Newton’s method is a common method for performing power flow calculations. The convergence and robustness of Newton’s method have been unsatisfactory when the system load is heavy [1].

The optimal multiplier method is considered a relatively successful algorithm for solving ill-conditioned power flow problems. The optimal multiplier method combines the calculation of the optimal multiplier with the conventional Newton’s method to make the algorithm’s convergence controllable [2]. However, this method does not overcome the problem of sensitivity to the initial value. Some scholars have proposed a load flow algorithm for load model admittance of heavy-load nodes. They replace reactive loads with ground admittance at heavily loaded nodes. This method changes the minimum norm eigenvalue of the Jacobian matrix near the critical point. At the same time, this method can better solve the convergence problem of heavy-load ill-conditioned power flow. However, due to the impedance transformation of the load, the convergence result is a certain distance from the initial load condition.

The results obtained by this algorithm can only be regarded as approximate solutions. Some scholars have proposed two different parameterization methods to solve the ill-conditioned trend [3]. These two methods make the power flow calculation converge at the critical point by shifting the singular point of the Jacobian correction matrix. But its calculation accuracy is low, and the calculation time is relatively long. Some scholars have conducted in-depth research on the convergence of the tensor method for solving nonlinear equations. Some scholars have introduced two methods for solving the power flow equation based on the tensor method. But it is only applicable to the case where the tensor equation has zero roots and the case where Cartesian coordinates represent the power flow equation. Some scholars have compared the optimal multiplier method for Cartesian and polar coordinates. They believe that the optimal multiplier method in polar coordinates can solve the power flow problem better.

The optimal multiplier method is the same as some load flow algorithms that preserve nonlinearity. These algorithms all make use of higher-order information about the step size. Therefore, the above algorithm has better convergence than Newton’s method, which only retains the first-order term. The optimal multiplier method obtains the Newton direction by performing a Newton iteration [4]. Then find the best step size. The step size is restricted to search only in the Newtonian direction. When the Jacobian matrix of the modified equation is singular or close to singular, the optimal multiplier method is very unreliable because the Newton step size or Newton direction cannot be obtained. Its iterations are also non-convergent. In this paper, two new methods based on the tensor method to solve the power flow of the power system are proposed. Method 1 is a power flow calculation method for obtaining a tensor correction amount based on the least-squares method. Method 2 is a power flow calculation method based on the direct tensor method in polar coordinates. The calculation results of several examples show that the method in this paper is effective.

^{r}

Equation (1) is a nonlinear equation system. The incremental

Where ^{(k)}) is the Jacobian matrix at the current point ^{(k)}. And then we get ^{(k+1)} as ^{(k+1)} = ^{(k)} + ^{k}. The ^{T}

The interpolation method is to uses the calculation results of several previous iterations. ^{(k)}) is not singular, we multiply both sides of equation (2) by

When the quadratic equation of equation (3) about _{i} has real roots, we can obtain the real roots _{i}

If we use two-point interpolation, its accuracy is sufficient in many cases.

It can be obtained by formula (4)

From formula (5), it can be seen that the increment _{a}^{-1}^{(k)}). It is Newton’s correction and it is the main part of _{b}^{-1}[(^{2})/2] is the tensor modifier. It is also a correction to _{a}^{(0)}, and the implementation steps of the method when iterating at step

Calculate the function value ^{(k)}) of the current point.

If the ^{(k)}) element with the largest amplitude satisfies the convergence accuracy condition, the algorithm ends.

Calculate the Jacobian matrix ^{(k)}) of the current point.

We solve for Newton’s correction _{a} = -^{-1}^{(k)}).

If _{a} = -^{-1}[1/(2a^{2})]. At this time, the total correction amount is _{a} + _{b}. If equation (3) has no real root, we use Newton’s correction _{a}.

The above basic algorithm requires that the expression (3) is a quadratic equation of one variable with respect to _{i}^{(k)} + _{i}^{(k)} + _{i}^{m}^{(k)} + _{2} is minimal. At this point, we can find an approximate tensor correction. Its least squares problem can be expressed as

Where _{i}, ^{n×(n-p)} is a set of the orthonormal basis for the space spanned by the columns of matrix ^{-T}

If remember ^{T}(^{T}^{-1}^{T} ^{T} ^{-T} ^{(k)}) + ^{T} ^{-T} ^{2})/2. ^{2} Each element in is the square of the corresponding element of

For any ^{p} there is always ^{n}^{T} ^{(k)} + ^{T}

And because ^{-1} = ^{T} ), we can derive from equation (6) to get

We substitute equation (2) and the expression of

So have

The first part of equation (9) is the Newton correction _{a}^{-1}^{(k)}), and the second part is the tensor correction:

So far we have obtained a new solution method for tensor correction. It has nothing to do with whether the quadratic equation of equation (3) about _{i}^{(0)}, and when the

Calculate the function value ^{(k)}) of the current point.

If the ^{(k)}) element with the largest amplitude satisfies the convergence accuracy condition, the algorithm ends.

Calculate the Jacobian matrix ^{(k)}) of the current point.

We solve for Newton correction _{a} = -^{-1}^{(k)}).

If the tensor equation of equation (3) has a real number solution, solve _{b} = -^{-1}[1/(2^{2})]. If equation (3) has no real solution, then we solve the least-squares problem about _{b}, _{b} = -^{-1} (^{2}-^{-T} ^{-1}

Find the total correction amount, _{a} + _{b}.

Regardless of whether it is the basic algorithm or method 1, we need to perform interpolation calculation and corresponding ^{(k)} + _{a} is the Newton correction and _{b}_{a} ≫ _{b}, the equation (1) equals 0 can be transformed into

We can rewrite it as

In the formula, ^{T} ^{T}

When the system is lightly loaded, _{ij}_{ij}_{ij}^{T}^{(0)}. When step

Calculate the function value ^{(k)}) of the current point.

