear Programming NLP It seems that there is No Big Difference among Wellknown Solvers Nonlinear Programming (NLP) It seems that there is No Big Difference among Well-known Solvers!](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-15.jpg)
Nonlinear Programming (NLP) It seems that there is No Big Difference among Well-known Solvers! Why? !! M. Pourakbari Kasmaei, 2019 15
![Nonlinear Programming NLP Example 7 2 Economic Dispatch DNLP Let us consider a nonsmooth Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (DNLP) Let us consider a non-smooth](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-16.jpg)
Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (DNLP) Let us consider a non-smooth objective function: ü ED Problem with Valve-point Effect M. Pourakbari Kasmaei 16
![Nonlinear Programming NLP Ø Discontinuous Nonlinear Program DNLP ü Model with Nonsmooth Nonlinear terms Nonlinear Programming (NLP) Ø Discontinuous Nonlinear Program (DNLP) ü Model with Non-smooth Nonlinear terms](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-17.jpg)
Nonlinear Programming (NLP) Ø Discontinuous Nonlinear Program (DNLP) ü Model with Non-smooth Nonlinear terms with Discontinuous Derivatives, Derivatives but no discrete variables ü The same as NLP problems, except that the nonsmooth functions may appear as well. ü These models are more difficult to solve than normal NLP models and it is highly recommended to avoid using such models. M. Pourakbari Kasmaei 17
![Nonlinear Programming NLP Set i 12 Variable z Positive Variables Pgi Equations Eq Obj Nonlinear Programming (NLP) Set i /1*2/; Variable z; Positive Variables Pg(i); Equations Eq. Obj,](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-18.jpg)
Nonlinear Programming (NLP) Set i /1*2/; Variable z; Positive Variables Pg(i); Equations Eq. Obj, Eq 1, Eq 2, Eq 3, Eq 4, Eq 5; Table gendata(i, *) a b c e f upplim lowlim 1 0. 0003 10. 1 671 170 0. 091 50 0 2 0. 0013 8. 8 374 110 0. 078 40 0; Eq. Obj. . z =e= sum(i, gendata(i, 'a')*Power(Pg(i), 2) + gendata(i, 'b')*Pg(i) +gendata(i, 'c') +abs(gendata(i, 'e')*sin(gendata(i, 'f')*(gendata(i, 'lowlim')-Pg(i))))); Eq 1. . sum(i, Pg(i)) =e= 62; ü CONOPT Eq 2. . (2/3)*Pg('1') + Pg('2') =l= 68; ü IPOPT Eq 3. . 2*Pg('1') + 0. 3*Pg('2') =g= 50; ü KNITRO Eq 4(i). . Pg(i) =l= gendata(i, 'upplim'); ü SNOPT Eq 5(i). . Pg(i) =g= gendata(i, 'lowlim'); ü …. Option DNLP= NONLINEAR_SOLVER_NAME; Model DNLP_ED /ALL/; Solve DNLP_ED USING DNLP MINIMIZING Z; Example 7. 2: ED (DNLP) M. Pourakbari Kasmaei Set i /1*2/; Variable z; Positive Variables Pg(i); Equations Eq. Obj, Eq 1, Eq 2, Eq 3, Eq 4, Eq 5; Table gendata(i, *) a b c upplim lowlim 1 0. 0003 10. 1 671 50 0 2 0. 0013 8. 8 374 40 0; Eq. Obj. . z =e= sum(i, gendata(i, 'a')*Power(Pg(i), 2) + gendata(i, 'b')*Pg(i) + gendata(i, 'c')); Eq 1. . sum(i, Pg(i)) =e= 62; Eq 2. . (2/3)*Pg('1') + Pg('2') =l= 68; Eq 3. . 2*Pg('1') + 0. 3*Pg('2') =g= 50; Eq 4(i). . Pg(i) =l= gendata(i, 'upplim'); Eq 5(i). . Pg(i) =g= gendata(i, 'lowlim'); Model NLP_ED /ALL/; Solve NLP_ED USING NLP MINIMIZING Z; 18
![Nonlinear Programming NLP Example 7 2 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-19.jpg)
Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1912. 872 +INF 50. 000 12. 000 +INF Solution Using CONOPT MARGINAL. . . ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 LOWER 1045. 000 62. 000 -INF 50. 000 LEVEL 1045. 000 62. 000 45. 333 103. 600 UPPER MARGINAL 1045. 000 1. 000 62. 000 13. 919 68. 000. +INF. -INF 50. 000 12. 000 50. 000 40. 000 -1. 288. . . 50. 000 12. 000 +INF . . M. Pourakbari Kasmaei 19
