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

doi:10.1007/s11708-015-0374-6

Front. Energy

2015, Vol. 9

Issue (4) : 399-412 https://doi.org/10.1007/s11708-015-0374-6

RESEARCH ARTICLE

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

Shagufta KHAN(

), Suman BHOWMICK

Department of Electrical Engineering, Delhi Technological University, Delhi 110042, India

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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.

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

Corresponding Author(s): Shagufta KHAN

Issue Date: 04 November 2015

Cite this article:

Shagufta KHAN,Suman BHOWMICK. Impact of selection of DC base values and DC link control strategies on sequential AC-DC power-flow convergence[J]. Front. Energy, 2015, 9(4): 399-412.

URL:

https://academic.hep.com.cn/fie/EN/10.1007/s11708-015-0374-6
https://academic.hep.com.cn/fie/EN/Y2015/V9/I4/399

Fig.1 HVDC link between buses ‘i’ and ‘j’ of an existing power system network

Fig.2 Equivalent circuit diagram for AC-DC interconnection

Per- unit system 1	Per-unit system 2
$V dR = a R V i cos α R − X c I d$	$V dR = 32 π a R n b V i cos α R − 3 X c π n b I d$
$V dI = a I V j cos γ I − X c I d$	$V dI = 32 π a R n b V j cos γ I − 3 X c π n b I d$
$V dR = a R V i cos φ R$	$V dR = 32 π a R n b V i cos φ R$
$V dI = a I V j cos φ I$	$V dI = 32 π a I n b V j cos φ I$
	$I d = V dR − V dI R d$
	$P dR = V dR I d$

Tab.1 Basic HVDC equations in different per-unit systems

Control strategies	Specified quantities	Unknown quantities
1	$α R, P dR, γ I, V dI$	$a R, a I, V dR, φ R, φ I$
2	$a R, P dR, a I, V dI$	$α R, γ I, V dR, φ R, φ I$
3	$α R, I d, γ I, V dI$	$a R, a I, V dR, φ R, φ I$
4	$α R, P dR, a I, V dI$	$a R, γ I, V dR, φ R, φ I$
5	$a R, P dR, γ I, V dI$	$α R, a I, V dR, φ R, φ I$
6	$a R, P dR, γ I, a I$	$α R, V dR, V dI, φ R, φ I$
7	$a R, P dR, α R, γ I$	$a I, V dR, V dI, φ R, φ I$
8	$α R, V dR, γ I, P dI$	$a R, a I, V dI, φ R, φ I$

