# 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 |

## References

- Kazantsev, Y.V.; Glazyrin, G.V.; Khalyasmaa, A.I.; Shayk, S.M.; Kuparev, M.A. Advanced Algorithms in Automatic Generation Control of Hydroelectric Power Plants. Mathematics
**2022**, 10, 4809. [Google Scholar] [CrossRef] - Sibtain, D.; Gulzar, M.M.; Murtaza, A.F.; Murawwat, S.; Iqbal, M.; Rasool, I.; Hayat, A.; Arif, A. Variable structure model predictive controller based gain scheduling for frequency regulation in renewable based power system. Int. J. Numer. Model. Electron. Netw. Devices Fields
**2022**, 35, e2989. [Google Scholar] [CrossRef] - Tan, K.; Tian, Y.; Xu, F.; Li, C. Research on Multi-Objective Optimal Scheduling for Power Battery Reverse Supply Chain. Mathematics
**2023**, 11, 901. [Google Scholar] [CrossRef] - Grisales-Noreña, L.F.; Cortés-Caicedo, B.; Alcalá, G.; Montoya, O.D. Applying the Crow Search Algorithm for the Optimal Integration of PV Generation Units in DC Networks. Mathematics
**2023**, 11, 387. [Google Scholar] [CrossRef] - Gul, M.; Tai, N.; Huang, W.; Nadeem, M.H.; Ahmad, M.; Yu, M. Technical and Economic Assessment of VSC-HVDC Transmission Model: A Case Study of South-Western Region in Pakistan. Electronics
**2019**, 8, 1305. [Google Scholar] [CrossRef] [Green Version] - Bento, P.; Pina, F.; Mariano, S.; Calado, M.D.R. Short-term Hydro-thermal Coordination By Lagrangian Relaxation: A New Algorithm for the Solution of the Dual Problem. KnE Eng.
**2020**, 728–742. [Google Scholar] [CrossRef] - Gulzar, M.M.; Murawwat, S.; Sibtain, D.; Shahid, K.; Javed, I.; Gui, Y. Modified Cascaded Controller Design Constructed on Fractional Operator ‘β’to Mitigate Frequency Fluctuations for Sustainable Operation of Power Systems. Energies
**2022**, 15, 7814. [Google Scholar] [CrossRef] - Saini, O.; Chauhan, A. Optimal Operation of Short-Term Variable-Head Hydrothermal Generation Scheduling Using the Differential Evolution Method, Newton-Raphson Method and Heuristic Search Method. In Proceedings of the 1st International Conference on Recent Innovation in Electrical, Electronics and Communication System, RIEECS, Bhubaneswar, India, 27–28 July 2018. [Google Scholar]
- Wood, A.J. Power Generation, Operation, and Control, 3rd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2014. [Google Scholar]
- Jian, J.; Pan, S.; Yang, L. Solution for short-term hydrothermal scheduling with a logarithmic size mixed-integer linear programming formulation. Energy
**2019**, 171, 770–784. [Google Scholar] [CrossRef] - Gjerden, K.S.; Helseth, A.; Mo, B.; Warland, G. Hydrothermal scheduling in Norway using stochastic dual dynamic programming; a large-scale case study. In Proceedings of the 2015 IEEE Eindhoven PowerTech, Eindhoven, The Netherlands, 29 June–2 July 2015; pp. 1–6. [Google Scholar] [CrossRef]
- Fakhar, M.S.; Liaquat, S.; Kashif, S.A.R.; Rasool, A.; Khizer, M.; Iqbal, M.A.; Baig, M.A.; Padmanaban, S. Conventional and Metaheuristic Optimization Algorithms for Solving Short Term Hydrothermal Scheduling Problem: A Review. IEEE Access
**2021**, 9, 25993–26025. [Google Scholar] [CrossRef] - Iqbal, M.; Gulzar, M.M. Master-slave design for frequency regulation in hybrid power system under complex environment. IET Renew. Power Gener.
