Download Hybrid power flow analysis : Combination of AC and DC

Hybrid power flow analysis
: Combination of AC and DC models
Soobae Kim, Student Member, IEEE and Thomas J. Overbye, Fellow, IEEE
Department of Electrical and Computer Engineering
University of Illinois Urbana-Champaign
Urbana, IL 61801 USA
Email : [email protected] and [email protected]
Abstract—This paper presents a hybrid approach with AC and
DC power flow models for power flow analysis. Power flow
problems are solved with the AC model in a part of system,
whereas with the DC model in other parts. The proposed
methodology can obtain benefits from both AC (non-linear) and
DC (linear) models. In other words, this approach formulates
both nonlinear and linear algebraic equations, and deals with
tradeoff issues between accuracy of the AC model and faster
solution of the DC model. A case study with an IEEE 14-bus
system is provided to compare accuracy between the proposed
and the DC model. It is shown that the hybrid approach can be
used for applications requiring more accuracy than DC model
allows.
Index Terms—Power system analysis, Power flow model, AC
power flow, DC power flow
I. INTRODUCTION
Power flow analysis is used to solve steady-state operating
conditions of electric power systems. It is the most essential
tool for power system analysis and is applied to planning,
economic scheduling, system operation and control areas.
Therefore, it has been constantly required to develop the
efficient power flow analysis method with the widespread of
computer technologies [1] and especially, as the increase of
online application for wide area monitoring and control in the
field, it might be more stressed to develop the reliable and
efficient power flow methodology than before.
The classic power flow problem (AC model) can be
formulated based on Kirchhoff’s circuit law in electric power
networks and is a set of nonlinear algebraic equations
representing real, reactive power injection flow at each node.
The nonlinear equations can be solved with an iterative
algorithm. The Newton-Raphson (NR) method is the most
powerful algorithm using a linearizing technique. However,
because of inherent characteristics of nonlinear equations, the
AC model has convergence difficulties and is time consuming,
especially when contingency analysis is considered [2].
To overcome these problems, several approximate models
using physical properties of power systems have been studied
[3-6]. Decoupled and DC power flow models are representative.
The decoupled model is based on the observation that the
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interactions are weak between real power and voltage
magnitude and between reactive power and voltage angle. This
weakness allows related elements of the Jacobian matrix to be
ignored, and thus the computation expense for the power flow
solution can be reduced [5].
The DC model additionally assumes that voltage magnitudes
at all buses are flat and reactive power flows are neglected. This
further approximation of the DC model creates a linear problem
and thus it has a very fast solution. According to [7], the DC
model is 60 times faster than the AC model. Thanks to the fast
solution, the DC power flow model has become a commonly
used analysis technique in power systems and market
applications. Examples include contingency analysis,
calculation of power transfer distribution factor (PTDF), line
outage distribution factors (LODF), transmission interchange
limit analysis, market clearing engine (MCE) and financial
transmission right (FTR) [7].
However, because of inherent approximations, the DC model
may cause large errors in these applications [6]. In [8],
guidelines are presented to reduce the amount of error. Lu, et al.
[7] proposed method to enhance DC model with a correction
term, based on historical data while still maintaining its
computational
efficiency
and
linear
formulation.
Notwithstanding much effort, precision of the DC method is
fully dependent on configuration and operating conditions of
individual power systems. Thus it is crucial to carefully
implement DC model.
This paper presents a hybrid approach by combining the AC
with the DC models in the problem formulation. Both the time
requirements of the AC model and the inaccuracies of the DC
model are reduced. Thus, the hybrid model is stronger than
either the AC or DC models operating alone. The paper is
organized as follows. Section II presents a brief analytic basis
of the power flow problem. The proposed approach is presented
in Section III. And Section IV illustrates the results from the
IEEE 14-bus system. The conclusion is made in Section V.
II. FORMULATION OF THE POWER FLOW PROBLEM
This section briefly explains the formulation of the AC and
the DC power flow models.