Tab.2 HVDC control strategies

	HVDC link specification			Power flow solution
	HVDC link specification			AC terminal buses			HVDC variables
	Spec. values	Model 1	Model 2	ACSV	Model 1	Model 2	DCSV	Model 1	Model 2
Control strategy 1	$P dR$ /pu	0.5	0.5	$V 6$ /pu	0.99	0.99	$V dR$ /pu	1.005	2.3022
	$P dR$ /pu	0.5	0.5	$V 6$ /pu	0.99	0.99	$I dR$ /pu	0.4975	0.2172
	$V dI$ /pu	1.0	2.3	$θ 6 / (°)$	11.037	11.058	$a R$	1.0695	0.8798
	$V dI$ /pu	1.0	2.3	$θ 6 / (°)$	11.037	11.058	$a I$	1.1174	0.9223
	$α R / (°)$	5	5	$V 7$ /pu	0.9878	0.988	$cos φ R$	0.94	0.97
	$α R / (°)$	5	5	$V 7$ /pu	0.9878	0.988	$cos φ I$	0.90	0.93
	$γ I$ /(°)	18	18	$θ 7 / (°)$	11.150	11.165	NI	6	6
	$γ I$ /(°)	18	18	$θ 7 / (°)$	11.150	11.165	CT	5.15	4.98
Control strategy 2	$P dR$ /pu	0.5	0.5	$V 6$ /pu	1.0	1.0	$V dR$ /pu	1.005	2.3022
	$P dR$ /pu	0.5	0.5	$V 6$ /pu	1.0	1.0	$I dR$ /pu	0.4975	0.2172
	$V dI$ /pu	1	2.3	$θ 6 / (°)$	10.926	10.946	$α R / (°)$	12.419	15.393
	$V dI$ /pu	1	2.3	$θ 6 / (°)$	10.926	10.946	$γ I / (°)$	17.929	22.330
	$a R$	1.08	0.9	$V 7$ /pu	0.994	0.997	$cos φ R$	0.93	0.94
	$a R$	1.08	0.9	$V 7$ /pu	0.994	0.997	$cos φ I$	0.90	0.90
	$a I$	1.1	0.94	$θ 7 / (°)$	11.079	11.060	NI	8	6
	$a I$	1.1	0.94	$θ 7 / (°)$	11.079	11.060	CT	6.66	5.11
Control strategy 3	$I dR$ /pu	0.5	0.5	$V 6$ /pu	0.99	0.99	$/puV/pu dR$ /pu	1.005	2.305
	$I dR$ /pu	0.5	0.5	$V 6$ /pu	0.99	0.99	$P dR$ /pu	0.5025	1.1525
	$V dI$ /pu	1	2.3	$θ 6 / (°)$	11.035	10.626	$a R$	1.0697	0.9012
	$V dI$ /pu	1	2.3	$θ 6 / (°)$	11.035	10.626	$a I$	1.1147	0.9459
	$α R / (°)$	5	5	$V 7$ /pu	0.9878	0.9859	$cos φ R$	0.94	0.95
	$α R / (°)$	5	5	$V 7$ /pu	0.9878	0.9859	$cos φ I$	0.90	0.91
