# Coordinated Economic Operation of Hydrothermal Units with HVDC Link Based on Lagrange Multipliers

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## Abstract

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## 1. Introduction

- Formulation of complex hydrothermal scheduling problem.
- Modelling of AC grids to add network constraints using DC optimal power flow (DCOPF) in the existing scheduling problem.
- Induction of HVDC link with line flows limitation constraints in hydrothermal problem.
- Linearization of quadratic cost curves of thermal generators to deal with inequality constraints.
- Implementation of linear programming-based Lagrange multipliers methods on a case study to check the robustness of the proposed method.

## 2. Problem Formulation

#### 2.1. Hydrothermal Problem Formulation

#### 2.2. HVDC Line Flow Problem Formulation

#### 2.3. AC Network Problem Formulation

#### 2.4. Overall Problem Formulation

#### 2.5. Constraints

- Load balance constraints

- Thermal plant generation limit

- Hydro plant generation limit

- Water volume limit

- HVDC line limitation

**P**is the nodal power injections at all nodes, θ is nodal angle, ${P}_{B}$ is line flow for AC and HVDC system, D and E are node-arc matrices, and $-{P}_{B}^{max}$ and ${P}_{B}^{max}$ are line flow limits. The ${\mathrm{q}}_{\mathrm{T}\mathrm{O}\mathrm{T}}$ is the total water volume available for generation, ${N}_{H}$ is the number of hydro units, and ${\mathrm{q}}_{\mathrm{h}}$ is the water discharge rate to generate hydro power ${\mathrm{P}}_{\mathrm{h}}$ in an interval. Moreover, (15) is similar to (8) with constraints mentioned as Lagrange multipliers.

## 3. Research Methodology

#### 3.1. Objective Function

#### 3.2. Output Vector

#### 3.3. Equality Constraints

**x**= b) form of equality constraints requires dimensions. The dimensions of equality constraints based on the case study are:

- Number of columns: As the vector x has 13 × 1, matrix A should have 13 columns to multiply x.
- Number of rows: As there are five branches, four buses, and one water volume constraint, (18) and (19) will have to contribute a total of ten (10) rows to matrix A.

**E**, is given as (21):

#### 3.4. Inequality Constraints

## 4. Results and Discussion

#### 4.1. Scenario-1: Infinite HVDC Line Capacity

#### 4.2. Scenario-2: Limited HVDC Line Capacity

_{Line2–3}= P

_{HVDC}= 400 MW is added on the HVDC link shown in Figure 4. The load power demand during time intervals 19, 20, and 21 is 1450 MW, 1500 MW, and 1400 MW, respectively, as shown in Figure 5. During these time intervals, the power shared by a hydro unit is 500 MW, thermal-1 is 683 MW, 767 MW, and 600 MW, and thermal-1 is 267 MW, 233 MW, and 300 MW, as given in Figure 6. Due to HVDC line limitations, the power is diverted to other transmission lines of the network. Hence, the network does not consider a single bus which results in different buses’ nodal lambdas (λ) price and line (λ) price. This makes the nodal price at bus-1 = λ

_{1}= 15.4 $/MWh, bus-2 = λ

_{2}= 14.2 $/MWh, bus-3 = λ

_{3}= 16.1 $/MWh, and bus-4 = λ

_{4}= 15.7 $/MWh, as shown in Figure 7. The line limit price in three different time intervals increased from 0 $/MWh to 3.1 $/MWh and 5.1 $/MWh due to line congestion heating and losses. The line limit price is shown in Figure 7. Moreover, the optimal power generation of each generating station to meet the load demand in each interval with limited line capacity is shown in Figure 9.

_{HVDC}= 400 MW limit. However, thermal-1 unit in 24 h period costs $166,739.8 and the thermal-2 unit costs $112,082.5 without HVDC line limitation. The proposed approach optimally scheduled the outputs of thermal generating stations under both scenarios to minimize the operating cost. It can be concluded that limited capacity HVDC links may change the optimal operating points of all generating stations throughout the load intervals. Hence, the electricity market will affect generation companies (GENCOs) and transmission system operators (TSO). Therefore, the proposed approach will be helpful in complex hydrothermal scheduling including embedded HVDC lines in existing AC networks for GENCOs and TSO to find their power generation price, bus nodal price, and line limit price.

