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Transcript
1
TEMPUS OF ENERGY, Course: Distribution of Power Energy
Power flow analysis for distribution networks with DG
Jordan Radosavljević
Faculty of Technical Sciences, University of Priština in Kosovska Mitrovica, Serbia.
[email protected]
1. Introduction
Given all generators and loads in a system, a power flow calculation provides the voltages at
all the nodes in the system. Once these voltages are known, calculating the flows in all the branches
is straightforward. Power flow studies are simply the application of power flow calculations to a
variety of load and generation conditions and network configurations. The efficiency of the applied
power flow computation method in a radial distribution network is of an extreme importance. A fast
and efficient backward/forward sweep method for the computation of power flows is applied. It is
assumed that the distribution network is three-phase and balanced.
2. Branch numbering schemes
The solution method used for radial distribution networks is based on the direct application
of the KVL and KCL. To enhance the numerical performance of the backward/forward sweep
solution method, an efficient branch numbering scheme should be used. Figure 1 shows two typical
branch numbering schemes: (a) by branches and (b) by layers. These numbering schemes are very
simple and straightforward. The branch numbering scheme (a) has been implemented in the power
flow algorithm. The selection scheme does not affect the accuracy and speed of calculation.
Fig. 1 Branch numbering schemes: (a) by branches, (b) by layers.
3. Line modeling
The π equivalent circuit is most extensively used in practical applications. Figure 2 is the
nominal π equivalent circuit of the line or cable, between nodes m and i, where the serial impedance
0
Z Vi and shunt admittance Y Vi .
2
Fig. 2. The π equivalent circuit of lines and cables.
0
The lines in distribution networks are short and the shunt admittance Y Vi are usually
neglected in the power flow calculation.
4. Transformer modeling
To model the transformer the Г equivalent is used, as shown in Figure 3. Transformers are
modelled through the impedance Z Ti and the non-rated transmission ratio t  1  nt , where n is
an integer which represents the tap changer position. For manually controlled transformers n is
usually n  2 , and for transformers with automatic control n is usually equal to n  10  12 .
t could range 0.025, 0.015 or 0,0125. In the power flow calculations, the magnetizing admittance
o
of the transformer Y Ti are usually neglected.
Fig. 3. Equivalent model of transformer.
5. Load modeling
Behavior of loads can be modeled by studying the changes in their active and reactive power
requirements due to changes in system voltages, either analytically or experimentally. There are
three well established models for loads representation in power system studies:

Constant power: for which load active power and reactive power are assumed to be constant
irrespective of bus load voltage; which means they are usually voltage independent.
P  P0 , Q  Q0

(1)
Constant current: in which the active and reactive powers are assumed to be proportional to
the load bus voltage. Usually it represents the combination of constant impedance and
constant power loads.
3
V 
P  P0   ,
 V0 

V 
Q  Q0  
 V0 
(2)
Constant impedance: this is the traditional representation, in which, at system nominal
frequency, the active and reactive consumed powers are assumed to be proportional to the
square of supply voltage. The load is represented by a constant admittance added to the
system admittance matrix at the concerned load node.
2
V 
P  P0   ,
 V0 
V 
Q  Q0  
 V0 
2
(3)
where V0 represents the initial operating voltage and P0, Q0 are active and reactive power
corresponding to the initial operating voltage.
5.1 Uniformly distributed load lumped model
Fig. 4 shows the general configuration of the exact model of uniformly distributed loads.
The node at one-fourth of the way from the source end has been presented as a dummy bus.
Fig. 4. Exact lumped load model of uniformly distributed loads
6. Distributed generation modeling
The ways of connection of DG units to the grid are summarized as:

Wind turbines: The grid connected wind turbines are divided into fixed and variable speed
groups. In the first group, a propeller through a gear box rotates the rotor of a squirrel cage
induction generator, which is directly connected to the grid. In the second group, a doubly
fed induction generator or a synchronous generator, either permanent magnet or
conventional one, is used. The AC output power of these units is converted via a power
electronic based rectifier and inverter to grid compatible AC power.

Fuel cells: Fuel cells by electro-chemical process convert directly the stored chemical
energy in the fuel to electrical and thermal energy without any electric machine. The output
DC power of the fuel cell is converted via an inverter to grid compatible AC power.
4

Photovoltaic systems: Photovoltaic systems convert solar energy into electricity and like fuel
cells; their output DC power is converted via an inverter to grid compatible AC power.

Internal combustion engines: These units convert chemical energy derived from liquid or
gas fuels into mechanical one. Then it rotates either a synchronous or an induction generator
which is directly connected to the grid.

Gas turbines: These units convert the potential energy saved in fossil fuels from chemical to
heat and then heat to mechanical. Afterwards, it rotates a synchronous generator that is
directly connected to the grid.

