How to Run a Three-Phase Power Flow
This guide describes how to run a power flow on a three-phase, unbalanced model. MATPOWER 8 includes the core functionality required to build out three-phase unbalanced modeling. It also includes a few simple prototype three-phase elements: buses, lines, generators, loads, and transformers and even a bus-link element used to connect three-phase and single-phase parts of a model.
Note
Keep in mind these are prototype elements. That means that the data formats used, etc. are not final and there are many additions, enhancements and changes planned for future versions. So don’t count on future versions to be backward compatible with these prototypes.
The functionality is accessed through the MATPOWER extension mp.xt_3p and can be used for power flow (PF), continuation power flow (CPF), and optimal power flow (OPF), by passing the extension as input to the run_pf(), run_cpf(), and run_opf() functions, respectively.
Some relevant reference documentation can be found in Three-Phase Prototype Extension, and the test code in t_run_mp_3p.m includes numerous examples of three-phase cases and how to run them.
Below is an example run.
>> mpopt = mpoption('verbose', 2);
>> run_pf('t_case3p_a', mpopt, 'mpx', mp.xt_3p)
MATPOWER Version 8.0, 17-May-2024
Power Flow -- AC-polar-power formulation
it max residual max ∆x
---- -------------- --------------
0 2.375e+00 -
1 4.042e-01 2.287e-01
2 7.179e-02 7.961e-02
3 4.570e-03 1.967e-02
4 2.405e-05 1.400e-03
5 6.090e-10 6.908e-06
Newton's method converged in 5 iterations.
PF successful
PF succeeded in 0.17 seconds (0.16 setup + 0.01 solve)
================================================================================
| System Summary |
================================================================================
elements on off total
--------------------- ------- ------- -------
3-ph Buses 4 - 4
3-ph Generators 1 - 1
3-ph Loads 1 - 1
3-ph Lines 2 - 2
3-ph Transformers 1 - 1
Total 3-ph generation 6109.9 kW 4206.5 kVAr
Total 3-ph load 5450.0 kW 2442.6 kVAr
Total 3-ph line loss 561.5 kW 1173.8 kVAr
Total 3-ph transformer loss 98.4 kW 590.2 kVAr
================================================================================
| 3-ph Bus Data |
================================================================================
3-ph Phase A Voltage Phase B Voltage Phase C Voltage
Bus ID Status (kV) (deg) (kV) (deg) (kV) (deg)
-------- ------ ------- ------- ------- ------- ------- -------
1 1 7.1996 0.00 7.1996 -120.00 7.1996 120.00
2 1 7.1637 -0.14 7.1105 -120.18 7.0821 119.26
3 1 2.3055 -2.26 2.2547 -123.62 2.2028 114.79
4 1 2.1750 -4.12 1.9298 -126.80 1.8327 102.85
================================================================================
| 3-ph Generator Data |
================================================================================
3-ph 3-ph Phase A Power Phase B Power Phase C Power
Gen ID Bus ID Status (kW) (KVAr) (kW) (kVAr) (kW) (kVAr)
-------- -------- ------ ------- ------ ------- ------ ------- ------
1 1 1 1341.42 970.52 2096.10 1341.41 2672.34 1894.59
================================================================================
| 3-ph Load Data |
================================================================================
3-ph 3-ph Phase A Power Phase B Power Phase C Power
Load ID Bus ID Status (kW) (PF) (kW) (PF) (kW) (PF)
-------- -------- ------ ------- ------ ------- ------ ------- ------
1 4 1 1275.00 0.8500 1800.00 0.9000 2375.00 0.9500