	$γ I / (°)$	18	18	$θ 7 / (°)$	11.151	11.481	NI	6	6
	$γ I / (°)$	18	18	$θ 7 / (°)$	11.151	11.481	CT	5.16	5.15
Control strategy 4	$P dR$ /pu	0.5	0.5	$V 6$ /pu	1.00	1.00	$V dR$ /pu	1.005	2.3022
	$P dR$ /pu	0.5	0.5	$V 6$ /pu	1.00	1.00	$I dR$ /pu	0.4975	0.2172
	$V dI$ /pu	1	2.3	$θ 6 / (°)$	10.92	10.946	$a R$	1.0588	0.871
	$V dI$ /pu	1	2.3	$θ 6 / (°)$	10.92	10.946	$γ I / (°)$	12.862	20.779
	$α R / (°)$	5	5	$V 7$ /pu	0.997	0.997	$cos φ R$	0.94	0.97
	$α R / (°)$	5	5	$V 7$ /pu	0.997	0.997	$cos φ I$	0.92	0.91
	$a I$	1.08	0.93	$θ 7 / (°)$	11.038	11.067	NI	6	6
	$a I$	1.08	0.93	$θ 7 / (°)$	11.038	11.067	CT	5.25	5.08
Control strategy 5	$P dR$ /pu	0.5	0.5	$V 6$ /pu	0.99	0.99	$V dR$ /pu	1.005	2.3022
	$P dR$ /pu	0.5	0.5	$V 6$ /pu	0.99	0.99	$I dR$ /pu	0.4975	0.2172
	$V dI$ /pu	1	2.3	$θ 6 / (°)$	11.036	11.057	$α R / (°)$	12.200	13.127
	$V dI$ /pu	1	2.3	$θ 6 / (°)$	11.036	11.057	$a I$	1.114	0.95
	$a R$	1.09	0.9	$V 7$ /pu	0.9908	0.9908	$cos φ R$	0.93	0.97
	$a R$	1.09	0.9	$V 7$ /pu	0.9908	0.9908	$cos φ I$	0.90	0.93
	$γ I / (°)$	18	18	$θ 7 / (°)$	11.109	11.131	NI	6	6
	$γ I / (°)$	18	18	$θ 7 / (°)$	11.109	11.131	CT	5.01	4.98
Control strategy 6	$P dR$ /pu	0.5	0.5	$V 6$ /pu	1.00	1.00	$V dR$ /pu	0.9946	2.5026
	$P dR$ /pu	0.5	0.5	$V 6$ /pu	1.00	1.00	$I dR$ /pu	0.5027	0.1998
	$a R$	1.05	0.95	$θ 6 / (°)$	10.92	10.948	$V dI$ /pu	0.9896	2.506
	$a R$	1.05	0.95	$θ 6 / (°)$	10.92	10.948	$α R / (°)$	5.6583	8.0293
	$a I$	1.1	1.0	$V 7$ /pu	0.994	0.9945	$cos φ R$	0.94	0.95
	$a I$	1.1	1.0	$V 7$ /pu	0.994	0.9945	$cos φ I$	0.90	0.93
	$γ I / (°)$	18	18	$θ 7 / (°)$	11.079	11.096	NI	6	6
	$γ I / (°)$	18	18	$θ 7 / (°)$	11.079	11.096	CT	5.17	5.16

