MODIFIED NEWTON-RAPHSON LOAD FLOW ANALYSIS FOR INTEGRATED AC/DC
POWER SYSTEMS
A. Panosyan, B. R. Oswald
Institute of Electric Power Systems, University of Hannover, Germany
ABSTRACT
The significant increase in planned offshore wind parks and the tendency towards large parks in considerable distances
offshore, make the well established HVDC technology a favorable solution for the connection of these large & distant
offshore wind parks to the main power grid onshore. It is therefore necessary to adequately model the HVDC transmission links and integrate them in the load flow analysis of the complete a.c.-d.c. system. In this paper, the well known
Newton-Raphson method for the load flow analysis is modified to achieve compatibility for a.c.-d.c. systems with integrated d.c. links in the a.c. network. The elements of the residual vector and the Jacobian matrix for the a.c. network are
kept unchanged and are merely complemented by a new vector and a new matrix, which represent the modifications due
to the d.c. link. The modified Jacobian equation includes the d.c. real and reactive power at the a.c.-d.c. buses, and their
dependency on the a.c. system variables. The modified Newton-Raphson method is evaluated on an a.c.-d.c. test system
with a load flow computation in MATLAB and the results are presented
Keywords: Power Systems Modeling, HVDC, Load Flow, Newton-Raphson
1 INTRODUCTION
The significant increase in planned offshore wind parks
and the tendency towards large parks in considerable
distances offshore poses a new challenge to grid operators. Although, present pilot farms with limited capacity can be connected to the main grid onshore relatively
easily and inexpensively using conventional a.c. transmission, connecting larger wind farms further offshore
is considerably more difficult. The main disadvantage
of an a.c. connection to these remote offshore parks is
the great amount of charging current produced by long
a.c. submarine cables due to the high capacitance of
a.c. cables, which reduces the active current-carrying
capacity of the cable and requires reactive power compensation. Thus, the greater the distance of these large
wind farms to the grid onshore become and the higher
the transmission voltage is, the more convenient technical and economical solution High Voltage Direct
Current (HVDC) transmission becomes for connecting
distant large offshore wind parks to the main grid.
Although the new Voltage-Sourced-Converter (VSC)
based HVDC technology allows higher flexibility over
the conventional Current-Source-Converter (CVC)
based HVDC and an independent reactive power control in addition to other advantages, the VSC transmission technology is still in its early stages and is still
economically not feasible at power levels of 500MW
or more due to the high cost of multiple converters and
cables that are required. The conventional HVDC will
therefore be the more logical choice at the present for
the connection of the planned offshore wind farms to
the main grid.
These large wind farms connected to the main grid
through two-terminal HVDC links represent a challenge for the simulation of analysis of the overall a.c.d.c. power system. Therefore, these new network elements should be appropriately modeled and represented in the different simulation programs. In this
paper, the adequate modeling of a two-terminal HVDC
link and its integration into the well known NewtonRaphson method for the load flow analysis is looked
into, taking into account the control strategies of the
HVDC converter stations.
2 GENERAL CONSIDERATIONS
The simplest way of integrating a d.c. link into the a.c.
load flow is representing it by constant active and reactive power injections at the two terminal buses in the
a.c. systems. Thus the two terminal a.c.-d.c buses are
represented as a PQ-bus with a constant voltage independent active and reactive power. However this is
clearly an inadequate representation where the links
contribution to a.c. system reactive power and voltage
conditions is significant, since the accurate operating
modes of the link and its terminal equipment are ignored.
A more advanced and accurate method is representing
buses connected to an HVDC as a PQ-bus with a voltage dependant active and reactive power. However, the
voltage dependency of the active and reactive power at
these a.c.-d.c. buses do not obey the general rules of
the conventional PQ-buses in a.c. systems. We have
thus a new type of PQ-bus, which we define as PQDC-
bus. In our approach the real and reactive power equations for the PQDC-buses, with their dependency on
both the a.c. voltages at the terminal buses and the
characteristics of the d.c. converters and their control
strategies, are derived and integrated into the a.c. load
flow algorithm.
3 MODELLING THE D.C. LINK
For the analysis of the steady state converter operation,
some basic and generally valid assumptions can be
made. Firstly, the a.c. voltages at the terminal buses are
perfectly balanced and sinusoidal. Thus, a perfect a.c.
filtering of all harmonic currents and voltages generated by the converters is assumed. Correspondingly, a
perfect filtering on the d.c. side too is assumed and the
d.c. current and voltage contain no a.c. components.