![Nonlinear Programming NLP Example 7 2 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-20.jpg)
Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1912. 872 +INF 50. 000 12. 000 +INF Solution Using MINOS MARGINAL. . . ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 LOWER 1045. 000 62. 000 -INF 50. 000 LEVEL 1045. 000 62. 000 45. 333 103. 600 UPPER MARGINAL 1045. 000 1. 000 62. 000 13. 919 68. 000. +INF. -INF 50. 000 12. 000 50. 000 40. 000 -1. 288. . . 50. 000 12. 000 +INF . . M. Pourakbari Kasmaei 20
![Nonlinear Programming NLP Example 7 2 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-21.jpg)
Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1912. 872 +INF 50. 000 12. 000 +INF Solution Using IPOPT MARGINAL. 1. 817 E-11 7. 576 E-11 ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 M. Pourakbari Kasmaei LOWER 1045. 000 62. 000 -INF 50. 000 LEVEL 1045. 000 62. 000 45. 333 103. 600 UPPER MARGINAL 1045. 000 1. 000 62. 000 13. 919 68. 000 -4. 01 E-11 +INF 1. 695 E-11 -INF 50. 000 12. 000 50. 000 40. 000 -1. 288 -3. 25 E-11 . . 50. 000 12. 000 +INF 1. 817 E-11 7. 575 E-11 21
![Nonlinear Programming NLP Example 7 2 Economic Dispatch NLP Failed in Finding the Solution Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) Failed in Finding the Solution!](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-22.jpg)
Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) Failed in Finding the Solution! Solution Using BARON M. Pourakbari Kasmaei 22
![Nonlinear Programming NLP Example 7 2 Economic Dispatch NLP Failed in Finding the Solution Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) Failed in Finding the Solution!](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-23.jpg)
Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) Failed in Finding the Solution! Solution Using MOSEK M. Pourakbari Kasmaei 23
![Nonlinear Programming NLP Example 7 2 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-24.jpg)
Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1778. 239 +INF MARGINAL. 22. 000 40. 000 -6. 78 E-10 EPS +INF Solution Using KNITRO ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 M. Pourakbari Kasmaei ---- VAR z ---- VAR Pg 1 2 LOWER 1045. 000 62. 000 -INF 50. 000 LOWER -INF. . LEVEL UPPER MARGINAL 1912. 872 +INF. 50. 000 12. 000 +INF 1. 817 E-11 7. 576 E-11 LEVEL UPPER MARGINAL 1045. 000 1. 000 62. 000 3. 647 54. 667 68. 000 -2. 03 E-10 56. 000 +INF 1. 8410 E-8 -INF 22. 000 40. 000 50. 000 40. 000 EPS -3. 321 . . 22. 000 40. 000 +INF EPS 24
![Nonlinear Programming NLP Example 7 2 Economic Dispatch NLP VAR z VAR Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-25.jpg)
Nonlinear Programming (NLP) Example 7. 2: Economic Dispatch (NLP) ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER 1729. 285 +INF 34. 523 27. 477 +INF MARGINAL. . . Solution Using LINDO/LINDOGLOBAL ---- EQU Eq. Obj ---- EQU Eq 1 ---- EQU Eq 2 ---- EQU Eq 3 ---- EQU Eq 4 1 2 ---- EQU Eq 5 1 2 M. Pourakbari Kasmaei ---- VAR z ---- VAR Pg 1 2 LOWER -INF. . LEVEL UPPER MARGINAL 1778. 239 +INF. 22. 000 40. 000 +INF -6. 78 E-10 EPS LOWER 1045. 000 62. 000 -INF 50. 000 LEVEL 1045. 000 62. 000 50. 492 77. 289 UPPER MARGINAL 1045. 000 1. 000 62. 000 4. 224 68. 000. +INF. -INF 34. 523 27. 477 50. 000 40. 000 . . 34. 523 27. 477 +INF . . 25