Tab.3 First study of IEEE 118-bus system (HVDC link: From bus No. 6 to bus No. 7; P_base=0.3386 pu)

Fig.3 Bus voltage profile for the case study of Table 3 with Control strategy 1 and Model 1

(a) Bus voltage magnitude without HVDC link; (b) bus voltage magnitude with HVDC link; (c) bus voltage magnitude difference

	HVDC link specification/pu			Power flow solution
	HVDC link specification/pu			AC terminal buses			HVDC variables
	Spec. values	Model 1	Model 2	ACSV	Model 1	Model 2	DCSV	Model 1	Model 2
Control strategy 1	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9803	0.9815	$V dR$ /pu	1.005	2.3022
	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9803	0.9815	$I dR$ /pu	0.4975	0.2172
	$V dI$ /pu	1.0	2.3	$θ 11 / (°)$	10.739	10.749	$a R$	1.08	0.8875
	$V dI$ /pu	1.0	2.3	$θ 11 / (°)$	10.739	10.749	$a I$	1.15	0.9463
	$α R / (°)$	5	5	$V 13$ /pu	0.9598	0.9632	$cos φ R$	0.94	0.97
	$α R / (°)$	5	5	$V 13$ /pu	0.9598	0.9632	$cos φ I$	0.90	0.93
	$γ I / (°)$	18	18	$θ 13 / (°)$	11.349	11.321	NI	6	6
	$γ I / (°)$	18	18	$θ 13 / (°)$	11.349	11.321	CT	5.14	5.06
Control strategy 2	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9801	1.00	$V dR$ /pu	1.005	2.3022
	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9801	1.00	$I dR$ /pu	0.4975	0.2172
	$V dI$ /pu	1	2.3	$θ 11 / (°)$	10.741	10.946	$α R / (°)$	9.1566	11.732
	$V dI$ /pu	1	2.3	$θ 11 / (°)$	10.741	10.946	$γ I / (°)$	17.969	20.934
	$a R$	1.09	0.9	$V 13$ /pu	0.9596	0.997	$cos φ R$	0.94	0.96
	$a R$	1.09	0.9	$V 13$ /pu	0.9596	0.997	$cos φ I$	0.90	0.91
	$a I$	1.15	0.95	$θ 13 / (°)$	11.351	11.060	NI	13	6
	$a I$	1.15	0.95	$θ 13 / (°)$	11.351	11.060	CT	10.61	5.07
Control strategy 3	$I dR$ /pu	0.5	0.5	$V 11$ /pu	0.9803	0.9741	$V dR$ /pu	1.005	2.305
	$I dR$ /pu	0.5	0.5	$V 11$ /pu	0.9803	0.9741	$P dR$ /pu	0.5025	1.1525
	$V dI$ /pu	1	2.3	$θ 11 / (°)$	10.737	10.179	$a R$	1.08	0.9159
	$V dI$ /pu	1	2.3	$θ 11 / (°)$	10.737	10.179	$a I$	1.1504	0.9836
	$α R / (°)$	5	5	$V 13$ /pu	0.9597	0.9481	$cos φ R$	0.94	0.95
	$α R / (°)$	5	5	$V 13$ /pu	0.9597	0.9481	$cos φ I$	0.90	0.91
	$γ I / (°)$	18	18	$θ 13 / (°)$	11.358	13.521	NI	6	6
	$γ I / (°)$	18	18	$θ 13 / (°)$	11.358	13.521	CT	5.09	5.06
Control strategy 4	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9843	1.00	$V dR$ /pu	1.005	2.3022
	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9843	1.00	$I dR$ /pu	0.4975	0.2172
	$V dI$ /pu	1	2.3	$θ 11 / (°)$	10.727	10.946	$a R$	1.0765	0.885
	$V dI$ /pu	1	2.3	$θ 11 / (°)$	10.727	10.946	$γ I / (°)$	17.868	20.881
	$α R / (°)$	5	5	$V 13$ /pu	0.9767	0.997	$cos φ R$	0.94	0.97
	$α R / (°)$	5	5	$V 13$ /pu	0.9767	0.997	$cos φ I$	0.90	0.91
	$a I$	1.13	0.95	$θ 13 / (°)$	11.099	11.067	NI	6	6
	$a I$	1.13	0.95	$θ 13 / (°)$	11.099	11.067	CT	5.15	5.14
Control strategy 5	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9847	0.9847	$V dR$ /pu	1.005	2.3022
	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9847	0.9847	$I dR$ /pu	0.4975	0.2172
	$V dI$ /pu	1	2.3	$θ 11 / (°)$	10.702	11.721	$α R / (°)$	10.678	11.732
	$V dI$ /pu	1	2.3	$θ 11 / (°)$	10.702	11.721	$a I$	1.1298	0.933
	$a R$	1.09	0.9	$V 13$ /pu	0.977	0.977	$cos φ R$	0.93	0.96
	$a R$	1.09	0.9	$V 13$ /pu	0.977	0.977	$cos φ I$	0.90	0.93
	$γ I / (°)$	18	18	$θ 13 / (°)$	11.069	11.094	NI	6	6
	$γ I / (°)$	18	18	$θ 13 / (°)$	11.069	11.094	CT	5.16	5.09
Control strategy 6	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9809	1.00	$V dR$ /pu	0.9076	2.4183
	$P dR$ /pu	0.5	0.5	$V 11$ /pu	0.9809	1.00	$I dR$ /pu	0.5509	0.2068
	$a R$	1	0.95	$θ 11 / (°)$	10.755	10.948	$V dI$ /pu	0.902	2.4162
	$a R$	1	0.95	$θ 11 / (°)$	10.755	10.948	$α R / (°)$	10.707	12.451
	$a I$	1.05	1	$V 13$ /pu	0.9624	0.9945	$cos φ R$	0.92	0.96
	$a I$	1.05	1	$V 13$ /pu	0.9624	0.9945	$cos φ I$	0.89	0.92
	$γ I / (°)$	18	18	$θ 13 / (°)$	11.335	11.096	NI	9	8
	$γ I / (°)$	18	18	$θ 13 / (°)$	11.335	11.096	CT	7.55	6.53

