On Optimistic and Pessimistic Bilevel Optimization Models for Demand Response Management
Abstract
:1. Introduction
2. Literature Review
3. Problem Definition
4. Preliminaries
4.1. The Continuous Knapsack Problem
 1.
 ${x}_{t}^{*}={x}_{t}^{l}$ for $t\in L$, ${x}_{t}^{*}={x}_{t}^{u}$ for $t\in U$;
 2.
 ${x}_{p}^{*}\in \{{x}_{p}^{l},{x}_{p}^{u},{D}_{p}^{l}{\sum}_{t\in L\cup U}{x}_{t}^{*},{D}_{p}^{u}{\sum}_{t\in L\cup U}{x}_{t}^{*}\}$.
4.2. General Properties of Optimal Solutions
 ${u}_{it}:={u}_{it}{q}_{t}^{l}$ for $i=1,\dots ,m$;
 ${c}_{t}:={c}_{t}{q}_{t}^{l}$;
 $Q:=Q{q}_{t}^{l}/T$;
 ${q}_{t}^{u}:={q}_{t}^{u}{q}_{t}^{l}$;
 ${q}_{t}^{l}:=0$.
5. Polynomially Solvable Special Cases with One Consumer Only
5.1. The Optimistic Variant
 If ${u}_{t}{q}_{t}^{\u2606}<{u}_{p}{q}_{p}^{\u2606}$, then $t\in L$, and ${q}_{t}^{\u2606}$ is decreased by $\u03f5\u03f5/T$, while ${q}_{\tau}^{\u2606}$ is increased by $\u03f5/T$ for all $\tau \ne t$, where $\u03f5={u}_{p}{q}_{p}^{\u2606}({u}_{t}{q}_{t}^{\u2606})$.
 If ${u}_{t}{q}_{t}^{\u2606}>{u}_{p}{q}_{p}^{\u2606}$, then $t\in U$, and ${q}_{t}^{\u2606}$ is increased by $\u03f5\u03f5/T$, while ${q}_{\tau}^{\u2606}$ increases by $\u03f5/T$ for all $\tau \ne t$, where $\u03f5={u}_{t}{q}_{t}^{\u2606}({u}_{p}{q}_{p}^{\u2606})$.
5.2. The Pessimistic Variant
6. The General Case with Multiple Consumers
6.1. Solution of the General Optimistic Variant
6.2. Solution of the General Pessimistic Variant
 Either ${u}_{i{\pi}_{i}\left(t\right)}{q}_{{\pi}_{i}\left(t\right)}>{u}_{i{\pi}_{i}(t+1)}{q}_{{\pi}_{i}(t+1)}$, or
 ${u}_{i{\pi}_{i}\left(t\right)}{q}_{{\pi}_{i}\left(t\right)}={u}_{i{\pi}_{i}(t+1)}{q}_{{\pi}_{i}(t+1)}$, and ${q}_{{\pi}_{i}\left(t\right)}{c}_{{\pi}_{i}\left(t\right)}\le {q}_{{\pi}_{i}(t+1)}{c}_{{\pi}_{i}(t+1)}$ for $t=1,\dots ,T1$.
 ${q}_{t}^{\prime}\ge 0$;
 If ${q}_{t}{c}_{t}>{q}_{{t}^{\prime}}{c}_{{t}^{\prime}}$ for some ${t}^{\prime}$, then ${q}_{t}^{\prime}{c}_{t}>{q}_{{t}^{\prime}}^{\prime}{c}_{{t}^{\prime}}$;
 If ${u}_{it}{q}_{t}>{u}_{i{t}^{\prime}}{q}_{{t}^{\prime}}$ for some ${t}^{\prime}$, then ${u}_{it}{q}_{t}^{\prime}>{u}_{i{t}^{\prime}}{q}_{{t}^{\prime}}^{\prime}$;
 If ${q}_{t}{c}_{t}>0$ then ${q}_{t}^{\prime}{c}_{t}>0$, and
 If ${u}_{it}{q}_{t}<0$ then ${u}_{it}{q}_{t}^{\prime}<0$ for each follower i.
 If ${u}_{i,{t}_{1}}{q}_{{t}_{1}}>{u}_{i,{t}_{2}}{q}_{{t}_{2}}$, then ${u}_{i,{t}_{1}}{q}_{{t}_{1}}^{\prime}>{u}_{i,{t}_{2}}{q}_{{t}_{2}}^{\prime}$ and the order of the two time periods does not change.
 If ${u}_{i,{t}_{1}}{q}_{{t}_{1}}={u}_{i,{t}_{2}}{q}_{{t}_{2}}$ and ${q}_{{t}_{1}}{c}_{{t}_{1}}\le {q}_{{t}_{2}}{c}_{{t}_{2}}$ then three cases can be distinguished:
 