## 5. Conclusions

_{h}= 500 MW) to obtain maximum efficiency. HVDC transmission line flow was successfully limited at P

_{HVDC}= 400 MW. The Lagrange multipliers method is used for optimal operation of hydrothermal plants on power networks. The proposed formulation is applied to two scenarios of a case study. In the first scenario of the case study, the total thermal generation cost comes out to $278,822.3. In the second scenario of the case study, the total cost of thermal generation is $279,025.4. The difference in cost in both scenarios is minimum. It is observed in both scenarios, with the change in load, that this algorithm optimally selects the thermal generator to redispatch to meet the load demand and other line constraints with minimum cost.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${P}_{hj}$ | Output power (MW) of ${h}^{th}$ hydro unit in ${j}^{th}$ period |

${P}_{tj}$ | Output power (MW) of ${t}^{th}$ thermal unit in ${j}^{th}$ period |

${F}_{t}\left({P}_{tj}\right)$ | Fuel cost rate ($/hour) for ${t}^{th}$ unit in ${j}^{th}$ period |

${{q}_{h}(P}_{hj}$) | Water flow rate (Acre-feet/hour) for ${h}^{th}$ unit in ${j}^{th}$ period |

${N}_{T}$ | Number of thermal power plants |

${N}_{H}$ | Number of hydro power plants |

${N}_{b}$ | Number of buses |

${N}_{M}$ | Number of lines (branches) |

${P}_{thj}{=P}_{tj}+{P}_{hj}$ | Total output power (MW) of ${t}^{th}$ thermal and ${h}^{th}$ hydro unit in ${j}^{th}$ period |

${J}_{max}$ | Maximum number of periods |

${n}_{j}$ | Number of hours in ${j}^{th}$ period |

${a}_{t},{b}_{t},{c}_{t}$ | Cost coefficients of ${t}^{th}$ thermal unit |

${x}_{h},{y}_{h},{z}_{h}$ | Water flow rate coefficients of ${h}^{th}$ hydro unit |

${q}_{TOT}$ | Total water volume available for power generation |

$\mathcal{L}$ | Lagrange function |

$\lambda ,\gamma ,\mu $ | Lagrange multipliers |

${P}_{flow}$ | HVDC line power flow limit |

${R}_{dc}$ | Resistance of HVDC line |

${\lambda}_{r}$ | Locational marginal Price (LMP) of rectifier bus |

${\lambda}_{i}$ | Locational marginal Price (LMP) of inverter bus |

PCC | Point of common coupling (PCC) |

Tr | Coupling transformer |

${V}_{{X}_{r}},{V}_{{C}_{r}}$ | Voltage at bus ${X}_{r}$ and ${C}_{r}$ on rectifier side |

${V}_{{X}_{i}},{V}_{{C}_{i}}$ | Voltage at bus ${X}_{i}$ and ${C}_{i}$ on inverter side |

C | DC link capacitor |

${\u2206P}_{k}$ | ${k}^{th}$ bus nodal power balance |

${P}_{gk}$ | Power generation on ${k}^{th}$ bus |

${P}_{dk}$ | Power demand on ${k}^{th}$ bus |

${P}_{k}^{cal}$ | Calculated nodal power on ${k}^{th}$ bus |

${G}_{km}$ | Conductance of line connecting bus node k and m |

${B}_{km}$ | Susceptance of line connecting bus node k and m |

$\theta $ | Nodal phase angle |

Subscript k, m | Indicate the nodal bus |

Subscript i, r | Indicate the inverter and rectifier, respectively |

V | Bus voltage magnitude |

$p.u$ | Per unit quantity |

DCOPF | Direct current optimal power flow |

HVDC | High voltage direct current |

${N}_{s}$ | Number of segments of quadratic cost function |

${S}_{a}$ | Slope of quadratic cost function |

T | Transpose of matrix |

W | Water volume |

${P}_{B}$ | Line power flow |

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Step-by-Step Implementation of Proposed Procedure for Coordinated/Optimal Economic Operation of Hydrothermal Units with HVDC Link Based on Lagrange Multipliers | |
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1 | Consider a power system network having ‘${N}_{b}$’ number of buses and ‘${N}_{M}$’ number of branches (shown in Figure 4). |

2 | Calculate the susceptance of each line following nodal power injection using (5), (6), and (7) of AC network and (4) for HVDC network or directly follow (10) which is common for both hydrothermal-based AC and HVDC power system. |

3 | Find slopes of cost functions of all thermal generators using (14) and define an objective function based on (16). |

