Impact of selection of DC base values and DC link control strategies on sequential AC-DC power-flow convergence

Abstract

This paper demonstrates the convergence of the integrated AC-DC power-flow algorithm as affected by the selection of different base values for the DC quantities and adoption of different control strategies for the DC link. For power-flow modeling of integrated AC-DC systems, the base values of the various DC quantities can be defined in several ways, giving rise to different sets of per-unit system equations. It is observed that different per-unit system models affect the convergence of the power-flow algorithm differently. In a similar manner, the control strategy adopted for the DC link also affects the power-flow convergence. The sequential method is used to solve the DC variables in the Newton Raphson (NR) power flow model, where AC and DC systems are solved separately and are coupled by injecting an equivalent amount of real and reactive power at the terminal AC buses. Now, for a majority of the possible control strategies, the equivalent real and reactive power injections at the concerned buses can be computed a-priori and are independent of the NR iterative loop. However, for a few of the control strategies, the equivalent reactive power injections cannot be computed a-priori and need to be computed in every NR iteration. This affects the performance of the iterative process. Two different per-unit system models and six typical control strategies are taken into consideration. This is validated by numerous case studies conducted on the IEEE 118-bus and 300-bus test systems.

Graphical abstract

Keywords

AC-DC power-flow / Newton-Raphson method / high voltage direct current (HVDC) control strategy

1 Introduction

Ever since the first high voltage direct current (HVDC) link was installed between the Swedish mainland and Gotland in 1954, there has been an increasing interest in HVDC transmission applications worldwide. With the ever-increasing load demands, system stability issues and different operating frequencies may render AC transmission infeasible. HVDC transmission allows power transmission between unsynchronized AC transmission systems, and can increase the system stability by preventing cascading failures caused by phase instability from propagating from one part of a wider power transmission grid to another. For lengths exceeding approximately 500 km, HVDC transmission is proving to be more economical than AC [1−3].

For planning, operation and control of power systems with HVDC links, the power-flow solution of power systems incorporated with HVDC links is required. This necessitates suitable power-flow models of integrated AC-DC systems [4−6]. For power-flow modeling of such systems, the base values of the various DC quantities can be defined in several ways, giving rise to different per-unit system models, each model comprising separate sets of system equations in per-unit. It is observed that different per-unit system models sets affect the convergence of the AC-DC power-flow algorithm in different ways.

Now, to solve the power flow equations in hybrid AC-DC systems, two different algorithms are generally reported. These are known as the unified and the sequential method, respectively. Some excellent research works on the unified method are presented [7−11]. Unlike the unified method, the sequential method is easier to implement and poses lesser computational burden due to the smaller size of the Jacobian matrix. Consequently, in this paper, only the sequential AC-DC power-flow algorithm is considered.

In the sequential AC-DC power-flow algorithm, the AC and DC systems are solved separately in each iteration and are coupled by injecting an equivalent amount of real and reactive power at the terminal AC buses. For a majority of the possible HVDC control strategies, the equivalent real and reactive power injections at the concerned buses can be computed a-priori and are independent of the Newton Raphson (NR) iterative loop. However, for a few of the control strategies, the equivalent reactive power injections cannot be computed a-priori and need to be computed in every NR iteration. This too affects the convergence of the algorithm. Some excellent research works on the AC-DC sequential power flow method are presented [12−15].

Now, it is observed that the convergence of the AC-DC sequential power-flow algorithm is affected by both the HVDC per-unit system model chosen as well as the particular control strategy adopted for the HVDC link. However, to the best of our knowledge, none of the research work in the published literature has addressed this important issue. Therefore, a detailed investigation of this aspect is conducted and numerous case studies are carried out on the IEEE 118-bus and 300-bus test systems to validate this.

2 System modeling

Figure 1 shows a typical AC-DC power system network in which a HVDC link is connected in the branch “i-j” between any two buses “i” and “j”, of the network. The two converters representing the rectifier and the inverter are connected to the AC system at buses “i” and “j” respectively, through their respective converter transformers. The HVDC link is accounted for as appropriate loads at the buses “i” and “j”. Figure 2 depicts the equivalent circuit for the network shown in Fig. 1.

Prior to the selection of variables and formulation of the equations, several basic assumptions are required which are generally accepted in the analysis of steady state DC converter operation [1−3]. These are

1) Three AC voltages at the terminal bus bar are balanced and sinusoidal.