Micro-turbines: These units work like gas turbines. The only difference is to rotate a highspeed permanent magnet synchronous generator. Hence, the generator is connected to the
grid via power electronic interface devices.
In accordance with the units mentioned above, the primary energy of DG units may be
injected to the grid via either a synchronous or an asynchronous electric machine which is directly
connected to the grid, a combination of an electric machine and a power electronic interface, or only
via a power electronic interface. If the electric machine is directly connected to the grid, its
operation determines the model of DG for power flow studies. In other cases, the characteristics of
the interface control circuit determine the DG model. These models are extracted in following.

Induction generator model: Generally, in an induction generator both active and reactive
powers are functions of slip:
P  P V , s 
Q  Q V , s 
(4)
where P and Q are produced active and reactive power, respectively, s is the slip of
induction generator speed, and V is the bus voltage. Assuming P is constant and neglecting
the very low dependency of reactive power to the slip, the expression (4) can be reduced as
follows:
P  Ps  const.
Q  f V 
(5)
The expressed model by (5), which is so-called Static Voltage Characteristic Model
(SVCM), is an appropriate model of squirrel cage induction generator for PFSs. Since the
bus voltages are near 1.0 p.u. in steady state cases, squirrel cage induction generator can be
modeled as a PQ bus for simplicity.
P  Ps  const .
Q  Qs  const

(6)
Synchronous generator model: Depending on the excitation system, synchronous generators
are divided into two categories: (A) with regulating excitation voltage and (B) with fixed
excitation voltage. The first one can also be subdivided into: (A1) voltage control mode
(constant terminal voltage) and (A2) power factor control mode (fixed power factor). The
DGs in sub-categories A1 and A2 are modeled by PV and PQ nodes, respectively. Consider
an example of a wound rotor synchronous generator to model the fixed excitation voltage
synchronous generators (category B). The following equation describes it:
5
 Eq
Q  
 Xd
2

V2
  P 2 
Xd

(7)
where, P and Q are the active and reactive power of the distributed generator respectively,
Eq is the no-load voltage and maintains constant, Xd is the synchronous reactance, V is the
generator terminal voltage. Assuming P is constant:
P  const.
Q  f V 
(8)
This expression is similar to (5), but Q is positive in expression (8), which means that
synchronous generators without excitation voltage regulation may inject reactive power to
the grid. Thus, the SVCM may be used as model of synchronous generators without
excitation voltage regulation too.

Power electronic interfaces: Generated power of photovoltaic systems, fuel cells, microturbines and some wind units are injected to the grid via power electronic interfaces. In such
cases, the DG model in PFSs depends on the control method which is used in the converter
control circuit. As a general rule, in case the control circuit of the converter is designed to
control P and V independently, the DG model shall be as a PV node and when it is designed
to control P and Q independently, the DG model shall be as a PQ node.
7. Power flow algorithm
The iterative algorithm for solving the radial system consists of four steps. These steps at
iteration k are:
1. Initialization: Let the root node be the slack node with known voltage magnitude and
angle, and let initial voltage for all other nodes be equal to the root node voltage.
Vi 0   1; θi0   0;
i  0,1, ... , N
(9)
2. Backward sweep to sum up line section current: starting from the end line section and
moving towards the root node, the current in line section i, in according with Fig. 5 is:
k 
k 
k 
 k 1
J i  I Pi  I Ci  Y i  V i
o

 J
k 

;
i  N, N-1, ... , 0; k  1,2, ...
 i
 i
Fig. 5. A part of the distribution network.
(10)
6
k 
k 
where: I Pi are current injections at node i corresponding to power load; I Ci are current
k 
are currents in line section  connected
injections at node i corresponding to capacitor; J 
to node i; Y i are sum of shunt admittances at node i;  i set of line sections connected to
o
k 1
node i; V i
are voltage at node i.
3. Forward sweep to update nodal voltage: starting from the first node and moving towards
the last node, the voltage at node i, in according with Fig. 5 is:
k 
k 
k 
Vi V m  Zi Ji
(11)
k 
k 
where: Z i is the impedance of branch i; V m , V i
k 
are voltages at nodes m and i; J i
is
the current in line i.
4. The voltage mismatches calculations: after above three steps are executed, the voltage
mismatch at each node (say node i) are calculated as below:
Vi k   Vi ( k )  Vi ( k 1) ,
i  0,1, ... , N
(12)
If any of these voltage mismatches is greater than a convergence criterion, the three steps are
repeated until convergence is achieved.
7.1 Incorporating DG units in power flow algorithm
Appreciation of DG in the power flow algorithm is achieved by modifying the expression
(10), according to Figure 6,
k 
k 
k 
k 
k 1
J i  I Pi  I Ci  I DGi  Y i  V i
o