4 | Formulate the DCOPF coordinated economic dispatch problem using (8) and (15) |

5 | Find a single matrix for all equality constraints, load balance, line power flows, and nodal power injections using (9), (18), and (19), respectively. This can be executed using node-arc incidence matrix product (D × E) using (22). |

6 | Embed hydropower plant variable and water volume constraint using (11) and (12). Then, develop a standard form of the matrix (Ax = b) using (24) for linear programming (LP). |

7 | Find parameters for b-matrix (of Ax = b) using load and generation buses of power system network (shown in Figure 4) and objection function given in (15). |

8 | Define inequality constraints of the power system network under study using (10), (11), and (13). |

9 | Apply the standard linear programming (LP) method using MATLAB software. |

10 | Check constraints. If all constraints are satisfied, then procedure is done. Otherwise, go to step 3. |

11 | Print the optimal operating schedule and nodal price of each bus. |

Unit | ${\mathit{a}}_{\mathit{t}}\left[\frac{\mathbf{\$}}{{\mathbf{M}\mathbf{W}}^{2}\mathbf{h}}\right]$ | ${\mathit{b}}_{\mathit{t}}\left[\frac{\mathbf{\$}}{\mathbf{M}\mathbf{W}\mathbf{h}}\right]$ | ${\mathit{c}}_{\mathit{t}}\left[\frac{\mathbf{\$}}{\mathbf{h}}\right]$ | ${\mathit{P}}_{\mathit{t}}^{\mathit{m}\mathit{i}\mathit{n}}\left(\mathbf{M}\mathbf{W}\right)$ | ${\mathit{P}}_{\mathit{t}}^{\mathit{m}\mathit{a}\mathit{x}}\left(\mathbf{M}\mathbf{W}\right)$ |
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Thermal-1 | 0.0033 | 10.8 | 1200 | 25 | 875 |

Thermal-2 | 0.003 | 12.6 | 1710 | 40 | 600 |

Unit | ${\mathit{x}}_{\mathit{h}}\left[\frac{\mathbf{A}\mathbf{F}}{{\mathbf{M}\mathbf{W}}^{2}\mathbf{h}}\right]$ | ${\mathit{y}}_{\mathit{h}}\left[\frac{\mathbf{A}\mathbf{F}}{\mathbf{M}\mathbf{W}\mathbf{h}}\right]$ | ${\mathit{z}}_{\mathit{h}}\left[\frac{\mathbf{A}\mathbf{F}}{\mathbf{h}}\right]$ | ${\mathit{P}}_{\mathit{h}}^{\mathit{m}\mathit{i}\mathit{n}}\left(\mathbf{M}\mathbf{W}\right)$ | ${\mathit{P}}_{\mathit{h}}^{\mathit{m}\mathit{a}\mathit{x}}\left(\mathbf{M}\mathbf{W}\right)$ | ${\mathit{q}}_{\mathit{T}\mathit{O}\mathit{T}}\left(\mathbf{A}\mathbf{F}\right)$ |
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Hydro | 0 | 4.9 | 50 | 0 | 500 | 30,000 |

Parameters | Proposed Method | Interior-Point Method | Dual-Simplex Method |
---|---|---|---|

No. of iterations | 6 | 10 | 292 |

Computational time (sec) | 1.05 | 1.58 | 2.93 |

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## Share and Cite

**MDPI and ACS Style**

Ahmad, A.; Kashif, S.A.R.; Ashraf, A.; Gulzar, M.M.; Alqahtani, M.; Khalid, M. Coordinated Economic Operation of Hydrothermal Units with HVDC Link Based on Lagrange Multipliers. *Mathematics* **2023**, *11*, 1610.
https://doi.org/10.3390/math11071610

**AMA Style**

Ahmad A, Kashif SAR, Ashraf A, Gulzar MM, Alqahtani M, Khalid M. Coordinated Economic Operation of Hydrothermal Units with HVDC Link Based on Lagrange Multipliers. *Mathematics*. 2023; 11(7):1610.
https://doi.org/10.3390/math11071610

**Chicago/Turabian Style**

Ahmad, Ali, Syed Abdul Rahman Kashif, Arslan Ashraf, Muhammad Majid Gulzar, Mohammed Alqahtani, and Muhammad Khalid. 2023. "Coordinated Economic Operation of Hydrothermal Units with HVDC Link Based on Lagrange Multipliers" *Mathematics* 11, no. 7: 1610.
https://doi.org/10.3390/math11071610