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Nonlinear Programming (NLP) Choosing Proper Solver do Matter! M. Pourakbari Kasmaei, 2019 26
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Mixed-Integer NLP (MINLP) ü The nature of problem is Nonlinear. ü At least one Integer Decision Variable Exists. ü Integer variable can be a part of Objective Function and/or the Feasible Region. M. Pourakbari Kasmaei, 2019 27
![an Obj d e m or ctiv ig F e h t ea F an Obj d e m /or ctiv ig F e h t ea F](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-28.jpg)
an Obj d e m /or ctiv ig F e h t ea F co sib un Va nta le ctio ria in Re n bl Int gio es eg n er Formulating MINLP Problems (General Form) M. Pourakbari Kasmaei, 2019 28
![MixedInteger NLP MINLP ü Like MILP Problems we have MixedBinary NLP MBNLP Problem Mixed-Integer NLP (MINLP) ü Like MILP Problems, we have: • Mixed-Binary NLP (MBNLP) Problem](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-29.jpg)
Mixed-Integer NLP (MINLP) ü Like MILP Problems, we have: • Mixed-Binary NLP (MBNLP) Problem • Integer NLP (INLP) Problems • Binary NLP (BNLP) Problems M. Pourakbari Kasmaei, 2019 29
![MixedInteger NLP MINLP Example 7 3 Economic Dispatch MINLP Cannot Be Handled By Commercial Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) Cannot Be Handled By Commercial](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-30.jpg)
Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) Cannot Be Handled By Commercial Solvers M. Pourakbari Kasmaei 30
![MixedInteger NLP MINLP Example 7 3 Economic Dispatch MINLP ü Instead of having linear Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) ü Instead of having linear](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-31.jpg)
Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) ü Instead of having linear objective function, we have a quadratic one. 18 25 M. Pourakbari Kasmaei, 2019 40 43 31
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Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) ü Assign Binary Variables to each Operating Zone 40 43 18 25 Only One Zone should be selected M. Pourakbari Kasmaei, 2019 32
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Mixed-Integer NLP (MINLP) Set i /1*2/; Variable z; Positive Variables Pg(i); Equations Eq. Obj, Eq 1, Eq 2, Eq 3, Eq 4_1, Eq 4_2, Eq 4_3, Eq 5; Table gendata(i, *) a b c e f upplim lowlim 1 0. 0003 10. 1 671 170 0. 091 50 0 2 0. 0013 8. 8 374 110 0. 078 40 0; Eq. Obj. . z =e= sum(i, gendata(i, 'a')*Power(Pg(i), 2) + gendata(i, 'b')*Pg(i) +gendata(i, 'c')); Eq 1. . sum(i, Pg(i)) =e= 62; Eq 2. . (2/3)*Pg('1') + Pg('2') =l= 68; Eq 3. . 2*Pg('1') + 0. 3*Pg('2') =g= 50; Eq 4(i). . Pg(i) =l= gendata(i, 'upplim'); Eq 4_1. . Pg('1') =l= 18*u 1 + 40*u 2 + 50*u 3; Eq 4_2. . Pg('1') =g= 25*u 2 + 43*u 3; Eq 4_3. . u 1 + u 2 + u 3 =e= 1; Eq 5(i). . Pg(i) =g= gendata(i, 'lowlim'); Option MINLP= NONLINEAR_SOLVER_NAME; Model MINLP_ED /ALL/; Solve DNLP_ED USING MINLP MINIMIZING Z; Example 7. 3: Economic Dispatch (MINLP) M. Pourakbari Kasmaei 33
![MixedInteger NLP MINLP Example 7 3 Economic Dispatch MINLP Solution Using KNITRO VAR Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) Solution Using KNITRO ---- VAR](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-34.jpg)