================================================================================
| 3-ph Line Data |
================================================================================
--> Current Injections at "From" Bus
3-ph 3-ph Bus 3-ph Bus Phase A Current Phase B Current Phase C Current
Line ID From ID To ID Status (A) (deg) (A) (deg) (A) (deg)
-------- -------- -------- ------ ------ ------ ------ ------ ------ ------
1 1 2 1 229.97 -35.9 345.66 -152.6 455.00 84.7
2 3 4 1 689.64 -35.9 1036.34 -152.6 1364.10 84.7
<-- Current Injections at "To" Bus
3-ph 3-ph Bus 3-ph Bus Phase A Current Phase B Current Phase C Current
Line ID From ID To ID Status (A) (deg) (A) (deg) (A) (deg)
-------- -------- -------- ------ ------ ------ ------ ------ ------ ------
1 1 2 1 230.06 144.1 345.72 27.4 455.06 -95.3
2 3 4 1 689.66 144.1 1036.35 27.4 1364.11 -95.3
--> Power Injections at "From" Bus
3-ph 3-ph Bus 3-ph Bus Phase A Power Phase B Power Phase C Power
Line ID From ID To ID Status (kW) (kVAr) (kW) (kVAr) (kW) (kVAr)
-------- -------- -------- ------ ------ ------ ------ ------ ------ ------
1 1 2 1 1341.4 970.5 2096.1 1341.4 2672.3 1894.6
2 3 4 1 1323.5 881.1 2043.4 1133.3 2598.7 1508.6
<-- Power Injections at "To" Bus
3-ph 3-ph Bus 3-ph Bus Phase A Power Phase B Power Phase C Power
Line ID From ID To ID Status (kW) (kVAr) (kW) (kVAr) (kW) (kVAr)
-------- -------- -------- ------ ------ ------ ------ ------ ------ ------
1 1 2 1 -1337.2 -963.4 -2074.4 -1319.1 -2652.4 -1830.6
2 3 4 1 -1275.0 -790.2 -1800.0 -871.8 -2375.0 -780.6
================================================================================
| 3-ph Transformer Data |
================================================================================
--> Current Injections at "From" Bus
3-ph 3-ph Bus 3-ph Bus Phase A Current Phase B Current Phase C Current
Xfrm ID From ID To ID Status (A) (deg) (A) (deg) (A) (deg)
-------- -------- -------- ------ ------ ------ ------ ------ ------ ------
1 2 3 1 230.06 -35.9 345.72 -152.6 455.06 84.7
<-- Current Injections at "To" Bus
3-ph 3-ph Bus 3-ph Bus Phase A Current Phase B Current Phase C Current
Xfrm ID From ID To ID Status (A) (deg) (A) (deg) (A) (deg)
-------- -------- -------- ------ ------ ------ ------ ------ ------ ------
1 2 3 1 689.64 144.1 1036.34 27.4 1364.10 -95.3
--> Power Injections at "From" Bus
3-ph 3-ph Bus 3-ph Bus Phase A Power Phase B Power Phase C Power
Xfmr ID From ID To ID Status (kW) (kVAr) (kW) (kVAr) (kW) (kVAr)
-------- -------- -------- ------ ------ ------ ------ ------ ------ ------
1 2 3 1 1337.2 963.4 2074.4 1319.1 2652.4 1830.6
<-- Power Injections at "To" Bus
3-ph 3-ph Bus 3-ph Bus Phase A Power Phase B Power Phase C Power
Xfmr ID From ID To ID Status (kW) (kVAr) (kW) (kVAr) (kW) (kVAr)
-------- -------- -------- ------ ------ ------ ------ ------ ------ ------
1 2 3 1 -1323.5 -881.1 -2043.4 -1133.3 -2598.7 -1508.6
Data Format
The prototype examples included in MATPOWER 8 simply define a few extra fields in the standard MATPOWER case struct for the three-phase elements. The example above uses t_case3p_a.m. This is a case with 4 three-phase buses, connected via 2 lines and 1 transformer, with a three-phase generator representing the substation at bus 1 and a three-phase load at bus 4.
function mpc = t_case3p_a
%T_CASE3P_A Four bus, unbalanced 3-phase test case.
%
% This data comes from 4Bus-YY-UnB.DSS, a modified version (with unbalanced
% load) of 4Bus-YY-Bal.DSS [1], the OpenDSS 4 bus IEEE test case with
% grounded-wye to grounded-wye transformer.