A. AC power flow model
AC power flow equations can be formulated with four
variables at each bus: voltage magnitude, angle, and real and
reactive power injection. At each bus, two of four above
variables are known and the remaining two are obtained by
solving a set of nonlinear power flow equations. Based on nodal
analysis and with specified constraints on power and/or voltage
at bus k, the complex power can be derived as:
Sk = Pksp + jQksp = Vk I k * = Vk [∑YknVn ]*
(1)
As these complex equations are not analytic functions of the
complex voltages, it is necessary to separate their real and
imaginary parts in order to apply the NR method. Depending on
bus types: load or generator bus, the power problems are as
follows:
At load bus (PQ) k,
Pk = Vk ∑Vn [Gkn cos(θk − θn ) +Bkn sin(θk − θ n )]
(2)
Qk = Vk ∑Vn [Gkn sin(θk − θ n ) − Bkn cos(θk − θn )]
(3)
At generator bus (PV) k,
Pk = Vk ∑Vn [Gkn cos(θk − θn ) +Bkn sin(θk − θ n )]
(4)
Vk = Vksp
Pk = ∑ Bkn (θ k − θn )
(6)
The real power system does not always match these
assumptions. So, [6] analyzed the assumptions in order to apply
the DC power flow model to a real field. To contain error to be
within 5% on average, the paper reports that the ratio (X/R)
should be greater than 4 and the standard voltage deviation
should be less than 0.01. In addition, it showed that the
assumption of a small angle difference is quite reasonable.
However, even following the guidelines, some line power
flows have significant inaccuracy compared to AC results.
III. PROPOSED METHODOLOGY
The proposed approach formulates the power flow problem
by combining the AC with the DC models. In other words,
power flow problems are solved with the AC model for certain
buses and the DC model for others. Power flow equations
which require accurate solution are formulated with the AC
model and the rest are done with the DC model for a faster but
less accurate solution. Then, NR method can be used to solve
the reduced set of nonlinear equations. Figure 1 shows the
procedure.
(5)
Where Pk / Qk is a real/reactive power injection at bus k,
Ykn = G kn + jB kn is an element of the system admittance matrix,
Vk and Vn are the voltage magnitude of the two end buses of a
line, and θk / θn are the voltage angles at bus k/n respectively.
The NR algorithm is a common method to solve a set of
nonlinear equations based on the Taylor series expansion. It is
characterized by quadratic convergence. Most problems with
the NR method converge around five iterations [1]. The
computation time is fully dependent on system size, and a
single solution for a large system, such as NERC 43,000, may
take as long as a few seconds [2].
B. DC power flow model
The DC power flow model simplifies the AC model with the
following assumptions:
1. Voltage magnitudes on all buses are 1 p.u.
2. Voltage angle differences are small:
sin(θk − θn ) ≈ θk − θn , cos(θ k − θ n ) ≈ 1
3. Line resistance is negligible:
Rkn << X kn , Gkn ≈ 0
4. Reactive power injections on all buses are ignored.
Based on the above assumptions, the real power flow in (2)
can be approximated as:
Figure 1. Procedure for the hybrid approach
From this approach, we hope to take advantages from both
the AC and the DC models. Accuracy in areas of interest can be
maintained with exact nonlinear equations, and computation
expense can be reduced with approximate linear equations.
This could result in an improved power flow model with a
faster solution without sacrificing accuracy in areas of interest.
A. AC and DC Area selection
The area can be divided depending on the importance of a
certain section with a viewpoint of power system operation and
control. This work calls the important area the AC region and
the other the DC region. Some buses in the DC region which
have a connection with buses in the AC region are called BD
buses (Boundary bus). Figure 2 shows the AC/DC/BD division
of a simple power system. If we assume that Buses 1, 2 and 3
are in the AC region, then Buses 4 and 6 may be BD buses,
which are connected with one of buses in the AC region. Bus 5
is in DC region.