2) The converter operation is perfectly balanced.

3) The direct current and voltage are smooth.

4) The converter transformer is lossless and the magnetizing admittance is ignored.

Subsequently, for hybrid power flow calculations, the DC and AC equations are combined together. This necessitates the translation of the converter equations into the per-unit system in order to use them with AC system per-unit equations. The base values of the various DC quantities can be defined in several ways, giving rise to multiple per-unit HVDC system models, each comprising a different set of equations. Although several choices are feasible, in this paper, due to a shortage of space, only two different ways of defining the DC quantities are presented. These two different conventions are given in Appendix and culminate in two different per-unit HVDC system models, which are presented in Table 1.

From Table 1, it can be observed that six independent equations involving ten unknowns are present. Hence, for a complete solution of the HVDC quantities, four variables need to be specified. These are known as control variables. Several combinations of control variables are possible and each combination comprises a control mode or strategy, which is elaborated in the next section.

3 HVDC control strategies

As already discussed in the last section, several control modes or strategies are possible corresponding to different combinations of the four control variables. Some of these control strategies [1,2] are listed in Table 2. Although several case studies were conducted with these control strategies, due to lack of space, the results corresponding to only six typical ones (Control strategies 1, 2, 3, 4, 5 and 6) are presented in this paper, which are detailed below.

3.1 Control strategy 1

In this control strategy, the firing angles of both the converters are specified. The tap ratios of both the converter transformers ‘a_R’ and ‘a_I’ are calculated subsequent to the AC load flow.

3.2 Control strategy 2

In this control strategy, the firing angles of both the converters are computed while their transformer tap ratios are specified. As the injected reactive power representing the converters gets updated in every iteration, this control strategy is slightly harder to implement than the others.

3.3 Control strategy 3

This control strategy is also known as the constant current and voltage controlled mode. As in mode 1, both the converter transformer tap ratios ‘{\alpha }_{\mathrm{R}}’ and ‘{\alpha }_{\mathrm{I}}’ can be calculated subsequent to the AC load flow.

3.4 Control strategy 4

In this control strategy, the firing angle on the rectifier side is specified along with the tap ratio of the inverter side transformer. The tap ratio on the rectifier side transformer ‘{\alpha }_{\mathrm{R}}’ along with the inverter side firing angle can be calculated subsequent to the AC load flow.

3.5 Control strategy 5

In this control strategy, the extinction angle of the inverter is specified along with the tap ratio of the rectifier side. On the other hand, the firing angle of the rectifier and the inverter side transformer tap ratio are computed subsequent to the AC load flow.

3.6 Control strategy 6

In this control strategy, the firing angle and the DC voltage of the rectifier side are computed given the tap ratios of the converter transformers along with the extinction angle of the inverter. The equivalent reactive power injections on both the rectifier and inverter sides are updated in every iteration.

3.7 Control strategy 7

In this control strategy, both the firing angle of the rectifier and the extinction angle of the inverter are specified along with the tap ratio of the rectifier transformer. Only the rectifier side equivalent reactive power gets updated in every iteration.

3.8 Control strategy 8

In this control strategy, both the firing angle of the rectifier and the extinction angle of the inverter are specified. Both the converter transformer tap ratios ‘{\alpha }_{\mathrm{R}}’ and ‘{\alpha }_{\mathrm{I}}’ are calculated subsequent to the AC load flow.

4 AC-DC power-flow equations

As discussed earlier, the effect of the DC link is included in the power flow equations by injecting an equivalent amount of real and reactive power at the terminal AC buses connected to the converters. This results in appropriate modifications of the mismatch equations at the converter terminal AC buses. The DC power at the rectifier and inverter side is a function of AC and DC values [1−3].