If the ^{(k)}) element with the largest amplitude satisfies the convergence accuracy condition, the algorithm ends.

Calculate the Jacobian matrix ^{(k)}) of the current point.

We solve for Newton correction _{a} = -^{-1}^{(k)}).

We use equation (11) to solve the quadratic term

Calculate || _{b} || / || _{b}_{a} + _{b} when || _{b} || / || _{b}_{a}.

Assume ^{(k+1)} = ^{(k)} + d,

We use C language programming. We use the above algorithm to calculate the power flow for multiple systems on a computer with a CPU frequency of 2.0GHz and memory of 1G. Table 1 shows the system scale data of each example [10]. Calculation example 1 and calculation example 2 are the New England and IEEE118 systems, respectively. Calculation examples 3-6 use 39 and 118 node system networks of building systems, respectively. Example 7 is the actual North China network system.

Data from the test system

Examples | Total number of nodes | Total number of branches | Examples | Total number of nodes | Total number of branches |
---|---|---|---|---|---|

Example 1 | 39 | 46 | Example 5 | 648 | 1147 |

Example 2 | 118 | 179 | Example 6 | 810 | 1434 |

Example 3 | 226 | 293 | Example 7 | 1923 | 2280 |

Example 4 | 324 | 573 | Examples |

Example 2 compares the iteration times and calculation time of the four methods when the system is under normal load. N is the number of calculation iterations. In method 2, take 0.15. It can be seen from Table 2 that the calculation time of the basic tensor method and method 1 is the longest under the rated load [11]. The computation time of method 1 is longer than that of the basic tensor method. Mainly due to its use of optimization algorithms. The calculation time of method 2 is due to directly solving the quadratic term. Its calculation time is the same as Newton’s method.

Comparison of iteration number and computation time (rated load) for 4 methods

Test | Newton’s method | Basic tensor method | Method 1 | Method 2 | ||||
---|---|---|---|---|---|---|---|---|

Example 1 | 3 | 0.061 | 2 | 0.11 | 2 | 0.122 | 2 | 0.47 |

Example 2 | 3 | 0.077 | 3 | 0.14 | 3 | 0.153 | 3 | 0.066 |

Example 3 | 4 | 0.147 | 3 | 0.151 | 3 | 0.179 | 3 | 0.104 |

Example 4 | 4 | 0.292 | 4 | 0.51 | 4 | 0.755 | 4 | 0.259 |

Example 5 | 4 | 0.527 | 4 | 1 | 4 | 1.12 | 4 | 0.539 |

Example 6 | 4 | 0.65 | 4 | 1.2 | 4 | 1.49 | 4 | 0.793 |

Example 7 | 4 | 0.949 | 4 | 1.421 | 4 | 1.97 | 4 | 0.955 |

Method 1 has the advantage of convergence. It can still converge when the basic tensor method and Newton’s method do not converge. The computation time of method 1 is longer than that of the basic tensor method and Newton’s method. Since the quadratic term can be easily obtained directly, the convergence and convergence speed of method 2 is the best among the four methods.

Tensor-based algorithms, including the basic tensor method, method 1, and method 2, all have better convergence than Newton’s method. However, the calculation amount of the basic tensor method and method 1 is larger than that of the Newton method [12]. Compared with the basic tensor method, the calculation amount of method 1 is larger. The tensor method is similar to other nonlinear Newton methods that retain high valence, and they all require the Newton direction. The difference is that the tensor method corrects the Newton direction. And other Newton-like methods that retain higher-order nonlinear terms represented by the optimal multiplier method. Only the Newton step size is optimized, and the Newton direction is not changed. When the system load increases, the Jacobian matrix condition number of the system worsens, making Newton’s method and optimal multiplier method not convergent.

In this paper, two new methods of power flow calculation based on the tensor method are proposed. Method 1 employs a least-squares optimization algorithm when the tensor equation has no real solution. We obtain its corresponding tensor correction value so that the power flow calculation based on the interpolation tensor method has better convergence. Method 2 is a power flow calculation method called the direct tensor method in polar coordinates. The convergence and calculation speed of power flow calculation are improved because the quadratic term is easily taken into account. The calculation results of several examples show that the two algorithms proposed in this paper are effective.

#### Data from the test system

Examples | Total number of nodes | Total number of branches | Examples | Total number of nodes | Total number of branches |
---|---|---|---|---|---|

Example 1 | 39 | 46 | Example 5 | 648 | 1147 |

Example 2 | 118 | 179 | Example 6 | 810 | 1434 |

Example 3 | 226 | 293 | Example 7 | 1923 | 2280 |

Example 4 | 324 | 573 | Examples |

#### Comparison of iteration number and computation time (rated load) for 4 methods

Test | Newton’s method | Basic tensor method | Method 1 | Method 2 | ||||
---|---|---|---|---|---|---|---|---|

Example 1 | 3 | 0.061 | 2 | 0.11 | 2 | 0.122 | 2 | 0.47 |

Example 2 | 3 | 0.077 | 3 | 0.14 | 3 | 0.153 | 3 | 0.066 |

Example 3 | 4 | 0.147 | 3 | 0.151 | 3 | 0.179 | 3 | 0.104 |

Example 4 | 4 | 0.292 | 4 | 0.51 | 4 | 0.755 | 4 | 0.259 |

Example 5 | 4 | 0.527 | 4 | 1 | 4 | 1.12 | 4 | 0.539 |

Example 6 | 4 | 0.65 | 4 | 1.2 | 4 | 1.49 | 4 | 0.793 |

Example 7 | 4 | 0.949 | 4 | 1.421 | 4 | 1.97 | 4 | 0.955 |

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neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network