Tab.4 Second study of IEEE 118-bus system (HVDC link: From bus No. 11 to bus No. 13; P_base=0.4081 pu)

Fig.4 Bus voltage profile for the case study of Table 5 with Control strategy 1 and Model 2

	HVDC link specification			Power flow solution
	HVDC link specification			AC terminal buses			HVDC variables
	Spec. values	Model 1	Model 2	ACSV	Model 1	Model 2	DCSV	Model 1	Model 2
Control strategy 1	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.0187	1.0189	$V dR$ /pu	1.004	2.3017
	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.0187	1.0189	$I dR$ /pu	0.3984	0.1738
	$V dI$ /pu	1.0	2.3	$θ 3 / (°)$	6.6208	6.6248	$a R$	1.0286	0.8517
	$V dI$ /pu	1.0	2.3	$θ 3 / (°)$	6.6208	6.6248	$a I$	1.0776	0.8949
	$α R / (°)$	5	5	$V 1$ /pu	1.0147	1.015	$cos φ R$	0.95	0.98
	$α R / (°)$	5	5	$V 1$ /pu	1.0147	1.015	$cos φ I$	0.91	0.93
	$γ I / (°)$	18	18	$θ 1 / (°)$	6.4066	6.4112	NI	7	7
	$γ I / (°)$	18	18	$θ 1 / (°)$	6.4066	6.4112	CT	42.83	43.761
Control strategy 2	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.0186	1.0194	$V dR$ /pu	1.004	2.3017
	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.0186	1.0194	$I dR$ /pu	0.3984	0.1738
	$V dI$ /pu	1	2.3	$θ 3 / (°)$	6.6218	6.6178	$α R / (°)$	9.8257	12.905
	$V dI$ /pu	1	2.3	$θ 3 / (°)$	6.6218	6.6178	$γ I / (°)$	19.893	17.310
	$a R$	1.04	0.87	$V 1$ /pu	1.0145	1.0167	$cos φ R$	0.94	0.96
	$a R$	1.04	0.87	$V 1$ /pu	1.0145	1.0167	$cos φ I$	0.90	0.94
	$a I$	1.09	0.89	$θ 1 / (°)$	6.408	6.3986	NI	7	7
	$a I$	1.09	0.89	$θ 1 / (°)$	6.408	6.3986	CT	45.35	45.4123
Control strategy 3	$I dR$ /pu	0.4	0.4	$V 3$ /pu	1.0187	1.0179	$V dR$ /pu	1.004	2.304
	$I dR$ /pu	0.4	0.4	$V 3$ /pu	1.0187	1.0179	$P dR$ /pu	0.4016	0.9216
	$V dI$ /pu	1	2.3	$θ 3 / (°)$	6.6206	6.5818	$a R$	1.0288	0.8691
	$V dI$ /pu	1	2.3	$θ 3 / (°)$	6.6206	6.5818	$a I$	1.0777	0.9138
	$α R / (°)$	5	5	$V 1$ /pu	1.0147	1.0123	$cos φ R$	0.95	0.96
	$α R / (°)$	5	5	$V 1$ /pu	1.0147	1.0123	$cos φ I$	0.91	0.92
	$γ I / (°)$	18	18	$θ 1 / (°)$	6.4083	6.9945	NI	7	7
	$γ I / (°)$	18	18	$θ 1 / (°)$	6.4083	6.9945	CT	42.31	43.6478
Control strategy 4	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.017	1.0172	$V dR$ /pu	1.004	2.3017
	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.017	1.0172	$I dR$ /pu	0.3984	0.1738
	$V dI$ /pu	1	2.3	$θ 3 / (°)$	6.641	6.6457	$a R$	1.0303	0.8531
	$V dI$ /pu	1	2.3	$θ 3 / (°)$	6.641	6.6457	$γ I / (°)$	21.472	19.121
	$α R / (°)$	5	5	$V 1$ /pu	1.0158	1.0159	$cos φ R$	0.95	0.98
	$α R / (°)$	5	5	$V 1$ /pu	1.0158	1.0159	$cos φ I$	0.89	0.93
	$a I$	1.1	0.9	$θ 1 / (°)$	6.4176	6.4239	NI	7	7
	$a I$	1.1	0.9	$θ 1 / (°)$	6.4176	6.4239	CT	44.327	43.73
Control strategy 5	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.0194	1.0194	$V dR$ /pu	1.004	2.3017
	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.0194	1.0194	$I dR$ /pu	0.3984	0.1738
	$V dI$ /pu	1	2.3	$θ 3 / (°)$	6.611	6.6178	$α R / (°)$	6.2125	9.5726
	$V dI$ /pu	1	2.3	$θ 3 / (°)$	6.611	6.6178	$a I$	1.0754	0.9209
	$a R$	1.03	0.86	$V 1$ /pu	1.0167	1.0167	$cos φ R$	0.95	0.97
	$a R$	1.03	0.86	$V 1$ /pu	1.0167	1.0167	$cos φ I$	0.91	0.93
	$γ I / (°)$	18	18	$θ 1 / (°)$	6.3904	6.3986	NI	7	7
	$γ I / (°)$	18	18	$θ 1 / (°)$	6.3904	6.3986	CT	43.20	43.5601
Control strategy 6	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.0186	1.0187	$V dR$ /pu	1.0264	2.8311
	$P dR$ /pu	0.4	0.4	$V 3$ /pu	1.0186	1.0187	$I dR$ /pu	0.3897	0.1413
	$a R$	1.08	1.08	$θ 3 / (°)$	6.6228	6.6279	$V dI$ /pu	1.0225	2.8297
	$a R$	1.08	1.08	$θ 3 / (°)$	6.6228	6.6279	$α R / (°)$	14.424	15.884
	$a I$	1.1	1.1	$V 1$ /pu	1.0146	1.015	$cos φ R$	0.93	0.95
	$a I$	1.1	1.1	$V 1$ /pu	1.0146	1.015	$cos φ I$	0.91	0.93
	$γ I / (°)$	18	18	$θ 1 / (°)$	6.4085	6.414	NI	8	8
	$γ I / (°)$	18	18	$θ 1 / (°)$	6.4085	6.414	CT	44.706	50.267

Tab.5 First study of IEEE 300-bus system (HVDC link: From bus No. 3 to bus No. 1; P_base=0.2404 pu)

