# A Multi-Energy System Expansion Planning Method Using a Linearized Load-Energy Curve: A Case Study in South Korea

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## Abstract

**:**

## 1. Introduction

- The expansion planning problem for a multi-energy system, where considering various energy resources, including fuel-based generators and a renewable electrical power source, as well as a CHP unit and energy storage resources, is modeled as an MILP problem with linearized constraints.
- To model linearized constraints including a linearized load curve, we use a load-energy curve instead of the LDC, and apply a Douglas-Peucker algorithm that can approximate linear functions and minimize the distortion of the original demand function.
- To validate the use of a linearized load-energy curve, the results of optimization methods using the linearized load-energy curve and the stepwise representation of LDC are compared.
- A case study for a multi-energy system based on a benchmark case from Goyang city in South Korea is presented.

## 2. Multi-Energy System Overview

## 3. Basic Optimization Model

#### 3.1. Objective Function

#### 3.2. CHP Constraints

#### 3.3. Energy Storage Constraints

#### 3.4. Lifespan Constraints

## 4. Linearized Load-Energy Curve

#### 4.1. Load-Energy Curve and Linearization

#### 4.2. Impact of the Renewable Electrical Power Source

## 5. Optimization Process

#### 5.1. Proposed Optimization Method

#### 5.2. Comparison of Optimization Results Using Different Load Curves

## 6. Case Study

#### 6.1. Data and Assumptions

#### 6.2. Simulation Results

#### 6.2.1. Case 1

#### 6.2.2. Case 2

^{6}) and 1.5% ($1.37 × 10

^{7}), respectively, variable costs increase by 0.72% ($1.34 × 10

^{7}). Compared to Case 1, the installed capacity of the CHP is reduced by 100 MW, and the utilized energy for the CHP is also reduced. However, resources such as DG2, DG3, and HOB1 are newly added, or extended, since the shortfall due to the reduction of the CHP should be secured. Although the utilizing electrical energy of DG3 is larger than that of DG2, the allocated capacity of DG2 is larger than that of DG3. DG2 delivers more energy at a cheaper variable cost than DG3. The installed capacity and the utilizing heat energy of HOB1 increase, because the installed capacity and the utilized energy of CHP are reduced. We note that TES, in the last project year, is charged from CHP, which has a relatively low variable cost.

#### 6.2.3. Case 3

#### 6.2.4. Case 4

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Indices | |

$y$ | Project year index, from $\left[1:{N}_{Y}\right]$. |

$i$ | Electrical energy resource index, from $\left[1:{N}_{ER}\right]$. |

$j$ | Heat energy resource index, from $\left[1:{N}_{HR}\right]$. |

$c$ | Candidate unit index, from $\left[1:{N}_{C}\right]$. |

$elp$ | Electrical load pattern index, from $\left[1:{N}_{ELPAT}\right]$. |

$eap$ | Index for node point of approximated electrical load-energy curve, from $\left[1:{N}_{EAP}\right]$. |

$hap$ | Index for node point of approximated heat load-energy curve, from $\left[1:{N}_{HAP}\right]$. |

$si$ | Index for stack of electrical energy resource, from $\left[1:{N}_{EST}\right]$. |

$sj$ | Index for stack of heat energy resource, from $\left[1:{N}_{HST}\right]$. |

$eseg$ | Index for segment of approximated electrical load-energy curve, from $\left[1:{N}_{ESEG}\right]$. |

$hseg$ | Index for segment of approximated heat load-energy curve, from $\left[1:{N}_{HSEG}\right]$. |

Variables | |

$l{d}_{e}^{elp,eap,y}$ | Load demand for node point, $eap$, of the electrical load-energy curve, $elp$, in project year, $y$ (MW). |

$e{d}_{e}^{elp,eap,y}$ | Energy demand for node point, $eap$, of the electrical load-energy curve, $elp$, in project year, $y$ (MWh). |

$l{d}_{h}^{hap,y}$ | Load demand for node point, $hap$, of the heat load-energy curve in project year, $y$ (MW). |

$e{d}_{h}^{hap,y}$ | Energy demand for node point, $hap$, of the heat load-energy curve in project year, $y$ (MWh). |

