Institute Instituteof ofAdvanced AdvancedEngineering Engineeringand andScience Science International Journal of Electrical and Computer Engineering (IJECE) Vol.
No.
June 2016, pp.
1344 Ae 1352 ISSN: 2088-8708.
DOI: 10.
11591/ijece.
Rotational Load Flow Method for Radial Distribution Systems Diego Issicaba* and Jorge Coelho** Department of Electrical Engineering.
Federal University of Technology - Parana (UTFPR).
Curitiba-PR.
Brazil Department of Electrical Engineering.
Federal University of Santa Catarina (UFSC).
Florianopolis-SC.
Brazil
Article Info
ABSTRACT
Article history:
Received Feb 5, 2016 Revised Mar 24, 2016 Accepted Apr 5, 2016 This paper introduces a modified edition of classical CespedesAo load flow method to radial distribution system analysis.
In the developed approach, a distribution network is modeled in different complex reference systems and reduced to a set of connected equivalent subnetworks, each without resistance, while graph topology and node voltage solution are Active power losses are then not dissipated in the modeled subnetworks and active power flows can be obtained as a consequence of radiality.
Thus, the proposed method preprocesses a series of variable transformations concomitant to an iterative algorithm using a forward-backward sweep to arrive at the load flow solution.
The proposed approach has been tested using literature and actual distribution networks, and efficiency improvements are verified in comparison to CespedesAo load flow method.
Keyword:
Power engineering Power distribution systems Load flow analysis Radial networks Coordinate rotation Copyright c 2016 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Diego Issicaba Department of Electrical Engineering.
Federal University of Technology - Parana (UTFPR) Av.
Sete de Setembro, 3165.
Sector D.
Rebouas, 80230-910 Curitiba-PR.
Brazil
55 41 3310-4626
issicaba@utfpr.
INTRODUCTION
Load flow analysis .
, .
provides the steady-state condition for power systems and is one of the most important numerical tools to system planning and designing.
In order to design solutions to power distribution system real-time operation .
, modeling and analysis might take into account unbalanced operation and detailed features of each component connected to the networks.
On the other hand, in long-term planning stages, some hypotheses and simplifications can be assumed .
balanced operation, constant power loa.
due to, for instance, uncertainties regarding future load profiles and distributed generation productions.
This is the case in adequacy evaluations .
, .
, where hundreds of thousands of load flow analysis may be executed, while performance indices are estimated by modeling failure/repair cycles of components and load/generation profiles as stochastic processes.
Detailed distribution system modeling and analysis is a well established topic, covered in books such as .
Simplified/single-phase load flow modeling and analysis dates mostly to the 80Aos and 90Aos and can be divided into two groups: the first group comprises Newtonian based methods adjusted to distribution system analysis .
, .
and the second group includes methods based on iterative forwardAebackward sweep processes .
, 10, 11, 12, 13, .
The sweep based techniques are known by their efficiency and take advantage of the fact that distribution networks usually are radially operated.
ForwardAebackward sweep methods employ the following general procedures: .
assuming a flat start or an approximate solution, current or power downstream each node is estimated in a backward sweep .
rom end nodes towards the substation nod.
using estimates from the previous step, node voltages are updated in a forward sweep .
rom the substation node towards end node.
these two steps are repeated up to the convergence of node voltages.
Among the several variations of sweep techniques, the one proposed by Baran and Wu .
, .
stands out by employing a set of equations, known as Distflow branch equations, that recursively relates active power, reactive power and voltage magnitudes.
Cespedes .
steps ahead by addressing the use of a biquadratic equation to relate node voltages as functions of downstream active and reactive power flows and immediate upstream node voltages.
This paper brings out the fact that, although CespedesAo method is efficient and robust to distribution system analysis, it involves a non-negligible amount of unnecessary calculations associated to updating active and reactive Journal Homepage: http://iaesjournal.
com/online/index.
php/IJECE ww ww ww .
i i aa ee ss j j oo uu rr nn aa l l .
cc oo m
IJECE
ISSN: 2088-8708
power flows in network branches.
In fact, a finite and relatively small number of standardized conductor types are utilized in each distribution feeder, composing subnetworks with the same r jx E/km and r/x ratio.
As consequence, given a possible iterative load flow solution, there is a constant relation between active and reactive losses at each subnetwork branch, making the concomitant effort of updating both, unnecessary.
The proposed approach, of the Rotational Load Flow Method, goes beyond CespedesAo formulation by performing generalized complex reference system rotations and by taking advantage of radiality to improve the efficiency of sweep procedures.
The approach has been tested using academic and actual electrical distribution networks, such that efficiency improvements are verified in comparison to CespedesAo load flow method.
The paper is organized into five sections as follows.