	HVDC link specification/pu			Power flow solution
	HVDC link specification/pu			AC terminal buses			HVDC variables
	Spec. values	Model 1	Model 2	ACSV	Model 1	Model 2	DCSV	Model 1	Model 2
Control strategy 1	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.008	1.0017	$V dR$ /pu	1.004	2.4017
	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.008	1.0017	$I dR$ /pu	0.3984	0.1666
	$V dI$ /pu	1.0	2.4	$θ 270 / (°)$	−11.41	−11.41	$a R$	1.047	0.9029
	$V dI$ /pu	1.0	2.4	$θ 270 / (°)$	−11.41	−11.41	$a I$	1.0934	0.9467
	$α R / (°)$	5	5	$V 292$ /pu	1.00	1.00	$cos φ R$	0.95	0.98
	$α R / (°)$	5	5	$V 292$ /pu	1.00	1.00	$cos φ I$	0.91	0.93
	$γ I / (°)$	18	18	$θ 292 / (°)$	−10.69	−10.65	NI	7	7
	$γ I / (°)$	18	18	$θ 292 / (°)$	−10.69	−10.65	CT	44.1922	42.0610
Control strategy 2	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.0005	1.0031	$V dR$ /pu	1.004	2.4017
	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.0005	1.0031	$I dR$ /pu	0.3984	0.1666
	$V dI$ /pu	1	2.4	$θ 270 / (°)$	−11.41	−11.66	$α R / (°)$	10.174	12.514
	$V dI$ /pu	1	2.4	$θ 270 / (°)$	−11.41	−11.66	$γ I / (°)$	15.673	21.844
	$a R$	1.06	0.92	$V 292$ /pu	1.00	1.00	$cos φ R$	0.94	0.96
	$a R$	1.06	0.92	$V 292$ /pu	1.00	1.00	$cos φ I$	0.92	0.91
	$a I$	1.08	0.97	$θ 292 / (°)$	−10.68	−10.66	NI	10	7
	$a I$	1.08	0.97	$θ 292 / (°)$	−10.68	−10.66	CT	62.7693	43.072
Control strategy 3	$I dR$ /pu	0.4	0.4	$V 270$ /pu	1.008	0.9963	$V dR$ /pu	1.004	2.404
	$I dR$ /pu	0.4	0.4	$V 270$ /pu	1.008	0.9963	$P dR$ /pu	0.4016	0.9616
	$V dI$ /pu	1	2.3	$θ 270 / (°)$	−11.41	−11.42	$a R$	1.0935	0.9253
	$V dI$ /pu	1	2.3	$θ 270 / (°)$	−11.41	−11.42	$a I$	1.004	0.964
	$α R / (°)$	5	5	$V 292$ /pu	1.00	1.00	$cos φ R$	0.95	0.96
	$α R / (°)$	5	5	$V 292$ /pu	1.00	1.00	$cos φ I$	0.91	0.92
	$γ I / (°)$	18	18	$θ 292 / (°)$	−10.65	−10.54	NI	7	7
	$γ I / (°)$	18	18	$θ 292 / (°)$	−10.65	−10.54	CT	43.486	43.934
Control strategy 4	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.008	1.0017	$V dR$ /pu	1.004	2.4017
	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.008	1.0017	$I dR$ /pu	0.3984	0.1666
	$V dI$ /pu	1	2.4	$θ 270 / (°)$	−11.41	−11.41	$a R$	1.047	0.9029
	$V dI$ /pu	1	2.4	$θ 270 / (°)$	−11.41	−11.41	$γ I / (°)$	19.036	21.844
	$α R / (°)$	5	5	$V 292$ /pu	1.00	1.00	$cos φ R$	0.95	0.98
	$α R / (°)$	5	5	$V 292$ /pu	1.00	1.00	$cos φ I$	0.90	0.91
	$a I$	1.1	0.97	$θ 292 / (°)$	−10.69	−10.65	NI	7	7
	$a I$	1.1	0.97	$θ 292 / (°)$	−10.69	−10.65	CT	43.6352	43.45
Control strategy 5	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.0031	1.003	$V dR$ /pu	1.004	2.4071
	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.0031	1.003	$I dR$ /pu	0.3984	0.1666
	$V dI$ /pu	1	2.3	$θ 270 / (°)$	−11.42	−11.41	$α R / (°)$	7.6894	9.0385
	$V dI$ /pu	1	2.3	$θ 270 / (°)$	−11.42	−11.41	$a I$	1.0934	0.9436
	$a R$	1.05	0.91	$V 292$ /pu	1.00	1.00	$cos φ R$	0.95	0.97
	$a R$	1.05	0.91	$V 292$ /pu	1.00	1.00	$cos φ I$	0.91	0.93
	$γ I / (°)$	18	18	$θ 292 / (°)$	−10.69	−10.66	NI	7	7
	$γ I / (°)$	18	18	$θ 292 / (°)$	−10.69	−10.66	CT	44.0450	43.015
Control strategy 6	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.002	1.0007	$V dR$ /pu	1.0105	2.3446
	$P dR$ /pu	0.4	0.4	$V 270$ /pu	1.002	1.0007	$I dR$ /pu	0.3958	0.1706
	$a R$	1.08	0.91	$θ 270 / (°)$	−11.41	−11.41	$V dI$ /pu	1.0066	2.3429
	$a R$	1.08	0.91	$θ 270 / (°)$	−11.41	−11.41	$α R / (°)$	13.551	19.868
	$a I$	1.1	0.93	$V 292$ /pu	1.00	1.00	$cos φ R$	0.93	0.95
	$a I$	1.1	0.93	$V 292$ /pu	1.00	1.00	$cos φ I$	0.91	0.93
	$γ I / (°)$	18	18	$θ 292 / (°)$	−10.68	−10.65	NI	10	11
	$γ I / (°)$	18	18	$θ 292 / (°)$	−10.68	−10.65	CT	59.5337	70.0756