${\omega}_{e}^{elp,si,eap,y}$ | Special ordered sets of type 2 variable for approximated point, $eap$, in stack, $si$, of the electrical load-energy curve, $elp$, in project year, $y$. |

${\omega}_{h}^{sj,hap,y}$ | Special ordered sets of type 2 variable for approximated point, $hap$, in stack, $sj$, of the heat load-energy curve in project year, $y$. |

${p}_{e}^{si,y}$ | Electrical load demand point for determining stack, $si$, in project year, $y$ (MW). |

${e}_{e}^{si,y}$ | Electrical energy demand point for determining stack, $si$, in project year, $y$ (MWh). |

${p}_{h}^{sj,y}$ | Heat load demand point for determining stack, $sj$, in project year, $y$ (MW). |

${e}_{h}^{sj,y}$ | Heat energy demand point for determining stack, $sj$, in project year, $y$ (MWh). |

${P}_{e}^{i,si,y}$ | Desired capacity of the electrical energy resource, $i$, to which stack, $si$, is allocated in project year, $y$ (MW). |

${E}_{e}^{i,si,y}$ | Utilized energy of the electrical energy resource, $i$, to which stack, $si$, is allocated in project year, $y$ (MWh). |

${P}_{h}^{j,sj,y}$ | Desired capacity of the heat energy resource, $j$, to which stack, $sj$, is allocated in project year, $y$ (MW). |

${E}_{h}^{j,sj,y}$ | Utilizing energy of the heat energy resource, $j$, to which stack, $sj$, is allocated in project year, $y$ (MWh). |

${e}_{EES}^{si,y}$ | Charging electrical energy of stack, $si$, in project year, $y$ (MWh). |

${e}_{TES}^{sj,y}$ | Charging thermal energy of stack, $sj$, in project year, $y$ (MWh). |

Binary Variables | |

$b{s}_{e}^{elp,si,eseg,y}$ | Status of candidate segment, $eseg$, in stack, $si$, of the electrical load-energy curve, $elp$, in project year, $y$. |

$b{s}_{h}^{sj,hseg,y}$ | Status of candidate segment, $hseg$, in stack, $sj$, of heat load-energy curve in project year, $y$. |

${u}_{e}^{i,si,y}$ | Status of candidate electrical energy resource, $i$, to which stack, $si$, is allocated in project year, $y$. |

${u}_{h}^{j,sj,y}$ | Status of candidate heat energy resource, $j$, to which stack, $sj$, is allocated in project year, $y$. |

${v}_{e}^{i,c,y}$ | Status of candidate generating unit, $c$, of electrical energy resource, $i$, in project year, $y$. |

${v}_{h}^{j,c,y}$ | Status of candidate generating unit, $c$, of heat energy resource, $j$, in project year, $y$. |

${u}_{ELPAT}^{elp,y}$ | Status of candidate electrical load-energy curve, $elp$, in project year, $y$. |

Parameters | |

${C}_{e}^{i,c}$ | Capacity of candidate generating unit, $c$, of electrical energy resource, $i$. |

${C}_{h}^{j,c}$ | Capacity of candidate generating unit, $c$, of heat energy resource, $j$. |

$C{C}_{e}^{i}$ | Capital cost of electrical energy resource, $i$. |

$C{C}_{h}^{j}$ | Capital cost of heat energy resource, $j$. |

$FOM{C}_{e}^{i}$ | Fixed operation and maintenance cost of electrical energy resource, $i$. |

$FOM{C}_{h}^{j}$ | Fixed operation and maintenance cost of heat energy resource, $j$. |

$F{C}_{e}^{i}$ | Fuel cost of electrical energy resource, $i$. |

$F{C}_{e}^{j}$ | Fuel cost of heat energy resource, $j$. |

$VOM{C}_{e}^{i}$ | Variable operation and maintenance cost of electrical energy resource, $i$. |

$VOM{C}_{h}^{j}$ | Variable operation and maintenance cost of heat energy resource, $j$. |

$L{T}_{e}^{i}$ | Lifetime of electrical energy resource, $i$. |

$L{T}_{h}^{j}$ | Lifetime of heat energy resource, $j$. |

${\rho}_{\mathrm{CHP},e}^{i}$ | Index of CHP unit in electrical energy resource, $i$. |

