# Interval Optimization-Based Optimal Design of Distributed Energy Resource Systems under Uncertainties

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematic Models

#### 2.1. Deterministic MILP Model

_{Total}, which is the sum of the equipment capital investment cost C

_{ECapital}, energy cost C

_{Energy}, and operating and maintenance cost of the equipment C

_{EOM}, and then subtract the renewable energy subsidy R

_{subsidy}, as follows:

_{u}of energy form u, the amount of energy purchased EPG

_{k,u}at time k, and annual duration hours AH

_{k}.

_{i,j}of equipment type i with capacity j cannot exceed its maximum available number N

_{i,j}. γ

_{i,j}is a binary variable that denotes whether equipment type i with capacity j is selected (0 for not selected, 1 for selected). Each equipment type can choose only one type of capacity.

_{k,RE,el}is dependent on the surface area of the installed equipment WArea

_{jRE}, renewable energy intensity Intens

_{k,RE}, generation efficiency η

_{jRE}, and the number of pieces of operating equipment δ

_{k,RE}. Meanwhile, the generated electricity cannot exceed the rated capacity Rcap

_{RE,jRE}of the operating equipment.

_{k,i,u}and the purchased energy should not be smaller than the sum of the load demands LD

_{k,u}of end users, energy consumption of equipment x

_{k,u}, and energy flow in storage E

_{k,u}.

#### 2.2. Interval Optimization

^{I}represents a random variable that is distributed between the lower limit A

^{L}and the upper limit A

^{R}. It can be expressed as:

^{L}= A

^{R}, the interval number degenerates to a real number. The interval number can also be expressed as:

^{c}and A

^{w}are the midpoint and radius values of the interval number, which can be expressed as:

_{i}are the objective function and constraints that contain interval uncertainties, respectively; and X and U are the decision variables and interval uncertainties, respectively. ${b}_{i}^{I}$ is the right-hand side interval of the ith constraint, and l is the number of the constraints. Ω

^{n}is the range of X, and q is the dimension of U.

^{I}and B

^{I}, and their formulations are discussed in a previous work [30]. When interval number B

^{I}degenerates into a real number b, the probability degree of A

^{I}≤ b is expressed as:

^{I}degenerates into a real number a, the probability degree of a ≤ B

^{I}is expressed as:

_{i}(X,U) ≤ ${b}_{i}^{I}$, can be transformed into a deterministic constraint as follows:

_{i}(X,U) ≥ ${b}_{i}^{I}$, can be easily transformed into the form of ≤, such as ${b}_{i}^{I}$ ≤ g

_{i}(X,U), and further transformed into the deterministic form using the aforementioned method.

^{c}(X) and radius f

^{w}(X) of the objective interval f

^{I}(X) are defined as:

^{L}(X) and upper limit f

^{R}(X) of the objective interval can be obtained using:

#### 2.3. Interval Optimization Model for DER Systems

_{1}, the uncertain constraint of the operation characteristic of the renewable energy equipment can be transformed into the following form:

_{2}, the constraint of energy balance can be transformed into

## 3. Numerical Study

#### 3.1. Climate Characteristics

^{2}. Figure 2, Figure 3 and Figure 4 show the hourly values of temperature, solar radiation intensity, and wind energy density for the three typical days.

#### 3.2. Load Demands

^{2}. The hospital has cooling, heating, and electricity demands throughout the year, with a high daytime load and a relatively low nighttime load. The hourly load demand of the hospital during the three typical days is shown in Figure 5.

#### 3.3. Electricity and Gas Tariffs

^{3}and a price of 0.5505 $/Nm

^{3}. The hospital’s electricity is commercial electricity, and the local commercial electricity price is 0.1357 $/kWh.

#### 3.4. DER Equipment Options

## 4. Setting of Cases

_{1}and λ

_{2}are both set as 0.8, and the weighting coefficient β is also set as 0.8.

^{®}Pentium

^{®}CPU G620 (2.60 GHZ) with 4 GB RAM.

## 5. Results and Analysis

#### 5.1. Optimal System Configuration

#### 5.2. Economic Performance

#### 5.3. Optimal Operation Strategy

#### 5.4. Sensitivity Analysis

_{1}and λ

_{2}in Case 4. The table indicates that when β decreases, the midpoint value of the interval of total annual cost also decreases. Meanwhile, the radius of the interval increases. The weighting coefficient β indicates the degree of emphasis on the midpoint value and the radius of the objective interval. The midpoint value represents the average design performance, and the radius represents the sensitivity of the objective value to uncertainties. A small β value ensures the system achieve good expected performance, and a large β value enables the system to achieve better robustness to uncertainties.