k 
J
;
i  N, N-1, ... ,0; k  1,2,...
(13)
 i
 i
k 
where I DGi are current injections at node i corresponding to DG.
Fig. 6. A part of the distribution network with DG connected at node i.
The DG units, which are modeled as PQ nodes can be treated as negative PQ loads in power flow
algorithm without any problem:
7
k 
I DGi 
sp
sp
PDGi
 jQDGi
V
* k 1
i
sp
sp
PDGi
 const; QDGi
 const;
;
(14)
sp
sp
, Q DGi
where PDGi
are produced (specified) active and reactive power of the DG connected at node i.
However, handling PV nodes in the power flow algorithm requires some additional
processes. It is to be noted that the generator terminal voltage is typically controlled by the
specification of the scheduled voltage magnitude. So, for a PV node, the active power output and
voltage magnitude of the generator are specified. In order to handle PV nodes in a power flow
algorithm, the backward/forward sweeps are performed considering them as negative PQ loads.
When the power flow is converged the following three steps are done:
1. Calculate voltage magnitude mismatch for all PV nodes
k 
Vi k   V i  V i
sp
sp
where V i
(15)
is the specified voltage magnitude for node i. If any of these mismatches is
greater than a threshold, then perform the next step:
2. Calculate current injections at node i corresponding to DG,
k 
I DGi 
sp
k 1
PDGi
 jQDGi
* k 1
Vi
(16)
k 
The reactive power injection QDGi
required to maintain the voltage at the generator bus i on
the specified magnitude, can be calculated using:
  
1
k 
(k)
(k 1 )
QDGi
 QDGi
 Im V i  Z PVi  V i  V i
sp
sp
 
*
(17)
where: QDGi is the reactive power injection of the DG connected at node i; V i is the

calculated voltage at node i, V i  Vi  i
V
sp
i
 Vi sp i

;

;
sp
Vi
is the specified voltage at node i,
Z PVi is equal to the sum of the complex impedances of all line sections
between the PV node i and the root node (substation bus).
In case there are n PV nodes in the distributin network, the reactive power injections at these
PV nodes are determined by vector equation:
 
