# Turboelectric Uncertainty Quantification and Error Estimation in Numerical Modelling
## 2 Network Model Method  
A high-level set of electrical network architectures are designed to obtain the optimal power flow within the required post-process results. At the start of the designed network, the input data are considered first. These data are the values of the electrical parameters generated or intended for propulsion design. Then, these values are applied to [PERSON]'s and [PERSON]'s laws. The three principle laws are  
* _[PERSON]'s current law_ [PERSON], a German scientist, proposed a fundamental law, known as [PERSON]'s current law (KCL). This law states that the sum of the currents at an electric node needs to be zero, which is defined formally in [6,7]: \[\sum_{n=1}^{N}\,i_{n}=0\]
* _[PERSON]'s voltage law_ Voltage is the motion of electric charge between two points and can be defined as the total work per unit charge. Furthermore, the voltage was defined as a unit of energy per charge by the Italian physicist [PERSON] as part of his contribution with regard to the electric battery: \[\mathit{volt}=\ \frac{\mathit{joule}\ (\mathit{energy})}{\mathit{ coulomb}\ (\mathit{electric\ charge})}\]  
After several experimental works on voltage, [PERSON] proposed his second law called [PERSON]'s voltage law (KVL) based on the principle that no loss in energy can occur in an electric circuit. This implies that the sum of all voltage sources must be equal to the sum of load voltages, which results in a total of zero for any closed circuit, as in [6,7]:  
\[\sum_{n=1}^{N}\,v_{n}=0\]
* _[PERSON]'s law_ Material properties are the main factors that affect the resistance when electric current flows through the conductors or circuit elements. This resistance during the current flow in the circuit elements can be considered as energy loss, which may be converted to the form of heat. According to [PERSON]'s law, which deals with an ideal resistor, \[V=I\,R\]  
As R is the value of resistance with the unit Ohm (\(\Omega\)), it can be defined as the voltage across an element divided by the current flow through it:  
\[\Omega=\frac{V}{A}\]  
To optimise the power flow in this stage, three methods were included within the module, as described below:* _Levenberg-Marquardt_ In the early 1960s, the Levenberg-Marquardt algorithm (LMA) was introduced to solve the nonlinear least-squares problem. It minimises the sum of squares errors using a sequence of updates to the parameter values between the measured data points and the fit parameterised function. Moreover, the nonlinear problem implies that the parameters of the fit function are not linear. The algorithm by itself consists of two methods, combination and calculation, depending on the optimal value: the gradient descent method, which updates the parameters in the steepest-descent direction, and the Gauss-Newton method, which assumes that the least-squares function is locally quadratic by determining its minimum. Thus, when the optimal values for the parameters are distant, then the LMA adopts the gradient descent method, whereas, when the optimal values for the parameters are close to each other, then the LMA adopts the Gauss-Newton method [8; 9]. * _[PERSON]_ This method was proposed in 1685 by [PERSON] as a root-finding approximation using a straight tangent [10; 11]. Figure 1 illustrate Newton-Raphson method. * _Holomorphic embedding_ In 2012, [PERSON] proposed the holomorphic embedding load flow method (HELM) as a novel non-iterative and non-initial value algorithm. It uses the techniques of complex and deterministic analysis. In brief, the HELM mechanism comprises the following steps [13; 14]:  
1. Find a suitable complex holomorphic embedding with the help of a complex parameter. 2. Calculate the power series using a sequence of linear systems that yield the coefficients progressively. 3. Determine the solution of the complex parameter as the analytical continuation of the power series via Pad'e Approximants.