_{1}and λ

_{2}decrease, the midpoint and radius of the interval of total annual cost also decreases. The probability degree represents the probability that the constraints can be satisfied. Although a decrease in the probability degree can improve economic performance, the risk that constraints may be violated increases. A trade-off between objective performance and reliability of the solution should be made when choosing the probability degree values.

#### 5.5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Pepermans, G.; Driesen, J.; Haeseldonckx, D.; Belmans, R.; D’Haeseleer, W. Distributed generation: Definition, benefits and issues. Energy Policy
**2005**, 33, 787–798. [Google Scholar] [CrossRef] - Di Somma, M.; Yan, B.; Bianco, N.; Graditi, G.; Luh, P.B.; Mongibello, L.; Naso, V. Design optimization of a distributed energy system through cost and exergy assessments. Energy Procedia
**2017**, 105, 2451–2459. [Google Scholar] [CrossRef] - Yan, B.; Di Somma, M.; Bianco, N.; Luh, P.B.; Graditi, G.; Mongibello, L.; Naso, V. Exergy-based operation optimization of a distributed energy system through the energy-supply chain. Appl. Therm. Eng.
**2016**, 101, 741–751. [Google Scholar] [CrossRef] - Wu, Q.; Ren, H.; Gao, W.; Ren, J. Multi-objective optimization of a distributed energy network integrated with heating interchange. Energy
**2016**, 109, 353–364. [Google Scholar] [CrossRef] - Ren, H.; Zhou, W.; Nakagami, K.I.; Gao, W.; Wu, Q. Multi-objective optimization for the operation of distributed energy systems considering economic and environmental aspects. Appl. Energy
**2010**, 87, 3642–3651. [Google Scholar] [CrossRef] - Di Somma, M.; Yan, B.; Bianco, N.; Graditi, G.; Luh, P.B.; Mongibello, L.; Naso, V. Multi-objective design optimization of distributed energy systems through cost and exergy assessments. Appl. Energy
**2017**, 204, 1299–1316. [Google Scholar] [CrossRef] - Morvaj, B.; Evins, R.; Carmeliet, J. Optimization framework for distributed energy systems with integrated electrical grid constraints. Appl. Energy
**2016**, 171, 296–313. [Google Scholar] [CrossRef] - Di Somma, M.; Yan, B.; Bianco, N.; Graditi, G.; Luh, P.B.; Mongibello, L.; Naso, V. Operation optimization of a distributed energy system considering energy costs and exergy efficiency. Energy Convers. Manag.
**2015**, 103, 739–751. [Google Scholar] [CrossRef] - Ju, L.; Tan, Z.; Li, H.; Tan, Q.; Yu, X.; Song, X. Multi-objective operation optimization and evaluation model for cchp and renewable energy based hybrid energy system driven by distributed energy resources in china. Energy
**2016**, 111, 322–340. [Google Scholar] [CrossRef][Green Version] - Karmellos, M.; Georgiou, P.N.; Mavrotas, G. A comparison of methods for the optimal design of distributed energy systems under uncertainty. Energy
**2019**, 178, 318–333. [Google Scholar] [CrossRef] - Mavromatidis, G.; Orehounig, K.; Carmeliet, J. Design of distributed energy systems under uncertainty: A two-stage stochastic programming approach. Appl. Energy
**2018**, 222, 932–950. [Google Scholar] [CrossRef] - Mavromatidis, G.; Orehounig, K.; Carmeliet, J. Comparison of alternative decision-making criteria in a two-stage stochastic program for the design of distributed energy systems under uncertainty. Energy
**2018**, 156, 709–724. [Google Scholar] [CrossRef] - Yanıkoğlu, İ.; Gorissen, B.L.; den Hertog, D. A survey of adjustable robust optimization. Eur. J. Oper. Res.
**2019**, 277, 799–813. [Google Scholar] [CrossRef] - Beraldi, P.; Violi, A.; Carrozzino, G.; Bruni, M.E. A stochastic programming approach for the optimal management of aggregated distributed energy resources. Comput. Oper. Res.
**2018**, 96, 200–212. [Google Scholar] [CrossRef] - Zhou, Z.; Zhang, J.; Liu, P.; Li, Z.; Georgiadis, M.C.; Pistikopoulos, E.N. A two-stage stochastic programming model for the optimal design of distributed energy systems. Appl. Energy
**2013**, 103, 135–144. [Google Scholar] [CrossRef] - Marino, C.; Quddus, M.A.; Marufuzzaman, M.; Cowan, M.; Bednar, A.E. A chance-constrained two-stage stochastic programming model for reliable microgrid operations under power demand uncertainty. Sustain. Energy Grids Netw.
**2018**, 13, 66–77. [Google Scholar] [CrossRef] - Ben-Tal, A.; Ghaoui, L.E.; Nemirovski, A. Robust Optimization; Princeton University Press: Princeton, NJ, USA, 2009. [Google Scholar]
- Wang, L.; Li, Q.; Cheng, X.; He, G.; Li, G.; Wang, R. A robust optimization approach for risk-averse energy transactions in networked microgrids. Energy Procedia