${\rho}_{\mathrm{CHP},h}^{j}$ | Index of CHP unit in heat energy resource, $j$. |

${\rho}_{EES}^{i}$ | Index of electrical energy storage in electrical energy resource, $i$. |

${\rho}_{TES}^{j}$ | Index of thermal energy storage in heat energy resource, $j$. |

${\rho}_{RES,e}^{i}$ | Index of renewable electrical power source in electrical energy resource, $i$. |

${\alpha}_{HPR}$ | Heat-to-power ratio |

${\gamma}_{d}$ | Interest rate |

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**Figure 5.**Optimization process for determining the candidate resource: (

**a**) Determining the desired point using the SOS2 method; (

**b**) Allocating the capacity and utilized energy of a candidate resource.

**Figure 6.**Load curves used in analysis: (

**a**) LDC and stepwise representation of LDC; (

**b**) Piecewise linear load-energy curve.

**Figure 7.**Load profiles for the first project year: (

**a**) Electricity load; (

**b**) Heat load; (

**c**) Load duration curve for electricity load; (

**d**) Load duration curve for heat load.

**Figure 8.**Piecewise linear load-energy curves for the first project year: (

**a**) Electricity load; (

**b**) Heat load.

**Figure 10.**Estimated costs by case. (Note that, in the Case 3 and 4, (a) means a case with forced allocation of RES and (b) means a case without forced allocation of RES).

**Figure 11.**Results of planning schedules for Case 4: (

**a**) Installed capacity of electricity resources without forced allocation of RES; (

**b**) Installed capacity of electricity resources with forced allocation of RES; (

**c**) Utilized energy of electricity resources without forced allocation of RES; (

**d**) Utilized energy of electricity resources with forced allocation of RES; (

**e**) Installed capacity of heat resources without forced allocation of RES; (

**f**) Installed capacity of heat resources with forced allocation of RES; (

**g**) Utilized energy of heat resources without forced allocation of RES; (

**h**) Utilized energy of heat resources with forced allocation of RES.

Unit | Fixed Cost ($/MW) (Capital Cost + Fixed Operations & Maintenance Cost) | Variable Cost ($/MWh) (Fuel Cost + Variable Operations & Maintenance Cost) |
---|---|---|

G1 | 60,000 | 100 |

G2 | 10,000 | 200 |

Model No. | Type of Curve | Total Cost (M$) | Desired Capacity of G1 (MW) | Utilized Energy of G1 (GWh) | Desired Capacity of G2 (MW) | Utilized Energy of G2 (GWh) |
---|---|---|---|---|---|---|

0 | LDC | 858.9 | 1087.2 | 7885 | 126.0 | 19.9 |

1 | Stepwise representation of LDC | 866.8 (0.92%) | 1028.8 (5.37%) | 7780 (1.33%) | 184.4 (46.35%) | 126.1 (533.7%) |

2 | Piecewise linear load-energy curve | 861.5 (0.31%) | 1080 (0.66%) | 7855 (0.38%) | 133.2 (5.68%) | 49.7 (150.3%) |

Parameter | Value |
---|---|

Project lifetime (Year) | 7 |

Interest rate (%) | 3.91 |

Demand growth rate (%) | 2.5 |

Approximation error tolerance (%) | 1 |

Resource Type | Unit Name | Overnight Capital Cost ($/MW) | Fixed O&M Cost ($/MW) | Fuel Cost ($/MWh) | Variable O&M Cost ($/MWh) | Life Span (Yr) | Candidate Capacity (MW) |
---|---|---|---|---|---|---|---|

Fuel-based Power Generator | DG1 | 900,000 | 15,000 | 33.2925 | 6.1 | 20 | 800, 700, 600, 500, 400, 300 |

DG2 | 650,000 | 15,000 | 182.3 | 15 | 20 | 90, 80, 70, 60, 50, 40 | |

DG3 | 875,000 | 17,500 | 46.75 | 12.5 | 20 | 90, 80, 70, 60, 50, 40 | |

Heat Only Boiler | HOB1 | 720,000 | 12,000 | 26.634 | 4.88 | 20 | 500, 450, 400, 350, 300, 250 |