Furthermore, the losses and magnetizing admittance of
the two-winding converter transformers are ignored.
This approach is a combination of the two main methods most a.c.-d.c. load flow algorithms are based on:
the simultaneous solution method and the sequential
solution method. Unlike the sequential method, in
which successive iterations between the a.c. load flow
and d.c. the load flow are taken, the a.c. and d.c. equations are all solved in the same iteration step. Nevertheless the d.c. variables are not explicitly included in the
state vector as in the simultaneous solution method. In
the Newton-Raphson method for the load flow analysis, with the basic Jacobian equation
J ∆x = y
A basic schematic diagram of a two-terminal HVDC
link interconnecting buses "r" (rectifier) and "i" (Inverter) is illustrated in Figure 1. The symbols appearing
in the diagram are defined as follows:
U = primary line-to-line a.c. voltage (r.m.s.)
U d = direct voltage
I d = direct current
n = transformer ratio
P, Q = active and reactive power
(0)
this leads to modifications on the elements of the residual vector ( y ) and of the Jacobian matrix ( J ) affiliated to the a.c.-d.c. buses, without adding new elements
to the residual vector and the Jacobian matrix. In this
approach, the real and reactive power and the a.c. voltages at the converter buses are considered the interface
between the a.c. and d.c. equations in each iteration
step.
The basic converter equations, for both rectifier and
inverter operations, describing the relationship between
the a.c. and d.c. variables can be written as follows [1].
Rectifier equations
A further point to be determined is whether the a.c.
network components which are attached to the twoterminal HVDC links should be considered as part of
the a.c. system or included in the d.c. system equations.
These components are usually the a.c. filters and the
converter transformers. The a.c. filters can be included
in the a.c. network as fixed real and reactive power
injections at the a.c. buses connected to the HVDC
link. The tap-changing converter transformers are however included in d.c. network, since the tap-changing of
the transformers is an integral part of the d.c. control
systems. Consequently, the primary sides of the transformers are chosen as the interface buses between the
a.c. and d.c. systems. Leaving the transformers out of
the a.c. network helps getting around the need to update the bus admittance matrix of the a.c. network with
each tap-changing.
Id
Ur
Pr + jQr
Ir
U d0r = k nrU r
U dr = U d0r cos α r − Rr I dr
where U d0r
is the ideal no-load direct voltage,
k = 3 2 π , and α r is the ignition delay angle. Rr is
the so called equivalent commutating "resistance",
which accounts for the voltage drop due to commutation overlap and is proportional to the commutation
reactance, Rr = 3 X r π .
The active power at the rectifier is given by
Pr = U dr I dr
(4)
Since losses at the converters and transformers can be
ignored ( Pr = Pac ), the reactive power at the rectifier
Rl
ni
nr
U dr
(2)
(3)
U di
Ii
Ui
Figure 1 Two terminal HVDC link interconnecting buses "r" and "i"
Pi + jQi
can be determined as
Qr = Pr tan ϕ r
(5)
where ϕ r is the phase angle between the a.c. voltage
and the fundamental a.c. current, and by neglecting the
commutation overlap can be calculated as.
ϕ r = arccos(U dr U d0r )
(6)
Inverter equations
The inverter operation of a converter can be correspondingly described by the following equations
U d0i = k niU i
(7)
(8)
(9)
(10)
(11)
U di = U d0i cos γ i − Ri I di
Pi = U di I di
Qi = Pi tan ϕi
ϕ i = arccos(U di U d0i )
where γ i is the extinction advance angle.
Line equation
The interdependence of the two d.c. voltages can be
expressed by
U dr = U di + Rl I d
(12)
with the d.c. line resistance Rl .
Based on eqs. (3),(8) and (12), the equivalent circuit of
the two terminal HVDC link is shown in Figure 2.
Rr
U d0r cos α r
Id
U dr
− Ri
Rl
U di
U d0i cos γ i
Figure 2 HVDC equivalent circuit
4 D.C. POWER EQUATIONS AND
MODIFICATIONS ON JACOBIAN EQUATIONS
Equations (4),(5),(9) and (10) represent the PQDC-buses
in the a.c. system. The active and reactive power at the
PQDC-buses depend on the a.c. bus voltages, the characteristics of the converters and their operating modes.