 If ${q}_{{t}_{1}}^{\prime}={q}_{{t}_{1}}$ and ${q}_{{t}_{2}}^{\prime}<{q}_{{t}_{2}}$ then ${u}_{i,{t}_{1}}{q}_{{t}_{1}}^{\prime}<{u}_{i,{t}_{2}}{q}_{{t}_{2}}^{\prime}$. Hence, the order of periods ${t}_{1}$ and ${t}_{2}$ will change for ${q}^{\prime}$ in order to satisfy the optimality conditions.
 
 If ${q}_{{t}_{1}}^{\prime}<{q}_{{t}_{1}}$ and ${q}_{{t}_{2}}^{\prime}={q}_{{t}_{2}}$ then ${u}_{i,{t}_{1}}{q}_{{t}_{1}}^{\prime}>{u}_{i,{t}_{2}}{q}_{{t}_{2}}^{\prime}$. Hence, the order of ${t}_{1}$ and ${t}_{2}$ will not change for ${q}^{\prime}$.
 
 If ${q}_{{t}_{1}}{q}_{{t}_{2}}={q}_{{t}_{1}}^{\prime}{q}_{{t}_{2}}^{\prime}$, then the order of ${t}_{1}$ and ${t}_{2}$ will not change for ${q}^{\prime}$.
Algorithm 1. Pessimistic Solution 

7. Experimental Evaluation
7.1. Numerical Example
7.2. Computational Experiments
8. Conclusions and Managerial Implications
8.1. Managerial Implications
8.2. Directions for Future Research
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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t  1  2 

${c}_{t}$  10  50 
${q}_{t}^{l}$  20  20 
${q}_{t}^{u}$  40  40 
${u}_{t}$  10  30 
${x}_{t}^{l}$  0  0 
${x}_{t}^{u}$  1  1 
t  1  2 

${c}_{t}$  10  50 
${q}_{t}^{l}$  20  20 
${q}_{t}^{u}$  40  40 
${u}_{t}$  40  40 
${x}_{t}^{l}$  0  0 
${x}_{t}^{u}$  1  1 
m  T  Opt  Time [s]  Gap [%]  

Avg.  Max.  
5  12  10  0.08     
24  10  0.16      
36  10  0.72      
48  10  1.40      
10  12  10  0.28     
24  10  2.73      
36  10  5.06      
48  10  13.92      
15  12  10  1.83     
24  10  6.01      
36  10  30.85      
48  10  47.88      
20  12  10  4.39     
24  8  81.29  0.58  5.34  
36  9  66.10  0.20  2.03  
48  7  172.74  1.37  12.76  
25  12  10  5.00     
24  5  185.13  0.90  3.41  
36  5  250.15  2.12  11.81  
48  5  203.51  13.11  100.00 
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Kis, T.; Kovács, A.; Mészáros, C. On Optimistic and Pessimistic Bilevel Optimization Models for Demand Response Management. Energies 2021, 14, 2095. https://doi.org/10.3390/en14082095
Kis T, Kovács A, Mészáros C. On Optimistic and Pessimistic Bilevel Optimization Models for Demand Response Management. Energies. 2021; 14(8):2095. https://doi.org/10.3390/en14082095
Chicago/Turabian StyleKis, Tamás, András Kovács, and Csaba Mészáros. 2021. "On Optimistic and Pessimistic Bilevel Optimization Models for Demand Response Management" Energies 14, no. 8: 2095. https://doi.org/10.3390/en14082095