**2022**, 16, 3041–3057. [Google Scholar] [CrossRef] - Gulzar, M.M. Maximum Power Point Tracking of a Grid Connected PV Based Fuel Cell System Using Optimal Control Technique. Sustainability
**2023**, 15, 3980. [Google Scholar] [CrossRef] - Zheyuan, C.; Hammid, A.; Kareem, A.; Jiang, M.; Mohammed, M.; Kumar, N. A Rigid Cuckoo Search Algorithm for Solving Short-Term Hydrothermal Scheduling Problem. Sustainability
**2021**, 13, 4277. [Google Scholar] [CrossRef] - Liaquat, S.; Fakhar, M.S.; Kashif, S.A.R.; Rasool, A.; Saleem, O.; Padmanaban, S. Performance Analysis of APSO and Firefly Algorithm for Short Term Optimal Scheduling of Multi-Generation Hybrid Energy System. IEEE Access
**2020**, 8, 177549–177569. [Google Scholar] [CrossRef] - Zeng, X.; Hammid, A.T.; Kumar, N.M.; Subramaniam, U.; Almakhles, D.J. A grasshopper optimization algorithm for optimal short-term hydrothermal scheduling. Energy Rep.
**2021**, 7, 314–323. [Google Scholar] [CrossRef] - Hassan, T.U.; Alquthami, T.; Butt, S.E.; Tahir, M.F.; Mehmood, K. Short-term optimal scheduling of hydro-thermal power plants using artificial bee colony algorithm. Energy Rep.
**2020**, 6, 984–992. [Google Scholar] [CrossRef] - Azad, A.S.; Rahaman, S.A.; Watada, J.; Vasant, P.; Vintaned, J.A.G. Optimization of the hydropower energy generation using Meta-Heuristic approaches: A review. Energy Rep.
**2020**, 6, 2230–2248. [Google Scholar] [CrossRef] - Zhao, J.; Zhang, Y.; Liu, Z.; Hu, W.; Su, D. Distributed multi-objective day-ahead generation and HVDC transmission joint scheduling for two-area HVDC-linked power grids. Int. J. Electr. Power Energy Syst.
**2021**, 134, 107445. [Google Scholar] [CrossRef] - Nemati, M.; Braun, M.; Tenbohlen, S. Optimization of unit commitment and economic dispatch in microgrids based on genetic algorithm and mixed integer linear programming. Appl. Energy
**2018**, 210, 944–963. [Google Scholar] [CrossRef] - Bento, P.M.R.; Mariano, S.J.P.S.; Calado, M.R.A.; Ferreira, L.A.F.M. A Novel Lagrangian Multiplier Update Algorithm for Short-Term Hydro-Thermal Coordination. Energies
**2020**, 13, 6621. [Google Scholar] [CrossRef] - Ergun, H.; Dave, J.; Van Hertem, D.; Geth, F. Optimal Power Flow for AC–DC Grids: Formulation, Convex Relaxation, Linear Approximation, and Implementation. IEEE Trans. Power Syst.
**2019**, 34, 2980–2990. [Google Scholar] [CrossRef] - Castro, L.M.; González-Cabrera, N.; Guillen, D.; Tovar-Hernández, J.; Gutiérrez-Alcaraz, G. Efficient method for the optimal economic operation problem in point-to-point VSC-HVDC connected AC grids based on Lagrange multipliers. Electr. Power Syst. Res.
**2020**, 187, 106493. [Google Scholar] [CrossRef] - Nguyen, T.T.; Pham, L.H.; Mohammadi, F.; Kien, L.C. Optimal Scheduling of Large-Scale Wind-Hydro-Thermal Systems with Fixed-Head Short-Term Model. Appl. Sci.