1

(k 1 )
Q(k)
DG  Q DG  Im V  Z PV  V  V
sp
sp
k 
 
*
(18)
where:
Q DG  QDG1 , QDG 2 ..., QDGn  is the vector of the reactive power injections at PV nodes;
V is the vector of the calculated voltages at PV nodes;
V
sp
is the vector of the specified voltages at PV nodes;
Z PV is the PV node sensitivity matrix. The dimension of Z PV is equal to the number of PV
nodes. The diagonal entry, Z PVii , in Z PV is equal to the sum of the complex impedances of
all line sections between PV node i and the root node (substation bus). If two PV nodes, i
8
and j, have completely different paths to the root node, then the off-diagonal entry Z PVij is
zero. If i and j share a piece of common path to the root node, then Z PVij is equal to the sum
of the complex impedances of all line sections on this common path.
3. QDGi then is compared with the reactive power generation limits. If QDGi is within the
limits, ie.,
min
k 
max
QDGi
 QDGi
 QDGi
then the corresponding DG currents, are injected to PV node i according to (16). In
subsequent iterations, these currents will be combined with other nodal current injections.
Otherwise, if QDGi violates any reactive power generation limit, it will be set to that limit,
and combined with the reactive load at this node. Subsequently, the rows and columns in the
PV node sensitivity matrix, Z PV , corresponding to this node are removed and the LU
factors of Z PV are updated.
The iteration described in steps 1-3 will continue until the voltage magnitude mismatches
for all PV nodes as calculated in (15) become less than a threshold.
9
8. Example
The described power flow algorithm is applied on a radial distribution network shown in
Fig. 7. Data for branch and loads of the distribution network are reported in Table 1 and Table 2,
respectively.
Fig. 7. Distribution network.
Table 1. Branch data.
Branch
0–1
1-2
2-3
3-4
4-5
4–6
3–7
3–8
2–9
1 – 10
R
(p.u.)
0.000963
0.001073
0.006409
0.005106
0.007506
0.002467
0.003005
0.001143
0.002760
0.002106
X
(p.u.)
0.003219
0.000673
0.004609
0.002876
0.004229
0.001390
0.002612
0.000994
0.002399
0.001187
B
t
0
0
0
0
0
0
0
0
0
0
0.9875
Table 2. Load data.
Node
Type
0
1
2
3
4
5
6
7
8
9
10
BAL
PQ
PQ
PQ
PQ
PV
PQ
PQ
PQ
PQ
PQ
V
(p.u.)
1
/
/
/
/
1.01
/
/
/
/
/
PG
(p.u.)
/
0
0
0
0
2
0
0
0
0
0
QG
(p.u.)
/
0
0
0
0
(-1 ÷1)
0
0
0
0
0
Pp
(p.u.)
/
0
0.522
0.936
0.336
0
0.477
0.672
0.207
1.116
0.882
QP
(p.u.)
/
0
0.174
0.312
0.112
0
0.159
0.224
0.069
0.372
0.294
Sbase = 1 МVA, Vbase = 23 kV.
Table 3 shows the results for two cases: (а) With DG at node 5, (б) Without DG at node 5.
10
Table 3. Results of the power flow calculations.
(а) With DG at node 5
(б) Without DG at node 5
Node Voltages:
-------------------------Node
V
theta
[r.j.]
[deg]
1.0000
1.0035
-0.4811
2.0000
1.0000
-0.4738
3.0000
0.9909
-0.2749
4.0000
0.9957
0.0387
5.0000
1.0100
0.5711
6.0000
0.9943
0.0230
7.0000
0.9883
-0.3382
8.0000
0.9906
-0.2823
9.0000
0.9960
-0.5688
10.0000
1.0013
-0.5055
11.0000
1.0000
0
Node Voltages:
-------------------------Node
V
theta
[p.u.]
[deg]
1.0000
1.0018
-0.8551
2.0000
0.9962
-0.9320
3.0000
0.9746
-1.3156
4.0000
0.9696
-1.3735
5.0000
0.9696
-1.3735
6.0000
0.9681
-1.3900
7.0000
0.9720
-1.3811
8.0000
0.9743
-1.3233
9.0000
0.9921
-1.0277
10.0000
0.9996
-0.8796
11.0000
1.0000
0
Currents in Branch:
-------------------------Branch
!I!
fi
[p.u.]
[deg]
1.0000
3.7047 -30.7409
2.0000
2.8023 -34.6261
3.0000
1.2275 -56.9930
4.0000
1.2314 -160.6165
5.0000
1.9836 -176.0484
6.0000
0.5057 -18.4119
7.0000
0.7167 -18.7732
8.0000
0.2203 -18.7173
9.0000
1.1811 -19.0037
10.0000
0.9285 -18.9404
11.0000
3.7516 -30.7409
Currents in Branch:
-------------------------Branch
!I!
fi
[p.u.]
[deg]
1.0000
5.5178 -19.5966
2.0000
4.5877 -19.6538
3.0000
2.8497 -19.7889
4.0000
0.8847 -19.8181
5.0000
0
0
6.0000
0.5194 -19.8249
7.0000
0.7288 -19.8160
8.0000
0.2239 -19.7582
9.0000
1.1857 -19.4626
10.0000
0.9301 -19.3145
11.0000
5.5876 -19.5966
Power Flow:
-------------------------Branch
P
Q
[r.j.]
[r.j.]
1.0000
3.2113
1.8735
2.0000
2.3190
1.5732
3.0000
0.6675
1.0169
4.0000
-1.1568
0.4061
5.0000
-2.0000
0.1181
6.0000
0.4770
0.1590
7.0000
0.6720
0.2240
8.0000
0.2070
0.0690
9.0000
1.1160
0.3720
10.0000
0.8820
0.2940
11.0000
3.2245
1.9177
Power Flow:
-------------------------Branch
P
Q
[r.j.]
[r.j.]
1.0000
5.2346
1.7761
2.0000
4.3282
1.4669
3.0000
2.6343
0.8801
4.0000
0.8137
0.2714
5.0000
0
0
6.0000
0.4770
0.1590
7.0000
0.6720
0.2240
8.0000
0.2070
0.0690
9.0000
1.1160
0.3720
10.0000
0.8820
0.2940
11.0000
5.2640
1.8741
Power Loss in Branch:
-------------------------Branch
ploss
qloss
[p.u.]
[p.u.]
1.0000
0.0407
0.1359
2.0000
0.0253
0.0159
3.0000
0.0290
0.0208
4.0000
0.0232
0.0131
5.0000
0.0886
0.0499
6.0000
0.0019
0.0011
7.0000
0.0046
0.0040
8.0000
0.0002
0.0001
9.0000
0.0116
0.0100
10.0000
0.0054
0.0031
11.0000
0
0
Total active power loss: 0.230 [p.u.]
Total reactive power loss: 0.254 [p.u.]
Power Loss in Branch:
-------------------------Branch
ploss
qloss
[p.u.]
[p.u.]
1.0000
0.0902
0.3015
2.0000
0.0678
0.0425
3.0000
0.1561
0.1123
4.0000
0.0120
0.0068
5.0000
0
0
6.0000
0.0020
0.0011
7.0000
0.0048
0.0042
8.0000
0.0002
0.0001
9.0000
0.0116
0.0101
10.0000
0.0055
0.0031
11.0000
0
0
Total active power loss: 0.350 [p.u.]
Total reactive power loss: 0.482 [p.u.]
11
References
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