To determine the dependency of P and Q on the a.c.
bus-voltages for a given pair of converters, four control
specifications are required. These are usually the direct
current or the d.c. power at one of the converters, the
nominal d.c. voltage at one of the converters (usually
the inverter), and the optimum value of the control
angles, which are usually the minimum control angles
α min and γ min , since the ignition delay angle and
extinction advance angle are kept as close to their
minimum value as possible to keep the reactive power
consumption of the converters at its minimum[2].
With the given converter parameter and control specifications and with the latest updated a.c. bus voltages at
each iteration step, the transformer ratios nr and ni
corresponding the optimum value of the control angles
are calculated [3]. However, critical operation conditions may result in one or both converter transformer
ratios reaching their upper or lower limits. When one
of the transformers reaches a limiting ratio, the inverter
d.c. voltage (and subsequently the rectifier d.c. voltage) is freed and readjusted, keeping the direct current
fixed. And when both transformer ratios reach their
upper and lower limits respectively, α or γ is freed,
depending on whether the rectifier transformer ratio or
the inverter transformer ratio has reached its maximum
level, since α and γ can not fall below their minimum value. These five operating states of the converters are summarized in Table 1. In case the ignition and
extinction angles are to be kept fixed at their minimum
values, the direct current can be freed instead. [2]
Once the operating state of the converters and the corresponding transformer ratios are determined, the transformer ratios need to be readjusted to match one of the
actual tap position of the transformers, since the tap
ratios can vary only in steps. This is done by rounding
the calculated ratio to the next highest tap position,
ensuring that the control angles, which are correspondingly readjusted too, remain over their minimum values.
The active and reactive d.c. power at each converter,
corresponding to the operating state obtained above,
can now be obtained through d.c. power equations of
the rectifier and inverter. Thus, the residual vector y
in eq. 1 is modified as follows
y = yac + ydc
(13)
where yac is the residual vector for the a.c. network
without the d.c. link, and ydc is the modification vector
due to the active and reactive d.c. power at the converter buses, and thus contains zero elements at all
other buses in the a.c.-d.c. network.
Finally, the Jacobian matrix elements associated with
the converter buses are to be modified. The polar coordinates form of the Jacobian matrix is the more appropriate choice, since four of the eight elements associated to each converter are then zero. This is due to the
independence of the d.c. active and reactive power
Table 1 Converter operating states and the a.c. voltage dependency of d.c. active and reactive power
γ
nr
ni
Pr
Pi
Qr
α
Op. State
nr,min < nr < nr,max ni,min < ni < ni,max const. const.
α min
γ min
const.
State 1
Qi
const.
State 2
α min
γ min
nr,max/min
State 3
α min
γ min
nr,min < nr < nr,max
ni,max/min
f (U i )
f (U i )
f (U i )
f (U i )
State 4
α min
f (U r , U i )
nr,max
ni,min
f (U r )
f (U r )
f (U r )
f (U r , U i )
State 5
f (U r , U i )
γ min
nr,min
ni,max
f (U i )
f (U i )
f (U r , U i )
f (U i )
nr,min < nr < nr,max
f (U r )
f (U r )
f (U r )
f (U r )
from the phase angles of the terminal a.c. buses ( δ r
and δ i ). Hence, only the partial derivatives of P and
Q in respect to the a.c. bus voltages U r and U i are to
be calculated. In order to do this, the dependency of the
d.c. power on a.c. voltages, for the actual operating
state of the converters and transformer ratios, should
first be established.
The five possible operating states and the corresponding a.c. voltage dependency of the d.c. power are demonstrated in Table 1.
x = x0
y : ∆p, ∆q
J:H M N L
NO
YES
y = yac + ydc
J = J ac + J dc
When the residual vector and the Jacobian matrix are
modified for the HVDC link, the Jacobian equations
are then solved and the a.c. voltages and angles updated. For the next iteration step the updated a.c. voltages at the terminal buses are used to determine the
adequate operating state of the converters, and consequently the modifications on y and J , and so on till
convergence is obtained (Figure 3).
x = x + ∆x
To test the algorithm, a load flow program with the
integrated a.c.-d.c. algorithm has been written in
MATLAB. The program is applied on an a.c. test system with an integrated two terminal HVDC link. The
9-bus test system is thus modified by replacing the
transmission line from bus 4 to bus 5 with a hypothetical d.c. link (Figure 4). The fictitious characteristics of
the d.c. link are given in Table 1. The a.c. filters are not
taken into consideration here, but could be included in
the a.c. network. The rest of the system is the same as
the 9 bus system.