**2020**, 10, 2964. [Google Scholar] [CrossRef] - Ahmad, A.; Kashif, S.A.R.; Nasir, A.; Rasool, A.; Liaquat, S.; Padmanaban, S.; Mihet-Popa, L. Controller Parameters Optimization for Multi-Terminal DC Power System Using Ant Colony Optimization. IEEE Access
**2021**, 9, 59910–59919. [Google Scholar] [CrossRef] - Gulzar, M.M.; Sibtain, D.; Ahmad, A.; Javed, I.; Murawwat, S.; Rasool, I.; Hayat, A. An Efficient Design of Adaptive Model Predictive Controller for Load Frequency Control in Hybrid Power System. Int. Trans. Electr. Energy Syst.
**2022**, 2022, 7894264. [Google Scholar] [CrossRef] - Baradar, M.; Ghandhari, M. A Multi-Option Uni Fi Ed Power Flow Approach for Hybrid AC/DC Grids Incorporating. IEEE Trans. Power Syst.
**2013**, 28, 2376–2383. [Google Scholar] [CrossRef] - Gonzalez-Torres, J.C.; Damm, G.; Costan, V.; Benchaib, A.; Lamnabhi-Lagarrigue, F. A Novel Distributed Supplementary Control of Multi-Terminal VSC-HVDC Grids for Rotor Angle Stability Enhancement of AC/DC Systems. IEEE Trans. Power Syst.
**2020**, 36, 623–634. [Google Scholar] [CrossRef] - Sibtain, D.; Gulzar, M.M.; Shahid, K.; Javed, I.; Murawwat, S.; Hussain, M.M. Stability Analysis and Design of Variable Step-Size P&O Algorithm Based on Fuzzy Robust Tracking of MPPT for Standalone/Grid Connected Power System. Sustainability
**2022**, 14, 8986. [Google Scholar] [CrossRef] - Cao, J.; Du, W.; Wang, H.F.; Member, S. An Improved Corrective Security Constrained OPF for Meshed AC/DC Grids with multi-terminal VSC-HVDC. IEEE Trans. Power Syst.
**2015**, 31, 485–495. [Google Scholar] [CrossRef] - Al-Sakkaf, S.; Kassas, M.; Khalid, M.; Abido, M.A. An Energy Management System for Residential Autonomous DC Microgrid Using Optimized Fuzzy Logic Controller Considering Economic Dispatch. Energies
**2019**, 12, 1457. [Google Scholar] [CrossRef] [Green Version] - Salman, U.; Khan, K.; Alismail, F.; Khalid, M. Techno-Economic Assessment and Operational Planning of Wind-Battery Distributed Renewable Generation System. Sustainability
**2021**, 13, 6776. [Google Scholar] [CrossRef] - Alhumaid, Y.; Khan, K.; Alismail, F.; Khalid, M. Multi-Input Nonlinear Programming Based Deterministic Optimization Framework for Evaluating Microgrids with Optimal Renewable-Storage Energy Mix. Sustainability
**2021**, 13, 5878. [Google Scholar] [CrossRef] - Khalid, M. Wind Power Economic Dispatch—Impact of Radial Basis Functional Networks and Battery Energy Storage. IEEE Access
**2019**, 7, 36819–36832. [Google Scholar] [CrossRef] - Khalid, M.; Ahmadi, A.; Savkin, A.V.; Agelidis, V.G. Minimizing the energy cost for microgrids integrated with renewable energy resources and conventional generation using controlled battery energy storage. Renew. Energy
**2016**, 97, 646–655. [Google Scholar] [CrossRef] - Gulzar, M.M.; Iqbal, A.; Sibtain, D.; Khalid, M. An Innovative Converterless Solar PV Control Strategy for a Grid Connected Hybrid PV/Wind/Fuel-Cell System Coupled with Battery Energy Storage. IEEE Access
**2023**. [Google Scholar] [CrossRef] - Gulzar, M.M.; Sibtain, D.; Khalid, M. Cascaded Fractional Model Predictive Controller for Load Frequency Control in Multiarea Hybrid Renewable Energy System with Uncertainties. Int. J. Energy Res.
**2023**. [Google Scholar] [CrossRef]

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