%
% [1] https://sourceforge.net/p/electricdss/code/HEAD/tree/trunk/Distrib/IEEETestCases/4Bus-YY-Bal/4Bus-YY-Bal.DSS
%% MATPOWER Case Format : Version 2
mpc.version = '2';
%%----- Power Flow Data -----%%
%% system MVA base
mpc.baseMVA = 100;
mpc.bus = [];
mpc.gen = [];
mpc.branch = [];
mpc.gencost = [];
%%----- 3 Phase Model Data -----%%
%% system data
mpc.freq = 60; %% frequency, Hz
mpc.basekVA = 1000; %% system kVA base
%% bus data
% busid type basekV Vm1 Vm2 Vm3 Va1 Va2 Va3
mpc.bus3p = [
1 3 12.47 1 1 1 0 -120 120;
2 1 12.47 1 1 1 0 -120 120;
3 1 4.16 1 1 1 0 -120 120;
4 1 4.16 1 1 1 0 -120 120;
];
%% branch data
% brid fbus tbus status lcid len
mpc.line3p = [
1 1 2 1 1 2000/5280;
2 3 4 1 1 2500/5280;
];
%% transformer
% xfid fbus tbus status R X basekVA basekV
mpc.xfmr3p = [
1 2 3 1 0.01 0.06 6000 12.47;
];
%% load
% ldid ldbus status Pd1 Pd2 Pd3 ldpf1 ldpf2 ldpf3
mpc.load3p = [
1 4 1 1275 1800 2375 0.85 0.9 0.95;
];
%% gen
% genid gbus status Vg1 Vg2 Vg3 Pg1 Pg2 Pg3 Qg1 Qg2 Qg3
mpc.gen3p = [
1 1 1 1 1 1 2000 2000 2000 0 0 0;
];
%% line construction
% lcid R11 R21 R31 R22 R32 R33 X11 X21 X31 X22 X32 X33 C11 C21 C31 C22 C32 C33
mpc.lc = [
1 0.457541 0.15594 0.153474 0.466617 0.157996 0.461462 1.078 0.501648 0.384909 1.04813 0.423624 1.06502 15.0671 -4.86241 -1.85323 15.875 -3.09098 14.3254
];
The data for the individual elements are found in the fields bus3p, line3p, xfrm3p, load3p, and gen3p fields of mpc. The lc field contains the per-mile impedance parameters for different line construction configurations referenced by the individual lines.
Besides the fields for the individual elements, cases with three-phase elements are also expected to include freq, the system frequency in Hertz, and base_kVA, the system per-unit kVA base for three-phase portions of the network.
Details of the data for the individual three-phase elements are summarized in the tables below.
bus3p
Col |
Name |
Description — see also |
|---|---|---|
1 |
busid |
unique 3-phase bus ID (positive integer) |
2 |
type |
bus type: 1 = PQ, 2 = PV, 3 = reference (voltage ref + slack) |
3 |
basekV |
nominal voltage in kV |
4 |
Vm1 |
phase A voltage magnitude in p.u. |
5 |
Vm2 |
phase B voltage magnitude in p.u. |
6 |
Vm3 |
phase C voltage magnitude in p.u. |
7 |
Va1 |
phase A voltage angle in degrees (nominal 0) |
8 |
Va2 |
phase B voltage angle in degrees (nominal -120) |
9 |
Va3 |
phase C voltage angle in degrees (nominal 120) |
line3p
Col |
Name |
Description — see also |
|---|---|---|
1 |
brid |
unique 3-phase line ID (positive integer) |
2 |
fbus |
“from” bus ID |
3 |
tbus |
“to” bus ID |
4 |
status |
1 = in-service, 0 = out-of-service |
5 |
lcid |
line construction ID (to look up in |
6 |
len |
line length in miles |
xfmr3p
Col |
Name |
Description — see also |
|---|---|---|
1 |
xfid |
unique 3-phase transformer ID (positive integer) |
2 |
fbus |
“from” bus ID |
3 |
tbus |
“to” bus ID |
4 |
status |
1 = in-service, 0 = out-of-service |
5 |
R |
resistance (p.u. on transformer base) |
6 |
X |
reactance (p.u. on transformer base) |
7 |
basekVA |
transformer per-unit power base in kVA |
8 |
basekV |
transformer per-unit voltage base in kV |
load3p
Col |
Name |
Description — see also |
|---|---|---|
1 |
ldid |
unique 3-phase load ID (positive integer) |
2 |
ldbus |
bus ID |
3 |
status |
1 = in-service, 0 = out-of-service |
4 |
Pd1 |
phase A active power demand in kW |
5 |
Pd2 |
phase B active power demand in kW |