**2019**, 158, 6595–6600. [Google Scholar] [CrossRef] - Jeddi, B.; Vahidinasab, V.; Ramezanpour, P.; Aghaei, J.; Shafie-khah, M.; Catalão, J.P.S. Robust optimization framework for dynamic distributed energy resources planning in distribution networks. Int. J. Electr. Power Energy Syst.
**2019**, 110, 419–433. [Google Scholar] [CrossRef] - Akbari, K.; Nasiri, M.M.; Jolai, F.; Ghaderi, S.F. Optimal investment and unit sizing of distributed energy systems under uncertainty: A robust optimization approach. Energy Build.
**2014**, 85, 275–286. [Google Scholar] [CrossRef] - Zhang, B.; Li, Q.; Wang, L.; Feng, W. Robust optimization for energy transactions in multi-microgrids under uncertainty. Appl. Energy
**2018**, 217, 346–360. [Google Scholar] [CrossRef] - Moore, R.E.; Bierbaum, F. Methods and Applications of Interval Analysis; Society for Industrial & Applied Math: Philadelphia, PA, USA, 1979. [Google Scholar] [CrossRef][Green Version]
- Su, Y.; Zhou, Y.; Tan, M. An interval optimization strategy of household multi-energy system considering tolerance degree and integrated demand response. Appl. Energy
**2020**, 260, 114144. [Google Scholar] [CrossRef] - Wang, J.; Li, Y.; Zhou, Y. Interval number optimization for household load scheduling with uncertainty. Energy Build.
**2016**, 130, 613–624. [Google Scholar] [CrossRef] - Taghizadeh, M.; Bahramara, S.; Adabi, F.; Nojavan, S. Optimal thermal and electrical operation of the hybrid energy system using interval optimization approach. Appl. Therm. Eng.
**2020**, 169, 114993. [Google Scholar] [CrossRef] - Bai, L.; Li, F.; Cui, H.; Jiang, T.; Sun, H.; Zhu, J. Interval optimization based operating strategy for gas-electricity integrated energy systems considering demand response and wind uncertainty. Appl. Energy
**2016**, 167, 270–279. [Google Scholar] [CrossRef][Green Version] - Zhang, Y.; Huang, Z.; Zheng, F.; Zhou, R.; An, X.; Li, Y. Interval optimization based coordination scheduling of gas–electricity coupled system considering wind power uncertainty, dynamic process of natural gas flow and demand response management. Energy Rep.
**2020**, 6, 216–227. [Google Scholar] [CrossRef] - Wang, S.; Yuan, S. Interval optimization for integrated electrical and natural-gas systems with power to gas considering uncertainties. Int. J. Electr. Power Energy Syst.
**2020**, 119, 105906. [Google Scholar] [CrossRef] - Yang, Y.; Zhang, S.; Xiao, Y. Optimal design of distributed energy resource systems based on two-stage stochastic programming. Appl. Therm. Eng.
**2017**, 110, 1358–1370. [Google Scholar] [CrossRef] - Jiang, C.; Han, X.; Liu, G.R.; Liu, G.P. A nonlinear interval number programming method for uncertain optimization problems. Eur. J. Oper. Res.
**2008**, 188, 1–13. [Google Scholar] [CrossRef] - Yokoyama, R.; Ito, K. Optimal design of gas turbine cogeneration plants in consideration of discreteness of equipment capabilities. J. Eng. Gas. Turbines Power-Trans. ASME
**2006**, 128, 336–343. [Google Scholar] [CrossRef] - Li, H.; Nalim, R.; Haldi, P.A. Thermal-economic optimization of a distributed multi-generation energy system—A case study of beijing. Appl. Therm. Eng.
**2006**, 26, 709–719. [Google Scholar] [CrossRef] - Mehleri, E.D.; Sarimveis, H.; Markatos, N.C.; Papageorgiou, L.G. Optimal design and operation of distributed energy systems: Application to greek residential sector. Renew. Energ.
**2013**, 51, 331–342. [Google Scholar] [CrossRef] - Farmer, R. Simple cycle oem design ratings. In Gas Turbine World 2012 GTW Handbook; Pequot Publishing Inc: Southport, CT, USA, 2012; Volume 29, pp. 70–80. [Google Scholar]
- Becchio, C.; Bottero, M.C.; Corgnati, S.P.; Dell’Anna, F. Decision making for sustainable urban energy planning: An integrated evaluation framework of alternative solutions for a nzed (net zero-energy district) in turin. Land Use Policy
**2018**, 78, 803–817. [Google Scholar] [CrossRef]

