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The Trias holomorphic method (THM) is a novel type of load flow algorithm based on a holomorphic embedding technique. The flow problem, also known as power flow, consists in computing the steady state of three-phase balanced AC power networks with complex injections expressed totally or partially in terms of power. The load flow calculation is the most fundamental problem in the analysis of power systems and is the corner stone for almost all other tools used in power system simulation.

In contrast to most other load flow methods, which are based on numerical iterative schemes such as Gauss-Seidel, Newton-Raphson, or variants (such as homotopic continuation methods ), the holomorphic embedding load flow method is non-iterative, deterministic, and non-equivocal. Using techniques associated with Algebraic Curves, the method always ensures the computation of the correct solution to the (multi-valued) load flow problem. The method thus provides a solution to a long-standing problem of all iterative load flow methods, namely the unreliability of the iterations in finding the correct solution (or any solution at all), especially so under stressed grid conditions. While this problem is not so important in off-line studies, it becomes critical in real-time, and makes any sort of load flow exploratory analysis unreliable. Therefore the holomorphic embedding loadflow is particularly suited for real-time applications, and mandatory for any EMS software based on exploratory algorithms, such as contingency analysis, limits violations, and topology restoration solvers.

The method is grounded on a rigorous mathematical theory offering a rich set of results and new insights into the load flow problem, and in practical terms it could be summarized as follows:
 * 1) define an (holomorphic) embedding for the equations in terms of a complex parameter $s$, such that for $s =0$ the system has an obvious correct solution, and for $s =1$ one recovers the original problem.
 * 2) then it is possible to compute univocally power series for voltages as analytic functions of $s$. The correct load flow solution at $s =1$ will be obtained by analytic continuation of the known correct solution at $s =0$.
 * 3) perform the analytic continuation using algebraic approximants, which in this case are guaranteed to either converge to the solution if it exists, or not converge if the solution does not exist (voltage collapse).

The THM load flow was invented by Dr. Antonio Trias from Grupo AIA, and it is covered by two patents in the US. Its computational cost is similar and competitive in general with other state of the art iterative algorithms, such as the Fast Decoupled Load Flow. It has been implemented in industrial-strength EMS applications currently operating at several first-class large utilities in Mexico, Europe, and the US.

= Background = Traditional load flow algorithms were developed between the years 1956 up to the early eighties, and they include three main foundational approaches: the Gauss-Seidel method, which has poor convergence properties but very little memory requirements and it is straightforward to implement; the full Newton-Raphson method , which has fast (quadratic) convergence properties but it is computationally costly; and the Fast Decoupled Load Flow (FDLF) method , which is based on Newton-Raphson but greatly reduces its computational cost by means of a decoupling approximation that is valid in most transmission networks. After these seminal methods were developed many other incremental improvements have been published, mostly for special networks or network scenarios, but nevertheless the underlying technique in all of them remains an iterative solver, either of Gauss-Seidel or of Newton type. See Load Flow Methods for a more extensive account of the history of load flow methods.

There are two fundamental problems with all iterative schemes of this type: on the one hand, there is no guarantee that the iteration will always converge to a solution; on the other, since the system has multiple solutions, it is not possible to control which solution will be selected. As the network approaches the point of voltage collapse, spurious solutions get closer to the correct one, and the iterative scheme may be easily attracted to one of them because of the well-known phenomenon of Newton fractals: when the Newton method is applied to complex functions, the basins of attraction for the various solutions intermingle in a fractal way. In other words, no matter how close the initial seed is to the desired solution, there is always some non-zero chance of straying off to a different solution. These fundamental problems have been illustrated for the two-bus model. Although there exist homotopic continuation techniques which alleviate the problem, the fractal nature of the basins of attraction precludes a 100% reliable method.

The relevance of these problems in practice depends very much on the context of application. There are typically three scenarios:
 * 1) off-line load flows: the impact of the problem is minimal, since the analyst has time to investigate the solution and tweak the seed if necessary, until convergence to the correct solution is found.
 * 2) real-time load flows: the impact of the problem is much severe, as the operators do not have the time to explore different seeds when a problem is encountered. Most of the time convergence is ensured because the real-time solution is smoothly tracked over time, but sudden changes in the grid or stressed states may easily cause convergence problems.
 * 3) real-time exploratory load flows, such as “what-if” simulation, contingency studies, or automated explorations: the impact in this case is the highest, since many of the aternative states to the current real-time state are wildly different and may easily approach collapse.

The key differential advantage of the Trias Holomorphic Method load flow is that it is fully deterministic and unambiguous: it guarantees that the solution always corresponds to the correct operative solution, when it exists; and it signals the non-existence of the solution when the conditions are such that there is not any solution (voltage collapse). Additionally, the method is competitive with the FDNR method in terms of computational cost. From the theoretical point of view, it brings a solid mathematical treatment of the load flow problem that provides new insights not previously available with the iterative numerical methods.