HOB2 | 520,000 | 15,000 | 182.3 | 15 | 20 | 400, 350, 300, 250, 200, 150 | |

CHP | CHP | 1,150,000 | 5850 | 22.77 | 2.75 | 20 | 900, 800, 700, 600, 500, 400 (Heat-to-Power ratio: 0.92) |

Electrical Energy Storage | EES | 3,092,000 | 42,000 | 0 | 35 | 7 | 24, 20, 16, 12, 8, 4 |

Thermal Energy Storage | TES | 3,184,000 | 52,000 | 0 | 35 | 7 | 24, 20, 16, 12, 8, 4 |

Renewable Electrical Power Source | PV | 1,375,000 | 10,500 | 0 | 0 | 20 | 13.3 |

Case Number | Fuel-Based Power Generator | Heat Only Boiler | CHP | Storage | Renewable Electrical Power Source | |
---|---|---|---|---|---|---|

Electrical Energy | Thermal Energy | |||||

1 | ◯ | ◯ | ◯ | - | - | - |

2 | ◯ | ◯ | ◯ | ◯ | ◯ | - |

3 | ◯ | ◯ | ◯ | - | - | ◯ |

4 | ◯ | ◯ | ◯ | ◯ | ◯ | ◯ |

**Table 6.**Seven-year planning schedules for Cases 1, 2, and 3 (RES = renewable electrical power source).

Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
---|---|---|---|---|---|---|---|---|---|

Case No. | Forced Allocated of RES | Unit | Installed Capacity (MW)/Utilized Energy (GWh) | ||||||

1 | - | DG1 | 700/5553 | 700/5692 | 700/5834 | 700/5980 | 700/6110 | 700/6114 | 700/6118 |

DG2 | - | - | - | - | - | - | - | ||

DG3 | - | - | - | - | 40/18.83 | 40/168.3 | 40/321.5 | ||

HOB1 | 250/152.3 | 250/155.7 | 250/159.3 | 250/162.9 | 250/166.7 | 250/170.5 | 250/174.4 | ||

HOB2 | 250/9.488 | 250/10.062 | 250/10.65 | 250/11.25 | 250/11.87 | 250/12.51 | 250/13.16 | ||

CHP * | 700/2352(2164) | 700/2411(2218) | 700/2471(2273) | 700/2533(2330) | 700/2596(2388) | 700/2661(2448) | 700/2728(2509) | ||

2 | - | DG1 | 700/5553 | 700/5692 | 700/5840 | 700/6010 | 700/6110 | 700/6114 | 700/6118 |

DG2 | - | - | 60/0.2462 | 60/0.7965 | 60/1.349 | 60/1.915 | 60/2.576 | ||

DG3 | - | - | - | - | 50/75.33 | 50/252.1 | 50/417.8 | ||

HOB1 | 300/155.0 | 300/158.4 | 300/167.5 | 300/192.8 | 300/220.7 | 300/249.4 | 300/263.0 | ||

HOB2 | 250/6.792 | 250/7.366 | 250/8.264 | 250/9.619 | 250/11.01 | 250/12.43 | 250/15.63 | ||

EES | - | - | - | - | - | - | - | ||

TES | - | - | - | - | - | - | 8/14.08 | ||

CHP * | 600/2352(2164) | 600/2411(2218) | 600/2465(2267) | 600/2502(2302) | 600/2538(2335) | 600/2575(2369) | 600/2629(2418) | ||

3 | No | DG1 | 700/5553 | 700/5692 | 700/5834 | 700/5980 | 700/6110 | 700/6114 | 700/6118 |

DG2 | - | - | - | - | - | - | - | ||

DG3 | - | - | - | - | 40/18.83 | 40/168.3 | 40/321.5 | ||

HOB1 | 250/152.3 | 250/155.7 | 250/159.3 | 250/162.9 | 250/166.7 | 250/170.5 | 250/174.4 | ||

HOB2 | 250/9.488 | 250/10.062 | 250/10.65 | 250/11.25 | 250/11.87 | 250/12.51 | 250/13.16 | ||

CHP * | 700/2352(2164) | 700/2411(2218) | 700/2471(2273) | 700/2533(2330) | 700/2596(2388) | 700/2661(2448) | 700/2728(2509) | ||