(1)\mathrm{\Delta }{P}_i=P_i^{\mathrm{sp}}-P_i^{\mathrm{cal}}-P_{\mathrm{dR}},

(2)\mathrm{\Delta }{P}_j=P_j^{\mathrm{sp}}-P_j^{\mathrm{cal}}+P_{\mathrm{dI}},

(3)\mathrm{\Delta }{Q}_i=Q_i^{\mathrm{sp}}-Q_i^{\mathrm{cal}}-Q_{\mathrm{dR}},

(4)\mathrm{\Delta }{Q}_j=Q_j^{\mathrm{sp}}-Q_j^{\mathrm{cal}}-Q_{\mathrm{dI}},

where ‘P_dR’ and ‘P_dI’ are the equivalent real power injections at the AC terminal buses of the rectifier and inverter respectively, accounting for the DC link. Similarly, ‘Q_dR’ and ‘Q_dI’ are the corresponding reactive power injections. It is important to note the conventions of the signs of the DC link are equivalent to real and reactive power injections. This is because it is assumed that the rectifier consumes both real and reactive power from the AC grid while the inverter supplies real power and consumes reactive power from it. Also, for a majority of the possible HVDC control strategies, the equivalent real and reactive power injections at the concerned buses can be computed a-priori and are independent of the NR iterative loop. However, for a few of the control strategies, the equivalent reactive power injections cannot be computed a-priori and need to be computed in every NR iteration. This affects the performance of the iterative process.

5 Case studies and results

It is observed that the AC-DC power flow convergence is affected by the selection of the base values chosen for the various DC quantities. These base values can be defined in several ways, giving rise to multiple per-unit HVDC system models. In a similar manner, the control strategy adopted for the HVDC is also observed to affect the AC-DC power flow convergence. To validate this, several case studies were carried out with multiple HVDC links incorporated in the IEEE 118-bus and 300-bus test systems [16]. Although several choices are feasible, due to a shortage of space, only case studies pertaining to two different per-unit system models and six typical control strategies could be reported in this paper. Two comprehensive case studies on the IEEE 118-bus system and three on the IEEE 300-bus test systems are reported. For all the case studies, the commutating reactance and the DC link resistance are chosen as 0.1 pu and 0.01 pu, respectively. The number of bridges ‘n_b’ for all the converters is taken to be equal to 2 [7]. A convergence tolerance of 10−12 pu is uniformly adopted for all the case studies. In each of the case studies, ‘NI’ and ‘CT’ refer to the number of iterations and the computational time in seconds (with a 2.4 GHz, 4 GB RAM, Intel Core i3-3110 Processor based machine), respectively. All the case studies were carried out in MATLAB software.

5.1 Case1: First study with IEEE 118-bus system

In this study, a single HVDC link was incorporated in the transmission line between buses 6 and 7. The base case (without any HVDC link) active power flow in this line was found to be 33.86 MW. Subsequently, applying Control strategy 1 to the HVDC link, the active power flow is set to 50 MW. The rectifier firing angle and the inverter extinction angle are set to 5° and 18°, respectively. The inverter side DC voltages pertaining to the HVDC per-unit system models 1 and 2 are set to 1.0 pu and 2.3 pu, respectively. It may be noted that these values are different as two different formulations are used, which are detailed in columns 1 to 4 in Table 3. The power-flow solution corresponding to these specifications are also listed in columns 5 to 10 in Table 3. The state variables pertaining to the AC and DC systems are denoted as ACSV and DCSV, respectively. Although both the models require only six iterations to converge, Model 1 takes slightly more computational time (5.15 s) than Model 2 (4.98 s). In a similar manner, the HVDC link specifications corresponding to five other control strategies and their power-flow solution are also given in Table 3.

From Table 3, it is observed that similar to the case of Control strategy 1, for Control strategies 3 to 6, both the HVDC per-unit system models, namely Model 1 and Model 2 exhibit almost similar convergence characteristics, although Model 2 takes slightly less computational time than Model 1. However, for Control strategy 2, Model 2 demonstrates better convergence.

The bus voltage profile for the power-flow solution pertaining to the case with Control strategy 1 and Model 1 is illustrated in Fig. 3. It is observed that bus voltage profile hardly changes except the AC terminal buses connected to the rectifier and the inverter. Due to lack of space, the bus voltage profiles of the other case studies of Table 3 could not be accommodated.

5.2 Case 2: Second study of 118-bus system

In this study, a HVDC link is first incorporated in the transmission line between buses 11 to 13. The base case power flow in this line is 40.81 MW. The power-flow with the HVDC link is set to 50 MW. For all the six control strategies, the different HVDC link specifications along with the corresponding power-flow solutions are tabulated in Table 4.