Tab.6 Second study of IEEE 300-bus system (HVDC link: From bus No. 270 to bus No. 292; P_base=0.3652 pu)

	HVDC link specification			Power flow solution
	HVDC link specification			AC terminal buses			HVDC variables
	Spec. values	Model 1	Model 2	ACSV	Model 1	Model 2	DCSV	Model 1	Model 2
Control strategy 1	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.008	1.008	$V dR$ /pu	0.9044	2.2018
	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.008	1.008	$I dR$ /pu	0.4423	0.1817
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.19	−22.18	$a R$	0.9556	0.83
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.19	−22.18	$a I$	1.0025	0.87
	$α R / (°)$	10	10	$V 197$ /pu	1.0159	1.0164	$cos φ R$	0.93	0.96
	$α R / (°)$	10	10	$V 197$ /pu	1.0159	1.0164	$cos φ I$	0.88	0.91
	$γ I / (°)$	22	22	$θ 197 / (°)$	−22.63	−22.62	NI	7	8
	$γ I / (°)$	22	22	$θ 197 / (°)$	−22.63	−22.62	CT	42.9965	49.7154
Control strategy 2	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.00	1.00	$V dR$ /pu	0.9044	2.2018
	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.00	1.00	$I dR$ /pu	0.4423	0.1817
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.15	−22.13	$α R / (°)$	8.819	9.6773
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.15	−22.13	$γ I / (°)$	21.44	20.563
	$a R$	0.96	0.84	$V 197$ /pu	1.0124	1.0157	$cos φ R$	0.94	0.97
	$a R$	0.96	0.84	$V 197$ /pu	1.0124	1.0157	$cos φ I$	0.88	0.92
	$a I$	1	0.87	$θ 197 / (°)$	−22.64	−22.63	NI	9	7
	$a I$	1	0.87	$θ 197 / (°)$	−22.64	−22.63	CT	55.1537	42.9122
Control strategy 3	$I dR$ /pu	0.4	0.4	$V 199$ /pu	1.000	1.008	$V dR$ /pu	0.9040	2.2043
	$I dR$ /pu	0.4	0.4	$V 199$ /pu	1.000	1.008	$P dR$ /pu	0.3616	0.8816
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.65	−22.47	$a R$	0.9510	0.8505
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.65	−22.47	$a I$	0.9976	0.8976
	$α R / (°)$	10	10	$V 197$ /pu	1.0162	1.0127	$cos φ R$	0.94	0.95
	$α R / (°)$	10	10	$V 197$ /pu	1.0162	1.0127	$cos φ I$	0.88	0.89
	$γ I / (°)$	22	22	$θ 197 / (°)$	−22.65	−22.34	NI	7	7
	$γ I / (°)$	22	22	$θ 197 / (°)$	−22.65	−22.34	CT	43.62	42.35
Control strategy 4	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.000	1.000	$V dR$ /pu	0.9044	2.2018
	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.000	1.000	$I dR$ /pu	0.4423	0.1817
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.15	−22.13	$a R$	0.9633	0.8408
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.15	−22.13	$γ I / (°)$	21.625	20.653
	$α R / (°)$	10	10	$V 197$ /pu	1.0157	1.0157	$cos φ R$	0.93	0.96
	$α R / (°)$	10	10	$V 197$ /pu	1.0157	1.0157	$cos φ I$	0.88	0.92
	$a I$	1.0	0.87	$θ 197 / (°)$	−22.15	−22.63	NI	7	7
	$a I$	1.0	0.87	$θ 197 / (°)$	−22.15	−22.63	CT	42.4364	41.94
Control strategy 5	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.008	1.008	$V dR$ /pu	0.9044	2.2018
	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.008	1.008	$I dR$ /pu	0.4423	0.1817
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.18	−22.17	$α R / (°)$	7.839	8.218
	$V dI$ /pu	0.9	2.2	$θ 199 / (°)$	−22.18	−22.17	$a I$	0.991	0.97
	$a R$	0.95	0.83	$V 197$ /pu	1.0193	1.0193	$cos φ R$	0.94	0.91
	$a R$	0.95	0.83	$V 197$ /pu	1.0193	1.0193	$cos φ I$	0.83	0.93
	$γ I / (°)$	22	22	$θ 197 / (°)$	−22.63	−22.62	NI	7	7
	$γ I / (°)$	22	22	$θ 197 / (°)$	−22.63	−22.62	CT	38.5849	38.889
Control strategy 6	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.000	1.000	$V dR$ /pu	0.8985	2.3137
	$P dR$ /pu	0.4	0.4	$V 199$ /pu	1.000	1.000	$I dR$ /pu	0.4452	0.1729
	$a R$	0.95	0.89	$θ 199 / (°)$	−22.15	−22.14	$V dI$ /pu	0.8941	2.312
	$a R$	0.95	0.89	$θ 199 / (°)$	−22.15	−22.14	$α R / (°)$	6.947	12.51
	$a I$	1	0.93	$V 197$ /pu	1.00123	1.0127	$cos φ R$	0.94	0.96
	$a I$	1	0.93	$V 197$ /pu	1.00123	1.0127	$cos φ I$	0.88	0.90
	$γ I / (°)$	22	22	$θ 197 / (°)$	−22.64	−22.63	NI	9	7
	$γ I / (°)$	22	22	$θ 197 / (°)$	−22.64	−22.63	CT	52.7786	40.6926