13
12
14
6
9
C
11
5
10
G
C
1
4
2
7
8
3
G
C
Figure 3. IEEE 14-bus system
Figure 2. AC, BD and DC area selection
B. Power problem formulation and boundary assumption
For the problem formulation, this approach assumes that
each bus in the AC region has real and reactive power injection
or voltage magnitude information depending on the bus type.
The buses in the DC region have only real power injection as a
given condition. Then power flow equations in the AC and DC
regions can be formulated with (2), (3), (4), (5) and (6).
The BD bus has a crucial impact on the accuracy of the states
in AC region because they are fully dependent on each other in
power flow equations. But, the given condition for a BD bus is
only real power injection. Therefore, a proper guess for reactive
power injection or voltage magnitude of the BD bus is required
to improve accuracy in AC region. A best guess may be a flat
voltage approximation based on the power system operation
and control. In order to obtain accurate angle solution in the BD
bus, this work builds nonlinear real power flow balance
equations for the BD bus. In other words, the approach assumes
that the BD bus is considered to be a PV bus in AC region with
1 p.u. voltage magnitude.
All line power flows which are larger than 1 MW are tabulated
in Table 1. In order to compare the power flow errors, the mean
absolute percentage error (MAPE) of the active line power flow
are calculated by taking AC results as a reference:
MAPE =
From To
PAC − Ptest
× 100%
PAC
Real power flow (MW)
AC
DC
(9)
MAPE (%)
Hybrid
DC
Hybrid
1
2
159.6 147.9
157.9
7.4
1.1
1
5
75.3
71.1
74.5
5.6
1.1
2
3
75.9
70.1
75.5
7.7
0.6
2
4
55.7
55.2
55.2
0.8
0.9
2
5
41.1
40.9
40.4
0.4
1.6
3
4
-21.1 -24.1
-21.5
14.5
1.9
4
5
-60.0 -62.3
-60.6
3.9
1.0
4
7
28.0
29.6
27.9
5.8
0.6
IV. CASE STUDY
4
9
16.2
17.2
16.1
6.2
0.5
A case study has been conducted on an IEEE 14-bus system,
which has 2 generators, 3 synchronous condensers and 20
transmission lines. The test system is slightly modified by
removing the reactive power limit of the generators and
synchronous condensers. Fig. 3 shows a one-line diagram of the
test case system.
For the hybrid approach, buses 1, 4, 5 and 14 are selected for
the AC region and, according to the network configuration,
buses 2, 3, 6, 7, 9 and 13 are the BD bus and the remaining are
in the DC region.
5
6
44.1
45.2
42.2
2.3
4.5
6
11
7.4
6.3
6.0
14.3
18.3
6
12
7.8
7.5
7.3
3.3
6.1
6
13
17.8
17.0
17.6
4.2
0.8
7
9
28.0
29.0
27.9
3.4
0.6
9
10
5.2
6.2
4.5
18.3
13.9
9
14
9.4
9.9
9.9
5.0
5.1
10
11
-3.8
-2.8
-3.3
25.8
13.8
13
14
5.6
5.0
5.2
11.9
8.6
7.8
4.5
Average
Table 1. Line power flow test results
The average MAPE of all lines is 7.8% for the DC power
flow model and 4.5% for the hybrid approach. It is noted that
the mean error in the AC region is 2.4%. This is a 70%
reduction compared to the result from the DC model. Therefore,
this approach can give results in the areas of interest similar to
the approach with AC model in entire system.
V. CONCLUSION
This paper explores the development of power flow
algorithms for combining an accurate AC model with an
approximate DC model. This approach results in an improved
power-flow algorithm which has a fast solution without
sacrificing accuracy in areas of interest. The test results show
the hybrid approach is more accurate than using the DC model.
This approach can be utilized in a variety of power system
applications and could be a basis of further investigations for
the study of power system analysis.
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