1. Introduction
With the prospect of the increasing use of Direct Current (DC) transmission links in power systems, it has become increasingly important to have techniques to model the devices responsible for the AC-DC interconnection in power system analysis software, particularly in the power flow (PF), to allow correct network modeling as a whole to improve the quality of simulation results that could be used in studies of operation and transmission expansion planning of electric power systems.
DC transmission has become an alternative that is technically and economically competitive in the transport of large amounts of active power over long distances, in underwater crossings with the use of cables, and in asynchronous connections within a large variety of lengths, including zero, between two areas [
1,
2,
3,
4,
5].
The DC transmission links generally are characterized by the interconnection of two systems of Alternating Current (AC) by two converter stations: a rectifier and an inverter. The connection between these stations is established by one or more DC transmission lines: single or double polarity. The rectification and inversion are performed in the converter bridges composed by semiconductor elements.
High Voltage Direct Current (HVDC) transmission technology was used for the first time in Brazil in the 1980s to integrate the energy generated by the Itaipu Hydroelectric Power Plant, connecting the Foz do Iguaçu and Ibiúna converter stations, utilizing two dipoles at ±600 kV, over about 800 km. In 2009, Brazil resumed the use of HVDC transmission technology with the construction of the ±600 kV transmission system associated with the Rio Madeira hydroelectric complex (Santo Antônio and Jirau hydroelectric power plants), composed of two bipoles designed to transmit 6300 MW over approximately 2400 km, connecting Coletora Porto Velho and Araraquara stations. By the year 2020, an important initiative to consider is the establishment of the Belo Monte HVDC links (2539 km-long transmission), with the construction of the ±800 kV transmission system associated with the Belo Monte hydroelectric power plant.
From this context, it appears that the study of severe operating conditions is important to support studies on the reliability of hydroelectric plants. These studies can assist in assessing the Availability Factor (AFA) [
6]. Thus, modeling the electrical system as close as possible to reality is of fundamental importance. This is the case of the Santo Antônio hydroelectric plant, which may be impacted by the behavior of the HVDC transmission system.
The power flow software is the most frequently used tool in both operation and expansion planning studies of electrical power systems. It is used to determine the equipment rating, electrical equipment loading and system losses, bus voltage magnitudes and angles, and reactive power support requirements to maintain voltages within limits for a given scenario and a contingency list. For each case study, the network configuration and parameters, the bus active and reactive load, and MW generation must be specified. The increasing complexity of power systems, introduced by new large-scale AC and HVDC interconnections and by the application of Flexible AC Transmission System (FACTS) devices in such systems, have imposed new challenges to power system engineers and software developers. The reference [
7] describes a generic methodology for representing control devices and the associated challenges.
The commonly used techniques in studies of power flow alternate the convergence process between AC and DC system models. This method is carried out until the global convergence of the two systems is obtained. It is important to point out that the inclusion of DC links can cause, in general, a significantly slower convergence of power flow. One of the main reasons for this slow convergence is the different rates of convergence of the DC and AC systems, which interfere with each other, slowing system global convergence.
In reference [
8], a simultaneous (full Newton) methodology for the representation of DC transmission links based on the DC transmission link of the hydro electrical unit Itaipu (Brazil) was proposed by including equations that model the converters, the DC network (considering only two DC bus), and control modes to the set of equations that model the power system. The proposed model aims to (i) improve the convergence characteristics of systems with coupling between the AC-DC set of equations in the Newton-Raphson method and (ii) allow the adjustment of the control modes automatically during the iterative process.
In [
9], the authors present a versatile approach to AC/DC system representation, which is able to model several types of converter structures, including the conventional two-terminal AC-DC power flow, and to connect to more DC terminals. The sequential (alternate) method is used to solve the AC-DC power flow, in which the AC network solution is carried out through the conventional Newton-Raphson method, and in the DC network model, the system presents five variables per terminal. It is also proposed that the variation of taps be limited, which is an improvement compared to other methodologies. Another advantage is the possibility of solving DC systems connected in series by adjusting some equations of the DC network.