Update
state vector
Check convergence
NO
max( ∆x ) < ε
YES
Output Results
Figure 3 AC/DC-Newton-Raphson load-flow algorithm
G2
T2
2
T3
8
~
5 APPLICATION ON A TEST SYSTEM
Corresponding
modifications to
y and J
Solve
Jacobian equation
(14)
The modification matrix J dc is dependent on the operating state and is a zero matrix for state 1.
Form residual vector y
and Jacobian matrix J
DC Links?
After determining the converter operating state and
formulating ydc , the corresponding partial derivatives
of P and Q are calculated and the Jacobian matrix is
modified too
J = J ac + J dc
"Flat" start
G3
~
7
9
5
3
6
4
T1
1
~
G1
Figure 4 9-bus test system with integrated d.c. link
Table 2 Characteristics of the d.c. link
Rectifier
Bus Number
4
5.5 Ω
Commutation reactance
Minimum control angle
α = 7D
min
Transformer regulation range
Number of tap positions
Resistance of the DC line
Rated d.c. power at inverter
Rated d.c. voltage at inverter
±15%
27
Inverter
5
6.5 Ω
γ min = 10D
±15%
21
0.5 Ω
13MW
300 kV
Convergence within a tolerance of max( ∆x) = 0.001 is
achieved in 5 iteration steps, which is one step more
than the load flow for the 9 bus system without the d.c.
link. The a.c. and d.c. load flow results are given in
Table 2 and Table 3 respectively.
Table 3 a.c. load flow results
Bus
Voltage
Angle
Nr.
[kV]
[degree]
1
17.160
0.000
2
18.450
2.140
3
14.145
0.179
4
238.566
− 2.297
5
195.928 − 15.032
6
234.483
− 5.290
7
228.048
− 3.612
8
228.264
− 5.516
9
235.923
− 2.535
P
[MW]
− 74.834
− 163.00
− 85.000
12.976
112.024
90.000
0.000
100.000
0.000
Q
[MVar]
− 6.460
− 63.108
− 0.698
1.804
52.355
30.000
0.000
35.000
0.000
Table 4 d.c. load flow results
d.c. voltage [kV]
Control angles
Transformer tap position
Real power [MW]
Reactive power [MVar]
d.c. current [kA]
power is supplied from generator 2 instead, and thus
the increase in Q at bus 2.
To reduce the voltage drop at bus 5, VAr-compensators
would be used at the bus. The voltage drop could be
kept under 10% by compensating at least 30% of the
reactive load at the bus.
Rectifier
Inverter
299.497
299.476
7.596
10.000
0.938
1.150
12.978
12.977
1.804
2.355
0.0433
It can be clearly seen in the results that the inverter bus
5 sustains a considerable voltage drop due to the
HVDC link failing to transport the reactive power
consumed at the node from generator 1. The reactive
CONCLUSIONS
This paper presents a method for including d.c. links in
the Newton-Raphson a.c. load flow algorithm through
simple modifications only on the elements of the residual vector and the Jacobian matrix, which are associated with an a.c.-d.c. bus.
The advantages of this method over the sequential and
the simultaneous solution methods respectively are,
omitting a separate d.c. iteration at each a.c. iteration
step, and a simpler modification of the a.c. load flow
algorithm without the need to reshape the Jacobian
equation.
REFERENCES
1.
2.
3.
Kundur, P., Power System Stability and Control,
McGraw-Hill, 1994
Arrillaga, J., Bodger, P., Integration of h.v.d.c.
links with fast-decoupled load-flow solutions,
Proc. IEE, Vol. 124, No. 5, May 1977
Sanghavi, H.A., Banerjee, S.K., Load flow analysis of integrated a.c.-d.c. power systems, Fourth
IEEE Region 10 International Conference , 22-24
Nov. 1989
AUTHORS ADDRESS
The first author can be contacted at
Institute of Electric Power Systems
University of Hannover
Appelstr. 9A
30167, Hannover, Germany
email: panosyan@iee.uni-hannover.de