# 1. Introduction

Numerical methods based on iterative techniques have been used for a long time to analyse power flows in power systems. The most well-known ones include the methods of Gauss-Seidel, Newton-Raphson, Fast Decoupled Load Flow and other, which are most often variants of the previously mentioned methods. All of these methods are successfully used in power flow calculations by professional and publicly available programs alike.

However, the aforementioned iterative methods have some well-known imperfections with a greater or lesser impact on the calculation process and its results. The iterative methods’ most important imperfections include:

- no guarantee that the iterative process will always converge
- multiple solutions – since power system equations have many solutions, it cannot always be controlled which of them will actually converge.

At the same time, it is known that only one solution to a power flow equation corresponds to the concerned power system’s actual operating state. In this case, if the calculations starting point is not close enough to the solution sought, the iterative methods may not only not converge, but also may converge to a false solution.

The HELM method was developed to overcome these limitations of the classical methods used so far. HELM is a completely new and innovative method of solving equations describing steady states of power systems. The method employs complex analysis techniques. Its most important feature, however, is that:

- the solution, if it exists, corresponds to the actual state of the power system’s operation (regardless of the choice of the starting point)
- it unambiguously warns that if there is no solution, the system will experience a voltage avalanche (
*blackout*).

The method’s very important characteristic is that it is recurrent and not iterative like the classical algorithms.

The HELM method was developed by Antonio Trias. It was patented in the USA (2004–2011) as an integral part of a new system for power transmission and distribution grids monitoring and management [^{1}–^{4}].

The HELM method itself was published for the first time in 2012 [^{5}]. The publication described the mathematical foundations of the *holomorphic embedding method *used for a system with *PQ* type nodes. On an example of a 2-machine system, the principle and its possible applications is presented.

Subramanian et al. [^{6}] for the first time demonstrated how to model *PV* type nodes in the HELM method. They also presented a way to solve the solution accuracy problem in the case of holomorphic embedding of *PV* type nodes.

Baghsorkhi and Suetin [^{7}] presented the HELM method’s possible applications to calculate power flows in power grids with the *PV* nodes for which voltage constraints were determined. This issue is directly related to the possibilities of modelling voltage regulators in the discussed method.

Trias in [8] described in detail the theoretical foundations of the HELM method: how to build a holomorphic embedding to correctly solve equations describing power flows in the power system, how to use standard analytical techniques for practical calculations, how to extend the method in order to adapt it to variable control elements, such as *PV* type nodes.

Suetin and Baghsorkhi [^{9}] and Rao et al [^{10}] presented in an orderly manner: mathematical models of power system elements so far developed and used in the HELM method, the most frequently used methods of solving the equations forming the flow model, impact of selected holomorphic embeddings on building models of power system components, calculations of germ solutions, and operation of the recursive algorithm used in the HELM method.

Trias and Marín [^{11}] presented the HELM method’s possible applications for solving power flows in DC systems.

Wallace et al. in [^{12}] presented an alternative method for including *PV* type nodes in the HELM method.

Basiri-Kejani and Gholipour [^{13}] presented the possibilities of modelling of control devices in the discussed method. Their main considerations focused on FACTS type controls.

Liu et al. [^{14}] presented a concept of the multidimensional HELM method. Their main aim was to obtain an unambiguous approximation of the analytical power flow solution by finding a physical germ solution and the use of an arbitrary holomorphic embedding for each power generated and each load or groups of loads.

Santos et al [^{15}] and Sauter et al [^{16}] analytically compared the HELM method variants developed so far with classical flow algorithms.

Trias and Marín [^{17}] analysed the application of the Padé-Weierstrass technique to solving the power flow problem and its implications for improving the accuracy of the HELM method’s results.

Chiang et al. [^{18}] proposed a new version of the HELM method. They found their solution faster and more flexible in operation.

Feng and Tylavsky [^{19}] focused on the HELM method’s application to find flow equations solutions most interesting from the point of view of power system voltage stability.

Liu et al. [^{20}] proposed an Internet-based system for assessing voltage stability in a steady state, used to assess the probability of a voltage avalanche in the power system.

So far, the few publications described above, which refer to the presented method, have demonstrated that it is more efficient than and competitive to classical iterative methods. They showed the method’s great potential and possible uses in real-time applications for many operations related to the power system performance, such as: failure analysis, optimal power flows, building scenarios for system recovery after failures. This is particularly important from the point of view of the increasingly widespread introduction of smart real time applications, the main purpose of which is to support grid operators where there is no time to manually tune devices, until convergence of calculations is achieved.

This paper presents the main assumptions of the HELM method and how to build power system models using the complex analysis technique. The calculations are compared with classical iterative methods.