Tab.7 Third study of IEEE 300-bus system (HVDC link: From bus No. 199 to bus No. 197; P_base=0.3213 pu)

$S base$	Base MVA
$V ac ? base, I ac ? base, Z ac ? base$	AC base voltage, AC base current and AC base impedance
$V dc ? base, I dc ? base, Z dc ? base$	DC base voltage, DC base current and DC base impedance
$n b$	Number of bridges
$X c$	Commutating reactance
$V d, I d$	DC voltage and current
$φ R, φ I$	Power factor angles at the rectifier and inverter ends (Transformer’s primary side)
$α R, γ I$	Firing angle of the rectifier and extinction angle of the inverter
$V dR, V dI$	DC voltages at the rectifier and inverter sides
$a R, a I$	Converter transformer tap ratios on the rectifier and inverter sides
R_d	Resistance of DC link
$V i ∠ θ i$	AC bus voltage magnitude (rms) and phase angle at ith bus
$V j ∠ θ j$	AC bus voltage magnitude (rms) and phase angle at jth bus
$V doR, V doI$	No load direct voltages at the rectifier and inverter sides
$P dR, Q dR$	Active and reactive powers at the rectifier side
$P dI, Q dI$	Active and reactive powers at the inverter side

Convention 1	Convention 2
$V dc ? base = k V ac ? base$ where $k = 32 π n b$	$V dc ? base = V ac ? base$
$I dc ? base = 3 k I ac ? base$	$I dc ? base = 3 I ac ? base$
$Z dc ? base = k 2 Z ac ? base$	$Z dc ? base = Z ac ? base$
$R dc ? base = 3 π n b X c ? base$	$R dc ? base = X c ? base$

Table A1 Different base values for DC system

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