PV | - | - | - | - | - | - | - | ||

3 | Yes | DG1 | 700/5551 | 700/5690 | 700/5832 | 700/5978 | 700/6110 | 700/6114 | 700/6118 |

DG2 | - | - | - | - | - | - | - | ||

DG3 | - | - | - | - | 40/16.78 | 40/166.2 | 40/319.4 | ||

HOB1 | 250/152.3 | 250/155.7 | 250/159.3 | 250/162.9 | 250/166.7 | 250/170.5 | 300/177.1 | ||

HOB2 | 250/9.49 | 250/10.06 | 250/10.65 | 250/11.25 | 250/11.87 | 250/12.51 | 200/10.46 | ||

CHP * | 700/2352(2164) | 700/2411(2218) | 700/2471(2273) | 700/2533(2330) | 700/2596(2388) | 700/2661(2448) | 700/2728(2509) | ||

PV | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 |

Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
---|---|---|---|---|---|---|---|---|---|

Case No. | Forced Allocation of RES | Unit | Installed Capacity (MW)/Utilized Energy (GWh) | ||||||

4 | No | DG1 | 700/5553 | 700/5692 | 700/5840 | 700/6010 | 700/6110 | 700/6114 | 700/6118 |

DG2 | - | - | 60/0.2462 | 60/0.7965 | 60/1.349 | 60/1.915 | 60/2.576 | ||

DG3 | - | - | - | - | 50/75.33 | 50/252.1 | 50/417.8 | ||

HOB1 | 300/155.0 | 300/158.4 | 300/167.5 | 300/192.8 | 300/220.7 | 300/249.4 | 300/263.0 | ||

HOB2 | 250/6.792 | 250/7.366 | 250/8.264 | 250/9.619 | 250/11.01 | 250/12.43 | 250/15.63 | ||

EES | - | - | - | - | - | - | - | ||

TES | - | - | - | - | - | - | 8/14.08 | ||

CHP * | 600/2352(2164) | 600/2411(2218) | 600/2465(2267) | 600/2502(2302) | 600/2538(2335) | 600/2575(2369) | 600/2629(2418) | ||

PV | - | - | - | - | - | - | - | ||

4 | Yes | DG1 | 700/5551 | 700/5690 | 700/5838 | 700/6008 | 700/6110 | 700/6114 | 700/6118 |

DG2 | - | - | 60/0.246 | 60/0.798 | 60/1.352 | 60/1.920 | 60/2.582 | ||

DG3 | - | - | - | - | 50/73.24 | 50/250.0 | 50/415.7 | ||

HOB1 | 300/155.0 | 300/158.4 | 300/167.5 | 300/192.8 | 300/220.7 | 300/249.4 | 300/263.0 | ||

HOB2 | 250/6.793 | 250/7.367 | 250/8.264 | 250/9.619 | 250/11.01 | 250/12.43 | 250/15.63 | ||

EES | - | - | - | - | - | - | - | ||

TES | - | - | - | - | - | - | 8/14.08 | ||

CHP * | 600/2352(2164) | 600/2411(2218) | 600/2465(2267) | 600/2502(2302) | 600/2538(2335) | 600/2575(2369) | 600/2629(2418) | ||

PV | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 | 13.3/2.079 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ko, W.; Park, J.-K.; Kim, M.-K.; Heo, J.-H.
A Multi-Energy System Expansion Planning Method Using a Linearized Load-Energy Curve: A Case Study in South Korea. *Energies* **2017**, *10*, 1663.
https://doi.org/10.3390/en10101663

**AMA Style**

Ko W, Park J-K, Kim M-K, Heo J-H.
A Multi-Energy System Expansion Planning Method Using a Linearized Load-Energy Curve: A Case Study in South Korea. *Energies*. 2017; 10(10):1663.
https://doi.org/10.3390/en10101663

**Chicago/Turabian Style**

Ko, Woong, Jong-Keun Park, Mun-Kyeom Kim, and Jae-Haeng Heo.
2017. "A Multi-Energy System Expansion Planning Method Using a Linearized Load-Energy Curve: A Case Study in South Korea" *Energies* 10, no. 10: 1663.
https://doi.org/10.3390/en10101663