Another study of DC transmission links [
10] uses the eliminated variable method, which is a simple and reliable tool for the study of DC transmission links in multi-terminal systems, in which the DC and AC variables are treated separately. In this work, the active and reactive powers demanded by inverters are treated as voltage-dependent loads. Thus, the DC steady-state equations are solved, and the DC variables are eliminated from the PF equations. Voltage-dependent PQ buses were adopted in [
11] to represent the HVDC system in the power flow problem. The work [
12] proposed a modified power flow formulation considering the HVDC system as a voltage-dependent load on the AC side at both DC system sides in two-terminal systems.
In [
13], a nodal voltage-based universal steady-state PF formulation is proposed. The main goal is to consider the bipolar VSC-MTDC (Voltage Source Converter Multi-Terminal HVDC). A power flow alternating iterative method is proposed to obtain the positive/negative-pole DC network power flow. A series of nodal equivalent methods involving various control strategies are also proposed for the power flow algorithm. A comparative study of full Newton and sequential AC-DC power flow formulations for a VSC-MTDC is presented in the paper [
14].
In [
15], the authors use a sequential methodology to present a study where the solution of the AC-DC power flow is intrinsically related to the characteristics of a linear coefficient matrix
G, which integrates the information of the network and HVDC control modes. The main contribution of this paper is to demonstrate that a necessary condition to solve the HVDC system is that the coefficient matrix
G must be nonsingular. In the study, the conditions for the characteristic of a
G matrix under feasible combinations of control modes and the parameters of the HVDC systems are introduced. The reference [
16] proposed an alternative approach also based on a sequential method for the AC-DC power flow solution to handle multi-infeed DC systems.
The authors of [
17] present some methods based on the correlation analysis between the steady-state security region and operational constraints. The reference [
18] proposes an operation strategy of the hybrid multi-terminal high voltage DC to increase the utilization of the AC transmission corridors in parallel with HVDC systems considering the available transmission capacity (ATC) between two electrical areas. An analysis of distribution networks is carried out in the reference [
19]. This work proposed an integrated load flow approach for AC-DC distribution systems.
In this context, the main objective of this paper is to propose a new methodology for a full Newton AC-DC power flow formulation for an MTDC system with a generic methodology for representing the DC network in stationary studies. The main motivation is the increasing interest in the operational feasibility and potential application of multi-terminal HVDC systems. It is sought to demonstrate that the proposed methodology provides an efficient and generic way to represent any DC network in HVDC multi-terminal systems through a system of equations. Finally, some results are presented based on HVDC test systems to validate the effectiveness and robustness of the proposed methodology.
From the above, the main contributions of this paper are
to present an alternative power flow formulation based on a full Newton methodology for MTDC transmission systems, where all DC variables are considered in the formulation, allowing for a generic representation of HVDC control modes;
to describe the AC/DC base transformation in the Jacobian matrix calculation;
to propose a formulation that represents the generic DC network.
3. Proposed Methodology
The proposed method of solution consists of including the equations that model the HVDC systems in the conventional power flow formulation. In order to achieve this goal, six new state variables will be included for each converter: , , , , , and . The assumptions considered for Newton’s method to solve simultaneously the AC-DC PF, based on the system of equations , are presented below.
Considering the equations described in
Section 2, the system of equations related to the HVDC system (
) is structured as shown in
Table 1.
Based on
Table 1, the vector of state variables related to the HVDC system (
) will be updated according to the order presented in
Table 2.
The general linear system to be solved at each iteration in the solution process is shown in Equation (
16).
where:
By observing Equation (
16), the original Jacobian matrix of the problem is preserved. The new derivatives are located in additional rows and columns. The AC-AC block is the original Jacobian matrix and contains the derivatives of the equations of MW and MVAr power from the AC system in relation to the variables of the original system state. In the AC interface buses, injections of active and reactive power regarding the DC system should be considered. These injections are given by [
1]
From there, the residuals of the interface buses are given by
where:
Tutorial Example
This tutorial example shows with more details the basic principles of the proposed AC-DC power flow with a HVDC multiterminal system with three converter terminals (connected to infinity buses), one intermediate DC bus (called
X in this example), and three HVDC lines, shown in
Figure 2, which is composed of a DC network (represented by a generic methodology), with control modes that are shown below.
For the case in study, the converters are working under the control modes shown in
Table 3.
The system of equations that models, in a generic form, the DC network in this case is
The residuals equations related to the DC system (
) are
The DC variables of the problem are included in the following order: , , , , , , , , , , , , , , , , , , and .
The solution of the general linear system of Equation (
16) provides the values for the variables in the steady state of the AC/DC system.