Section 2 and 3 present mathematical background and the proposed method, respectively.
Section 4 shows numerical results for different test systems and an actual system.
Conclusions and final remarks are outlined in section 5.
MATHEMATICAL BACKGROUND
This section presents a brief review of the load flow formulation developed by Cespedes .
A discussion about complex reference system rotation, focused on distribution system analysis and applications, is also approached.
CespedesAo Load Flow Formulation Consider the radial distribution system schematic shown in Fig.
1, where distribution lines are modeled as series impedances Zi = zi OIi = ri jxi while complex voltages and complex load demands are represented, respectively, by Vi = vi Oi and SLi = sLi OLi = PLi jQLi .
OAi = 1, .
N .
Note that the index i is utilized to identify the node and the line upstream this node, depending on the variables involved.
In addition, ui denotes the node upstream node i.
Vui
Zi 1 Vi 1
SLi
SLi 1
VN Oe1 ZN VN
? branch N ? SLN Oe1
SLN
Figure 1.
Radial distribution network schematic.
The complex power flow injected at node i through an upstream branch, denoted by Si = si Oi , can be written as a function of downstream node voltages, loads and line losses.
This power flow can be also interpreted as an accumulated power at node i given by 2 Si = SLi SLk or, separating in real and imaginary terms, 2 Pk Q2k Pi = PL i PLk Qi = QLi QLk Pk2 Q2k where OIi denotes the set of nodes downstream node i and the set of lines downstream node i, depending on the variables involved.
Once accumulated active and reactive powers at node i are known, the voltage magnitude at node i can be obtained by solving the biquadratic equation .
vi4 Ai vi2 Bi = 0 Ai = 2 (Pi ri Qi xi ) Oe vu2 i .
Bi = Pi2 Q2i ri2 x2i Similarly, the angle at node i can be obtained through the following equation.
Rotational Load Flow Method for Radial Distribution Systems (Diego Issicaba and Jorge Coelh.
ISSN: 2088-8708
i = iOe1 tan Oe1 Pi xi Oe Qi ri Pi ri Qi xi vi2 Thus, assuming that the substation node complex voltage V0 = v0 O0o is specifed and node voltage magnitudes are initiated using a flat start or an approximate solution, the load flow solution can be found through iterative forwardAebackward sweeps as follows.
In a backward process, accumulated power flow at each node is calculated using .
, starting at end-nodes and stopping at the first node immediately downstream from the substation node.
In a forward process, voltage magnitudes are updated, away from the substation node, using .
Ae.
These procedures are continually repeated until the convergence of voltage magnitudes is reached.
The backward process is the most time consuming part of the algorithm and involves the calculation of power losses and power accumulation through network branches.
However, a finite and relatively small number of standardized conductor types are installed in a distribution network, each one with a specified r/x ratio.
These ratios can be mathematically controlled through a proper reference system rotation, as it will be exposed in the following Complex Reference System Rotation Assume an impedance Zi represented in an alternative complex reference system rotated by i, as illustrated in Fig.
Eci Ec Zii = rii jxi Zi = ri jxi Eui Eu Figure 2.
Impedance represented into two different complex reference systems phased by a i angle.
The complex value of the impedance in the alternative reference system can be obtained by Zii = Zi eji = rii jxi rii = ri cos i Oe xi sin i, i = ri sin i xi cos i .
and the updated r/x ratio given by ri cos i Oe xi sin i i = ri sin i xi cos i for xi i 6= 0 Analogously, a complex apparent load in the alternative reference system is given by SLii = SLi eji = PLii jQi PLii = PLi cos i Oe QLi sin i.
Qi Li = PLi sin i QLi cos i .
Once all impedances and loads are rotated by .
, the system model is represented in the alternative reference system.
Hence, the complex power injected at node i through an upstream branch is also rotated by i, such Sii = Si eji = Pii jQi IJECE Vol.
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ISSN: 2088-8708
Qi i = Pi sin i Qi cos i Pii = Pi cos i Oe Qi sin i, .
Since the system model is represented in the alternative reference system, the load flow voltage solution is also rotated by i, though voltage magnitudes remain unaltered.
In fact, using .
Ae.
, the biquadratic voltage equation can be rewritten in the alternative reference system as i 2 ii )4 Ai i ) Bi = 0 Bii = (Pii )2 (Qi i i i i i 2 Ai i = 2 (Pi ri Qi xi ) Oe .
ui ) , .
ii )2 .
By substituting .
, as well as by assuming vuii = vui , we obtain that Ai i = Ai and Bi = Bi .
consequence vi = vi , causing that voltage magnitude solutions are preserved.