6 |
Pd3 |
phase C active power demand in kW |
7 |
lpf1 |
phase A load power factor |
8 |
lpf2 |
phase B load power factor |
9 |
lpf3 |
phase C load power factor |
gen3p
Col |
Name |
Description — see also |
|---|---|---|
1 |
genid |
unique 3-phase generator ID (positive integer) |
2 |
gbus |
bus ID |
3 |
status |
1 = in-service, 0 = out-of-service |
4 |
Vg1 |
phase A voltage setpoint in p.u. |
5 |
Vg2 |
phase B voltage setpoint in p.u. |
6 |
Vg3 |
phase C voltage setpoint in p.u. |
7 |
Pg1 |
phase A active power injection in kW |
8 |
Pg2 |
phase B active power injection in kW |
9 |
Pg3 |
phase C active power injection in kW |
10 |
Qg1 |
phase A reactive power injection in kVAr |
11 |
Qg2 |
phase B reactive power injection in kVAr |
12 |
Qg3 |
phase C reactive power injection in kVAr |
lc
Col |
Name |
Description — see also |
|---|---|---|
1 |
lcid |
unique 3-phase line construction record ID (positive integer) |
2 |
R11 |
resistance (p.u. per mile), element (1,1) of 3x3 matrix |
3 |
R21 |
resistance (p.u. per mile), elements (2,1) & (1,2) of 3x3 matrix |
4 |
R31 |
resistance (p.u. per mile), elements (3,1) & (1,3) of 3x3 matrix |
5 |
R22 |
resistance (p.u. per mile), element (2,2) of 3x3 matrix |
6 |
R32 |
resistance (p.u. per mile), elements (3,2) & (2,3) of 3x3 matrix |
7 |
R33 |
resistance (p.u. per mile), element (3,3) of 3x3 matrix |
8 |
X11 |
reactance (p.u. per mile), element (1,1) of 3x3 matrix) |
9 |
X21 |
reactance (p.u. per mile), elements (2,1) & (1,2) of 3x3 matrix |
10 |
X31 |
reactance (p.u. per mile), elements (3,1) & (1,3) of 3x3 matrix |
11 |
X22 |
reactance (p.u. per mile), element (2,2) of 3x3 matrix |
12 |
X32 |
reactance (p.u. per mile), elements (3,2) & (2,3) of 3x3 matrix |
13 |
X33 |
reactance (p.u. per mile), element (3,3) of 3x3 matrix |
14 |
C11 |
capacitance (nF per mile), element (1,1) of 3x3 matrix |
15 |
C21 |
capacitance (nF per mile), elements (2,1) & (1,2) of 3x3 matrix |
16 |
C31 |
capacitance (nF per mile), elements (3,1) & (1,3) of 3x3 matrix |
17 |
C22 |
capacitance (nF per mile), element (2,2) of 3x3 matrix |
18 |
C32 |
capacitance (nF per mile), elements (3,2) & (2,3) of 3x3 matrix |
19 |
C33 |
capacitance (nF per mile), element (3,3) of 3x3 matrix |
buslink
Col |
Name |
Description — see also |
|---|---|---|
1 |
linid |
unique 3-phase buslink ID (positive integer) |
2 |
busid |
bus ID of single-phase bus |
3 |
bus3pid |
bus ID of three-phase bus |
4 |
status |
1 = in-service, 0 = out-of-service |
Example Cases
The following cases are included in lib/t.
Name |
Description |
3-phase |
1-phase |
bus links |
|---|---|---|---|---|
4 bus 3-phase case |
4 |
– |
– |
|
6 bus hybrid case |
4 |
2 |
1 PQ |
|
6 bus hybrid case |
4 |
2 |
1 PV (1-phase side) |
|
6 bus hybrid case |
4 |
2 |
1 PV (3-phase side) |
|
5 bus hybrid case |
4 |
1 |
1 REF (1-phase side) |
|
21 bus hybrid case |
12 |
9 |
3 PQ |
|
21 bus hybrid case |
12 |
9 |
3 REF-PQ, PV-PQ, PQ-PQ |
|
21 bus hybrid case |
12 |
9 |
3 REF-PQ, PQ-PV, PQ-PQ |
The data for the 4 bus, three-phase system in t_case3p_a.m comes from 4Bus-YY-UnB.DSS, a modified version (with unbalanced load) of 4Bus-YY-Bal.DSS, the OpenDSS 4-bus IEEE test case with grounded-wye to grounded-wye transformer. [1]
The five and six bus cases connect this 4-bus three-phase case to 1 or 2 single-phase buses. And the 21-bus cases are based on connecting three copies of the 4 bus three-phase case to the single-phase case9.m.