Mixed-Integer NLP (MINLP) Example 7. 3: Economic Dispatch (MINLP) Solution Using KNITRO ---- VAR z ---- VAR Pg 1 2 ---- VAR u 1 ---- VAR u 2 ---- VAR u 3 LOWER -INF. . . LEVEL UPPER 1625. 067 +INF 25. 000 37. 000. 1. 000. +INF 1. 000 Solution Using LINDO MARGINAL. EPS EPS 30. 470 52. 408 ---- VAR z ---- VAR Pg 1 2 ---- VAR u 1 ---- VAR u 2 ---- VAR u 3 M. Pourakbari Kasmaei LOWER -INF. . . LEVEL UPPER 1625. 067 +INF 25. 000 37. 000. 1. 000. +INF 1. 000 MARGINAL. . . 0. 110 30. 580 52. 519 34
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Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) Non-smooth Terms M. Pourakbari Kasmaei 35
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Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) ü Transform these terms into integer-based terms. 18 25 M. Pourakbari Kasmaei, 2019 40 43 36
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Mixed-Integer NLP (MINLP) Example 7. 4: Economic Dispatch (MINLP) Solution Using KNITRO ---- VAR z ---- VAR Pg 1 2 ---- VAR u 1 ---- VAR u 2 ---- VAR u 3 LOWER -INF. . . LEVEL UPPER 1729. 285 +INF 34. 523 27. 477. 1. 000. +INF 1. 000 Solution Using LINDO MARGINAL. EPS -14. 151 -31. 446 -39. 307 ---- VAR z ---- VAR Pg 1 2 ---- VAR u 1 ---- VAR u 2 ---- VAR u 3 M. Pourakbari Kasmaei LOWER -INF. . . LEVEL UPPER 1729. 285 +INF 34. 523 27. 477. 1. 000. +INF 1. 000 MARGINAL. EPS EPS 37
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Some Linearization Techniques Linearization Of Product Terms & SOS 2 Technique M. Pourakbari Kasmaei, 2019 38
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Some Linearization Techniques 1) Product of Two Binary Variables M. Pourakbari Kasmaei, 2019 39
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Some Linearization Techniques 2) Product of One Binary and One Continuous M. Pourakbari Kasmaei, 2019 40
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Some Linearization Techniques 3) Product of Two Continuous Variables M. Pourakbari Kasmaei, 2019 41
![Some Linearization Techniques Special Order Set 1 and 2 SOS 1 and SOS 2 Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-42.jpg)
Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2) ü SOS 1 is a set of variables for which at most one variable might be non-zero. ü SOS 2 is a set of variables for which at most two of variables might be non-zero, and if so, these two variables must be adjacent • Therefore, instead of selecting only one breakpoint, two breakpoints are selected and the weighting average of these points is the output. In other words, the value can be between these two breakpoints. M. Pourakbari Kasmaei, 2019 42
![Some Linearization Techniques Special Order Set 1 and 2 SOS 1 and SOS 2 Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2)](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-43.jpg)
Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2) M. Pourakbari Kasmaei, 2019 43
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Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2) M. Pourakbari Kasmaei, 2019 44
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Some Linearization Techniques Special Order Set 1 and 2 (SOS 1 and SOS 2) M. Pourakbari Kasmaei, 2019 45
![Thanks M Pourakbari Kasmaei 2019 46 Thanks! M. Pourakbari Kasmaei, 2019 46](https://slidetodoc.com/presentation_image/5c7a80bce79c303f8eb7487eaecf5ee2/image-46.jpg)
Thanks! M. Pourakbari Kasmaei, 2019 46
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