# 2. Mathematical model of the HELM method

For any node *i* of power grid consisting of *N* nodes, the equation binding the basic electrical quantities can be written in the form of:

(1)

where: the quantities in it indicate complex values, respectively: *S _{i}* – power in node

*i*,

*V*– voltage in node

_{i}*i*,

*I*– current in node

_{i}*i*,

*Y*– elements of the admittance matrix mapping connections in the grid. Indices (*) indicate conjugate values.

_{ik}Equation (1) had become the basis for the formulation of a new power flow calculation method based on the holomorphic embedding method. In the method’s first versions [^{1}–^{5}], power system nodes were modelled as *PQ* type nodes. For such a case equation (1), after appropriate transformations, can be written as:

(2)

Equation (2) represents the basic record of flow equations describing the state of *PQ* type nodes. Although in practice only a few nodes in an extensive power system are described thus, the above case can be treated as a starting point to describe the HELM method’s operating principle and to develop its model.

The holomorphic function is a function defined on an open subset of the plane of complex numbers ℂ with values contained in this set, which is differentiable in complex terms at every point of this subset. The function’s holomorphicity is a condition much stronger than differentiability in real terms, because a function with this property is infinitely differentiable, which makes it representable by Taylor’s formula (series).

Equation (2) is an algebraic equation the solution of which is the searched-for vector of nodal voltage with components *V _{i}*. However, an equation in this form is not holomorphic, because the Cauchy-Riemann conditions are not met due to the occurrence of complex quantities.

To change this situation, the HELM method proposes embedding the original algebraic equations in their functional holomorphic extension. With this treatment many properties of the complex analysis, unavailable or limited in solving algebraic equations, can be used.

Embedding is a multi-valued mapping of object A into object B, which preserves the properties of the embedded object (the properties concerned depend on the considered theory). Embedding implies the existence in object B of a subset “identical” to object A.

The embedding proposed in the HELM method consists in entering complex variable *z* into equation (2) in such a way that voltages *V _{i}*,

*V*become functions of this new variable. Embedding can be done in any way. For the

_{k}*PQ*type nodes described by equation (2), the holomorphic embedding may take the following form:

(3)

where: is a holomorphic, conjugate function of the conjugate variable *z*. This expression is not equivalent to a function !

The functional dependence in equation (3) of nodal voltage on complex variable *z* is a holomorphic function. In addition, the voltages in the system nodes meet the following dependencies, resulting from the holomorphic embedded applied:

** ** (4)

** ** (5)

** ** (6)

It should be noted that the holomorphic embedding used in equation (3) implies the following limit cases:

- solution for
*z*= 0 represents the grid operation without load and generation in the power system nodes – this is called a*germ solution*. - solution for
*z*= 1 represents the determination of the grid operating point for the full flow model.

The method’s additional characteristic is that it is recurrent and not iterative like the classical algorithms.

# 3. Problem solving methods

The holomorphic embedding function *V(z)* a holomorphic function of complex variable *z*. In practice, this means that the problem of power flows in the power system is solved by the HELM method in a function space in which functions and variables are complex numbers. One of the methods used to solve this type of problem is the power series method.

**Power series method **

Using one of the fundamental features of holomorphic functions, function *V*(*z*) can be represented as Maclaurin series, which is a particular form of Taylor series. In general, it is a power series with coefficients that are complex functions, dependent on complex variables of this series. Such a series is formulated as follows:

** ** (7)

In addition, for holomorphic function *V*(*z*) the following dependencies are met, which are obligatory when it is developed into the Maclaurin series:

** ** (8)

** ** (9)

Since in equation (3) function *V**(*z**) is in the denominator, it is convenient to enter function *W*(*z*) determined as follows:

** ** (10)

From equation (10) it follows directly that:

** ** (11)

By multiplying the power series in equation (10), the formulas can be determined that allow calculating the values of functional coefficients *W*[*n*]:

** ** (12)

** ** (13)

Based on the above relationships, the functional values of coefficients *V*[*n*] can be calculated after formula:

** ** (14)

Thus equation (3) of the state of *PQ*, type nodes, will take the form:

** ** (15)

Another approach applied to calculations by the HELM method is *continued fraction* approximation.

**Continued fraction approximation**

Theoretically, there are many ways to transform the original power series described by equation (7) into a continued fraction which approximates this series. One possible form of this transformation is as follows:

** ** (16)

The last expression in equation (16) can be written in the form of a recursive expression that allows calculating the value of the sought function *V*(*z*):

** ** (17)

By reference to the previous comments it should be noted that the value of function *V*(*z*), defining the operating voltage in all *N* grid nodes represented in the flow model, is obtained directly from equation (17), assuming value *z *= 1.

There are many other methods that can be used to solve the issues described in the HELM method. Only the methods most popular and most widely applied in the studies published so far are described above.