= Holomorphic Embedding = Consider the following general form for the load flow equations:

where the given (complex) parameters are the admittance matrix$N$, the bus shunt admittances $2^{ N }$, and the bus power injections $Y_{ik}$. For the purposes of the discussion, we will omit the treatment of controls, but the method can accommodate all types of controls. The method uses an embedding technique by means of a complex parameter $Y_{i} ^{sh}$ such that for $S_{i}$  the system has an obvious correct solution, and for $s$ one recovers the original problem.

But the first key ingredient in the method lies in requiring the embedding to be holomorphic, that is, that the system of equations for voltages $s =0$ is turned into a system of equations for functions $s =1$ in such a way that the new system defines $V$ as holomorphic functions (i.e. complex analytic) of the new complex variable s. The aim is to be able to use the nice and extensive properties of holomorphic functions in the complex plane, above all the process of analytical continuation which will allow the calculation of $V(s)$ at $V(s)$. Looking at equations ($$), a necessary condition for the embedding to be holomorphic is that $V(s)$ is replaced under the embedding with $s =1$, not $V^{*}$. This is because complex conjugation itself is not a holomorphic function. On the other hand, it is easy to see that the replacement $V^{*}(s^{*})$ does allow the equations to define a holomorphic function $V^{*}(s)$. However, for a given arbitrary embedding, it remains to be proven that $V^{*}(s^{*})$ is indeed holomorphic.

Taking into account all these considerations, an embedding of this type is proposed:

This particular choice is not only influenced by the need to have a trivially solvable system at $V(s)$, but also by physical intuition and the mechanics of the method as it will be clearly apparent below.

Now using classical techniques for variable elimination in polynomial systems (results from the theory of Resultants and Gröbner basis it can be proven that equations ($$) do in fact define as holomorphic functions.  More significantly, they define  as algebraic curves.  This last fact opens up a plethora of powerful results, in particular some very nice properties on the analytical continuations of the function, as explored in the next section. For now, let us notice that knowledge of the solution at $V(s)$  determines uniquely the solution everywhere (except on a finite number of branch cuts), thus getting rid of the multi-valuedness of the load flow problem.

The technique to obtain the coefficients for the power series expansion (on $s =0$ ) of voltages $s =0$ is quite straightforward once one realizes that Eqs. [eqref:lf_embedded] can be used to obtain them, order by order, by using the power series expansion for $s =0$, since their coefficients are related by the convolution formulas derived from the following identity:

The particular choice of the embedding then allows to successively obtain the coefficients of the voltages order by order, by solving linear systems (in which the matrix remains constant!) whose right-hand-sides are determined by the calculation of the coefficients for at the previous order.

A more detailed discussion about this procedure is included in the Holomorphic Embedding Wikipedia page, and a complete treatment suitable for software implementations is offered in Ref. .

= Analytic Continuation = Once the power series at $V$ are calculated to the desired order, the problem of calculating them at $1/V$  becomes one of analytic continuation. It should be strongly remarked that this does not have anything in common with the techniques of homotopic continuation. Homotopy is a powerful concept since it only makes use of the concepts of continuity and thus it is applicable to general smooth nonlinear systems, but on the other hand it does not always provide a reliable method to approximate the functions (as it relies on iterative schemes such as Newton-Raphson).

It can be proven that algebraic curves are complete global analytic functions, that is, knowledge of the power series expansion at one point (the so-called germ of the function) uniquely determines the function everywhere on the complex plane, except on a finite number of branch cuts. Stahl’s extremal domain theorem further asserts that there exists a maximal domain for the analytic continuation of the function, which corresponds to the choice of branch cuts with minimal logarithmic capacity measure. In the case of algebraic curves the number of cuts is finite, therefore it would be feasible to find maximal continuations by finding the combination of cuts with minimal capacity. However, things get even better: Stahl’s theorem on the convergence of Padé Approximants states that the diagonal and supra-diagonal Padé (or equivalently, the continued fraction approximants to the power series) converge to the maximal analytic continuation. The zeros and poles of the approximants remarkably accumulate on the set of branch cuts having minimal capacity.

These propereties confer the load flow method with the ability to unequivocally detect the condition of voltage collapse: the algebraic approximations are guaranteed to either converge to the solution if it exists, or not converge if the solution does not exist.

= Performance and implementation details = The Trias holomorphic load flow method has been experimentally shown to be competitive in terms of computational performance with the FDLF method. The matrix for the linear systems solved at each order of the calculation of the power series are fixed (only the right hand side varies), therefore only one factorization is needed, without introducing approximations. Modern sparse techniques make this very fast. On the other hand, the algorithm requires the calculation of the algebraic approximants. However, this is a naturally parallelizable process, as the approximant at each bus can be calculated completely independent of any other once the coefficients of the power series are known.

= Example: the two-bus model = The key ideas of the method are best exemplified on the simplest power system, the two-bus model. This model has the advantage that it is possible to solve it exactly in closed form, so that it is possible to compare the results of the method in full detail. The approximants themselves can also be calculated in closed form, and thus all the properties by Stahl’s theorem (e.g. how the zeros and poles of the approximants accumulate on the branch cuts with minimal capacity) can be explicitly exposed and verified. See Two-bus holomorphic embedding load flow.

= References =

= External links =
 * Power flow study
 * Power system simulation