Equipment Type | Rated Capacity (kW) | Rated Efficiency/COP | Unit Capital and Installation Cost ($/kW) | Unit Fixed O&M Cost ($/kW/year) | Unit Variable O&M Cost ($/kWh) | Load Regulation Range | Lifetime (year) |
---|---|---|---|---|---|---|---|

GT | 5200 | 29.4% | 596 | 0 | 0.004 | 0.4–1 | 15 |

4345 | 28.3% | 616 | 0 | 0.004 | 0.38–1 | 15 | |

GE | 5200 | 40.3% | 655 | 0 | 0.009 | 0.4–1 | 15 |

6000 | 41.2% | 635 | 0 | 0.009 | 0.4–1 | 15 | |

WT | 20 | 35% | 1017 | 0 | 0.0084 | - | 25 |

PV | 28 | 17% | 1694 | 16.7 | 0.02 | - | 25 |

WB | 700 | 78% | 130 | 0 | 0.0027 | 0.26–1 | 15 |

1000 | 78% | 130 | 0 | 0.0027 | 0.38–1 | 15 | |

GB | 700 | 83% | 100 | 0 | 0.0027 | 0.48–1 | 15 |

1041 | 83% | 100 | 0 | 0.0027 | 0.26–1 | 15 | |

HS | - | - | 23 $/kWh | - | 0.0013 | - | 20 |

AC | 1454 | 1.417 | 172 | 0 | 0.001 | 0.05–1.15 | 25 |

872 | 1.419 | 172 | 0 | 0.001 | 0.05–1.15 | 25 | |

CC | 3520 | 4.73 | 102 | 0 | 0.0015 | 0.1–1 | 25 |

1230 | 4.3 | 102 | 0 | 0.0015 | 0.1–1 | 25 | |

CS | - | - | 23 $/kWh | - | 0.0013 | - | 20 |

**Table 2.**Midpoint, radius, lower limit and upper limit of interval of total annual cost in the four cases ($).

Midpoint | Radius | Lower limit | Upper limit | |
---|---|---|---|---|

Case 1 | 5,816,459 | — | — | — |

Case 2 | 5,829,397 | 907,611 | 4,921,786 | 6,737,007 |

Case 3 | 5,842,010 | 910,266 | 4,931,744 | 6,752,275 |

Case 4 | 6,582,206 | 1,002,055 | 5,580,151 | 7,584,261 |

**Table 3.**Interval of total annual cost with different weighting coefficients and probability degrees ($).

β | λ_{1} | λ_{2} | Midpoint | Radius | Lower Limit | Upper Limit |
---|---|---|---|---|---|---|

0.9 | 0.8 | 0.8 | 6,758,731 | 969,499 | 5,789,232 | 7,728,231 |

0.8 | 0.8 | 0.8 | 6,582,206 | 1,002,055 | 5,580,151 | 7,584,261 |

0.7 | 0.8 | 0.8 | 6,490,883 | 1,026,250 | 5,464,633 | 7,517,133 |

0.6 | 0.8 | 0.8 | 6,485,701 | 1,028,585 | 5,457,116 | 7,5142,86 |

0.8 | 0.9 | 0.9 | 6,745,561 | 1,057,121 | 5,688,440 | 7,802,681 |

0.8 | 0.7 | 0.7 | 6,283,870 | 977,995 | 5,305,875 | 7,261,865 |

0.8 | 0.6 | 0.6 | 6,040,841 | 945,174 | 5,095,668 | 6,986,015 |

0.8 | 0.5 | 0.5 | 5,831,009 | 907,913 | 4,923,096 | 6,738,921 |

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**MDPI and ACS Style**

Li, D.; Zhang, S.; Xiao, Y. Interval Optimization-Based Optimal Design of Distributed Energy Resource Systems under Uncertainties. *Energies* **2020**, *13*, 3465.
https://doi.org/10.3390/en13133465

**AMA Style**

Li D, Zhang S, Xiao Y. Interval Optimization-Based Optimal Design of Distributed Energy Resource Systems under Uncertainties. *Energies*. 2020; 13(13):3465.
https://doi.org/10.3390/en13133465

**Chicago/Turabian Style**

Li, Da, Shijie Zhang, and Yunhan Xiao. 2020. "Interval Optimization-Based Optimal Design of Distributed Energy Resource Systems under Uncertainties" *Energies* 13, no. 13: 3465.
https://doi.org/10.3390/en13133465