Once the same voltage value is chosen for the substation node in both reference systems, i.
v0i = v0 , the condition vuii = vui is always satisfied because of the order in which the nodes are visited in the ladder iterative proce- Therefore, a network represented in the alternative reference system and other represented in the conventional reference system present both the same load flow voltage magnitudes, independently of the rotation angle chosen.
addition, the load flow procedures will present the same voltage iterates in both reference systems.
One important application of reference system rotation is that a distribution network can now be represented in a coordinate system where the r/x ratios assume low figures, improving the Newtonian based load flow algorithms in terms of robustness and efficiency.
The proposed approach goes beyond such concept by introducing generalized complex reference system rotations to take advantage of distribution network radiality in CespedesAo ladder iterative ROTATIONAL LOAD FLOW METHOD This section concerns the proposed approach and is organized for didactic reasons into two cases: a case comprising networks with a single conductor type and a general case with different conductor types installed through distribution networks.
Single cable type case Consider a network with only one conductor type installed.
By hypothesis, complex impedances present the same r/x ratio and I angle.
As exposed previously, r/x ratios can be controlled by a complex reference system rotation.
For the general schematic shown in Fig.
1, by conveniently choosing the rotation angle i = A2 Oe I and by KirchhoffAos current law we obtain i O Vui Oe Vi = Zii = jzi Moreover, using .
, the rotated accumulated complex power at node i can be computed by Sii = SLi eji SLk e Zk e = SLii SLik Separating the real and imaginary terms, and denoting Li Qi as the rotated reactive network losses at branch i, we have Pii = PLii PLik Lk LQi i = QLi Qi = koIi Using the variable transformation, the accumulated active power at node i can be directly obtained summing rotated active load demands downstream from node ui .
Therefore, there will not be any active losses found and active Rotational Load Flow Method for Radial Distribution Systems (Diego Issicaba and Jorge Coelh.
ISSN: 2088-8708 power flow calculation through network branches becomes straightforward.
Consequently, load and network data can be stored in a convenient coordinate reference system and the sweep algorithm can now be formulated to concern only the calculation of reactive losses, as described in Fig.
1: Read node and network data, both rotated by i = A2 Oe I, using .
2: Initiate node voltages assuming vi = v0 .
OAi = 1, .
N .
3: Compute accumulated active power using .
at all network nodes.
4: while Changes in vi Ou tolerance.
OAi do for all i, following the backward direction do Calculate accumulated reactive power at network nodes using .
end for for all i, following the forward direction do Obtain node voltage magnitudes starting from the first node using .
Ae.
end for 11: end while 12: Print result reports.
Figure 3.
Rotational Load Flow algorithm Ae Single conductor type case with constant power load modeling.
The biquadratic equation becomes simplified by removing resistances from the formulation, as follows.
OeAi
i (Ai ) Oe 4Bi
1/2 !1/2
i i Ai i = 2xi Qi Oe vui , i 2 Bii = (Pii )2 Qi .
i ) .
By eliminating all these calculations, non-negligible CPU time can be saved.
General case Consider a general distribution network with lateral branches and voltage dependent load demands.
Notice that this network can be divided into a set of connected subnetworks with the same conductor type.
These subnetworks can be represented in a convenient coordinate system to eliminate the line resistances from modeling, as illustrated in Fig.
conductor type II conductor type I q Angle marks conductor type i Ne Ne Ne ejiI jiI,II conductor type IV jiI,i jiI,IV Figure 4.
Rotational Load Flow principle Ae General case.
By preserving the topological structure and load flow voltage solution, the power will flow through the subnetworks assuming different representations for each reference system.
Hence, the load flow formulation can be simplified to eliminate the active losses from modeling by mathematically handling the differences of coordinate reference systems among subnetworks.
This accomplishment is obtained through a generalized complex reference system rotation algorithm, shown in Fig.
5, in which stored data is changed and connections between subnetworks are assigned by angle marks.
The rotation procedures are identical for the case of modeling other series components by series impedances.
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ISSN: 2088-8708
1: Read node and network data.
2: Si = Si .
OAi = 1, .
N .
3: counter Ia 0.
4: Start a list u = u0 = .
with the first node immediately downstream of the substation.
5: while counter O N do for all node i OO u do ii Ia A/2 Oe Ii .
Sii Ia Sii ejii .
Zii Ia Zii ejii .
for all k OO OIi do Ski Ia Ski ejii .
Zki Ia Zki ejii .
end for counter Ia counter 1.
end for u Ia .
, such that m OO ui .
OAi OO .
14: end while Figure 5.
Generalized complex reference system rotation.
1: Read rotation angles, as well as the node and network data, both obtained by the the algorithm shown in Fig.
2: Initiate node voltages assuming vi = v0 .
OAi = 1, .
N .
3: while Changes in vi Ou tolerance.