From Table 4 it can again be observed that both Model 1 and Model 2 exhibit almost similar convergence characteristics for all the control strategies except Control strategy 2 and 6, where the number of iterations taken to converge is more. This is due to the fact that with Control strategies 2 and 6, the equivalent reactive power injections (at the terminal buses connected to the rectifier and the inverter) are updated in every iteration. On the other hand, for Control strategies other than 2 and 6, the reactive power injections can be computed a-priori and are independent of the iterative loop. It is also observed from Table 4 that with Control strategies 2 and 6, the convergence of Model 2 is better than that of Model 1.

5.3 Case 3: First study of IEEE 300-bus system

In this study, a HVDC link is first incorporated in the transmission line between buses 3 and 1. The base case power flow in this line is 24.04  MW. The power-flow with the HVDC link is set to 40 MW. For all the six control strategies, the different HVDC link specifications along with the corresponding power-flow solutions are presented in Table 5.

From Tables 4 and 5, it can be observed that with the HVDC link incorporated, the IEEE 300-bus system takes more number of iterations to converge than the IEEE 118-bus system. Besides, both Model 1 and Model 2 exhibit almost identical convergence characteristics for all the control strategies. In addition, for Control strategy 6, the number of iterations to converge is more for both models, as expected. It is also observed that unlike the previous case studies, Model 1 fares better than Model 2 in respect of computational time for all the control strategies except Control strategy 4.

The bus voltage profile for the power-flow solution pertaining to the case with Control strategy 1 and Model 2 is displayed in Fig. 4. It is again observed that the bus voltage profile undergoes a very slight change at the AC terminal buses connected to the rectifier and the inverter.

5.4 Case 4: Second study of IEEE 300-bus system

In this study, a HVDC link is first incorporated in the transmission line between buses 270 to 292. The base case power flow in this line is 36.52 MW. The power-flow with the HVDC link is set to 40 MW. For all the six control strategies, the different HVDC link specifications along with the corresponding power-flow solutions are given in Table 4.

From Table 6, it can be observed that both Model 1 and Model 2 exhibit almost similar convergence characteristics for all the control strategies except Control strategies 2 and 6. For Control strategy 6, the number of iterations to converge is more for both models, as is expected. It is also observed that for Control strategy 2, Model 2 is faster while for Control strategy 6, Model 1 is faster. For Control strategies 1, 4 and 5, Model 2 takes lesser computational time while for Control strategy 3, Model 1 is faster.

5.5 Case 5: Third study of IEEE 300-bus system

In this study, a HVDC link is first incorporated in the transmission line between buses 199 and 197. The base case power flow in this line is 32.13 MW. The power-flow with the HVDC link is set to 40 MW. For all the six control strategies, the different HVDC link specifications along with the corresponding power-flow solutions are given in Table 7.

From Table 7, it can be observed that both Model 1 and Model 2 exhibit almost similar convergence characteristics for all the control strategies except Control strategies 2 and 6. For Control strategies 2 and 6, the number of iterations to converge is less for Model 2. Also, for Control strategy 1, Model 1 demonstrates better convergence than Model 2. In addition, for Control strategy 5, Model 1 is slightly faster while for Control strategies 3 and 4, Model 2 is slightly faster.

To minimize the reactive power requirement at rectifier and inverter terminals and overall system losses, the values of firing angles are kept low. The firing angle for rectifier is kept within 5° to 7° and the extinction angle for inverter is kept within 15° to 22°.

6 Conclusions

In integrated AC-DC systems, the selection of different base values for the DC quantities gives rise to different per-unit system models. Different per-unit system models affect the convergence of the power-flow algorithm differently. Likewise, the HVDC link control strategy adopted is also observed to affect the convergence of the power-flow algorithm. For a majority of the possible control strategies, the equivalent real and reactive power injections at the concerned buses can be computed a-priori and are independent of the NR iterative loop. However, for a few of the control strategies, the equivalent reactive power injections cannot be computed a-priori and need to be computed in every NR iteration. This affects the performance of the iterative process. In this paper, sequential method is used to solve the DC variables in the NR power flow model, where AC and DC systems are solved separately and are coupled by injecting an equivalent amount of real and reactive power at the terminal AC buses. Two different per-unit system models and six typical control strategies have been taken into consideration. Numerous case studies carried out on the IEEE 118-bus and 300-bus test systems validate this. In general, Model 2 demonstrates better convergence characteristics than Model 1. In a similar manner, the power-flow convergence is affected with Control strategies 2 and 6.

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