# 4. Consideration of actual states of the power system operation

Equation (2) and corresponding holomorphic embedding (3) describe the operating state of *PQ* type nodes. From a practical point of view, this approach is not sufficient to describe the full power flows in an extensive power system. The complete computational model must also represent other types of nodes and devices. Work on these issues is currently at the initial stage, but the first studies have already been developed that take into account the complexity of the power system operation. The forms of holomorphic embedding can be infinite. The most important of them, which have been implemented in practice, are presented below.

**Extended model of** *PQ***type nodes**

In this model the two following components were separated from elements of admittance matrix *Y _{ik}*: – component corresponding to “serial branches” and – component corresponding to “shunt elements

*”.*Such representation of

*PQ*type nodes allows for mapping of

*shunt elements*(reactors, capacitors, etc.) and facilitates modelling of transformers. Flow equations for the described case take the form:

** ** (18)

Holomorphic embedding for equation (18) will take the form:

** ** (18a)

Whereas the mathematical model used in the HELM method, corresponding to equation (18a), will take the form:

** ** (18b)

*PV***type node model**

For *PV* type nodes the voltage module |*V _{i}*| and the active power output

*P*are known. The unknown quantities are the voltage angle and reactive node power

_{i}*Q*. The respective holomorphic embedding equations that represent the reactive power calculation method can be formulated as follows:

_{i}** ** (19)

where: is the conjugate value of the constant (unchangeable) power *S _{i}* in node

*i*.

The mathematical model used in the HELM method, corresponding to equation (19), will take the form:

** ** (19a)

At the same time, the holomorphic embedding that represents the condition of voltage module |*V _{i}*|, can be formulated as:

** ** (20)

where: is the specified voltage magnitude in node *i*.

The mathematical model used in the HELM method, corresponding to equation (20), will take the form:

** ** (21)

The above equations are only a small representation of the mathematical models that make up the description of complex processes occurring in a real power system. Work on this has only just begun. It is to be hoped that with each new development, the library of available models allowing their representation and application in the HELM method will be enriched.

# 5. Calculation examples

In order to test the HELM method’s effectiveness, comparative analyses were carried out with a professional program for power flow calculation, PSS®E by Siemens PTI. Standard 3-, 14- and 118-node IEEE power system models were used for the calculations. The models were adapted to the specifics of the holomorphic embeddings developed and described before. For this reason, for example, the ability to control transformer ratios has been blocked in the IEEE models. All constraints and requirements for *PV* type nodes have been retained, such as fixed voltage levels and reactive power limits of generators. The calculations in Siemens PTI’s PSS®E program were performed using the full Newton-Raphson method.

The calculations were made on a computer with an Intel® Core- ™ i7-6700 HQ 2.6 GHz processor with a 64-bit MS Windows 10 Pro operating system. The HELM method algorithm was written in Python 3.6.

Results of the comparative calculations are presented in Tab. 1.

Tab. 1. Comparative analysis of flow calculations by PSS®E program and HELM method

# 6. Conclusions

The HELM method is a completely new and innovative method of solving equations describing steady states of power systems. The first theoretical works indicate the method’s great potential and applicability. This is also confirmed by the calculations carried out by this author. The results presented in Tab. 1 show that:

- calculations by the HELM method are highly accurate, regardless of the analysed grid size
- for a grid with a small number of nodes, the computation time is comparable or shorter than in classical methods
- with increase in the problem dimensions, the duration of computation by the HELM method increases significantly.

However, it should be remembered that the computation time in the HELM method is not of prime concern. Much more important are the method’s features due to the holomorphic embedding and the “transfer” of the power flow problem to the plane of complex numbers ℂ, while embedding the original algebraic equations into their functional holomorphic extension. The solution’s unambiguity (or the lack of it) obtained with such a transformation allows one to optimistically think, for example, about the HELM method’s applicability in real-time systems used to control the operation of a complex power system.

# 7. Future research directions

Theoretical and development work on the HELM method is at an early stage. So far, only some of the issues that are necessary to develop a fully functional power flow calculation method in real power systems have been worked out in a satisfactory way. Theoretical works should include the following critical elements:

- modelling of control elements, such as regulating transformers, phase shifters, FACTS devices, etc.
- modelling that reflect various workloads (current model, admittance model etc.).

Equally important as modelling power system components should be considered the need to search for new and more efficient methods of calculating functional variables, which are the solution to the power flow problem in the HELM method. Accuracy and speed of the function solution approximation play an important role in the calculation process and determine the entire method’s effectiveness and efficiency.

Finally, it should be noted that the HELM method has been commercialized and is now owned by Gridquant Inc. According to commercial information, this company offers a fully functional version of the program that allows calculating very large power grids. However, the way this program works, and the details of modelling individual grid components are business secrets of the company. An additional incentive to intensify research works is the very high price that Gridquant Inc. demands for the program, and this also applies to its academic version.