OAi do Compute load demands using an appropriate voltage dependent model.
for all i, following the backward direction do Calculate reactive losses at branch i using .
if ii = 0 then Calculate accumulated active and reactive powers through .
Obtain accumulated active and reactive powers using .
, as well as a proper coordinate rotation through .
end if end for for all i, following the forward direction do Obtain node voltage magnitudes starting from the first node using .
Ae.
end for 16: end while 17: Print result reports.
Figure 6.
Rotational Load Flow algorithm Ae General case.
The Rotational Load Flow method for radial distribution networks is presented in Fig.
The approach utilizes a sweep algorithm and performs complex reference system rotations of accumulated powers whereas a subnetwork connections is assigned.
In the forward procedure, rotations are not required and the basic biquadratic equation is simplified with the removing of line resistances from modeling.
RESULTS AND DISCUSSIONS
The proposed approach was implemented in MATLAB using an old computer .
,66 GHz.
Core Du.
, and it has been extensively tested with distribution networks obtained from literature and with actual distribution networks.
For all simulations, tolerance was set up in 0.
000001 and initial voltage magnitudes were assumed 1 pu.
Voltage dependent loads were modeled using the polynomial form PLi = PL0i P P vi P vi2 .
QLi = Q0Li Q Q vi Q vi2 where PL0i and Q0Li are the rated active and reactive loads, respectively.
The coefficients (P .
P .
P ) and (Q .
Q .
Q ) were set in .
8, 0.
1, 0.
CespedesAo method and the Rotational Load Flow approach were applied to solve load flow problems in a 12node network .
, 27-node network .
, 29-node network .
, 32-node network .
and a 69-node network .
Rotational Load Flow Method for Radial Distribution Systems (Diego Issicaba and Jorge Coelh.
ISSN: 2088-8708 Table 1 shows simulation results achieved with these comparative case-studies.
Table 1.
Simulation results for the rotational and CespedesAo methods using distribution networks obtained in the Runtime .
Saving Iter.
Ref.
RLFM
(%)
CM: CespedesAo method.
RLFM: Rotational Load Flow method.
E : Ratio between the number of changes on conductor types and the number of network nodes.
The comparative studies indicate that the proposed approach provides superior results in term of efficiency when compared to CespedesAo technique.
In fact, calculations of active losses have been eliminated in the proposed approach and sweep procedures were adapted to handle differences of complex reference system representations.
Node voltages are equally updated in both methods following that the number of iterations necessary for convergence are also preserved in the proposed approach.
Also, it was not found any sacrifice in solution quality by using the rotation technique.
A similar case study was devised for an actual distribution network, shown in Fig.
This network is composed of 236 nodes, 1337 kilometers of lines and 5 conductor types installed.
Simulation results for this network are presented in Table 2.
Figure 7.
An actual radial structured distribution network.
Table 2.
Simulation results for the rotational and CespedesAo methods using an actual distribution network Runtime .
RLFM Iter.
Saving (%) CM: CespedesAo method.
RLFM: Rotational Load Flow method.
E : Ratio between the number of changes on conductor types and the number of network nodes.
Extensive simulations pointed out runtime improvements depend upon the ratio between the number of changes in conductor types across the network and the number of network nodes.
This fact was expected since a large number of changes in conductor type might increase the number of rotations during the backward procedures, as formalized in Appendix A.
All outcomes, either using literature or actual networks, indicate that the proposed method provides improved results in comparison to CespedesAo method in term of efficiency.
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No.
June 2016: 1344 Ae 1352
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ISSN: 2088-8708
CONCLUSIONS AND FINAL REMARKS
Adequacy evaluations usually require a large number of load flow computations to estimate a small set of performance indices.
These evaluations are utilized in long-term planning stages, where simplifications such as balanced operation and constant power load demands can be assumed, while efforts are directed to accurately modeling the failure/repair cycles of network components and load/generation profiles as stochastic processes.
Aiming at achieving improved efficiency in adequacy evaluations, a modified version of CespedesAo load flow method to radial distribution systems is proposed.
The approach eliminates unnecessary calculations in CespedesAo load flow method by representing distribution systems in different and conveniently chosen coordinate reference systems.
For this accomplishment, a generalized complex reference systems rotation algorithm was introduced and the load flow formulation was adapted to handle differences in coordinate reference system representations.
Result analysis obtained from both literature and actual networks indicate that the proposed method provide improved efficiency in comparison to CespedesAo method.
Future works will investigate extensions of the proposed approach with regard to unbalanced distribution networks, dispersed generation, and weakly-meshed distribution systems.
ACKNOWLEDGMENTS
The authors would like to acknowledge the financial, technical and human support of the CNPq.
CAPES and INESC P&D Brasil.
REFERENCES