A Fuzzy Load Allocation Method for Distribution Expansion Planning
Maria Teresa Ponce de Leão
INESC – Instituto de Engenharia de Sistema e Computadores Pr. da República 93, 4050 Porto, Portugal
mleao@inescn.pt
Abstract
The operational distribution network, short and medium term, planning focuses on the analysis of network evolution under the guidelines set from the long-term planning. The major difficulty that turns out on this analysis is the massive data associated to these systems when compared with transmission and generation systems.
Moreover, the actual tendency of the independent production growth, in the MV network, introduces additional problems.
This approach presents a methodology to evaluate node data in order to feed a SCADA for distribution and uses a fuzzy methodology to minimise allocation errors.
Node injections are estimated taking into account the related uncertainty that derives from basic uncertainty together with uncertainty introduced by the method. Final results consist on the estimated injections for a given time period and on an evaluation of the configuration concerning operational limits.
1 Introduction
The electric distribution network consists, most of times, on a meshed network that operates radially. The system operator chooses the radial configuration to cope with adequate level of quality and reliability in the supply of node injections. Medium voltage (MV) distribution networks are characterized by a great number of nodes and branches. The nodes are the injection points of the network: consumer points (loads) and generation from independent producers. This means that the node injections can be positive or negative either if they are loads or independent generators. In what concerns the mathematical model both injections will be treated the same way.
The evolution of SCADA - Distribution Management Systems (DMS) [1] together with on line acquisition of field and existent stored data offer powerful distribution analysis tools. These recent facilities allow improvements in system profitability and flexibility provided that the software tools support on complete and reliable data. This leads to operating and long-term plans that are flexible, reliable and adequate to the competitive market rules.
Unfortunately, the uncertainty related to data is dealt with using moderate common sense and experienced knowledge. The results, from these definitions, are more or less scientific guesses, from the analysts, often without supporting on any systematic procedures.
This contribution presents a systematic fuzzy load allocation methodology that estimates injections to the MV nodes supporting on existent information which, most of times, consists on sparse measurements in the network (mainly in the outgoing feeders from primary substations)
and on technical data related to nodes. The system process will calculate on line active and reactive flows under any configuration.
To accomplish our objective we support on a methodology that bases on existent data and takes advantage from experience and additional available knowledge to model uncertainty associated to injections.
The available information will be taken into account when estimating loads. In the cases were information is scarce or missing, the module will assume default values and build membership functions to include the related basic uncertainty. The construction of the membership functions will support hierarchically on sets of rules driven from a legal basis, technical limits, estimated values from energies and available profile of consumption. The same kind of procedure will be applied to reactive values.
The distribution network dispersed generators will be treated as negative loads and related data will be modelled considering legal and technical rules as well as external influences from natural resources. The membership functions will be based on their actual rated power, on on- line measurements as well as on additional information.
The models presented in [2] will be applied. The description of the rules that lead to the evaluation of node injections (active and reactive power) will be presented in section 2.
In section 3 we describe a two step fuzzy allocation procedure. In the first one a rough allocation evaluates loads independently from the actual network configuration The process, fully described, will successively allocate the injection points with less uncertainty. After the primary load allocation is completed, one will have a set of provisional active and reactive fuzzy loads attributed to the nodes of the network. This set, however, will not be coherent with the Kirchhoff laws governing power flows on a network, because the active and reactive losses in the circuits have not been considered. In the second step an adjustment process is applied to explicitly consider the system’ active and reactive losses minimising the impact of the actual configuration on the losses evaluation. The final results provide the analyst with the fuzzy injections estimation prepared to feed any subsequent analysis tool.
To evaluate present network performance, in what concerns robustness for actual demands, an index of robustness will be provided.
2 Fuzzy Injections Model
Load estimation is difficult to model: most of times data is inexact or uncertain and probabilistic methods are not adequate as historical data is not available.
Nevertheless, uncertainty is often too large to be omitted.
Knowledge associated to data is vague and often derives from approximate reasoning and imprecision of natural
judgements about "typical loads". To include this vagueness in technical calculations of electricity distribution networks we use fuzzy models. These models are built to capture and join together the injection point’s available information.
Fuzzy numbers can assume different shapes nevertheless, once it facilitates mathematical operations, we approximate exact membership functions to triangular or trapezoidal ones without loosing The main related information.
The possibility functions seem appropriate to model qualitative typical estimations as: “about 20” or even
“more or less 100, but not more then 120 and not less then 80”. When representing a fuzzy class of consumption, to take into account the qualitative and vague nature of statements such as “industrial type”, we obtain the basic fuzzy load curve giving information about possible load values expected at each time step [7]. Performing this operation to every time step of the basic fuzzy load curves we obtain fuzzy load diagrams (figure 1). Each time step is associated with a possibility distribution that maintains the original fuzzy shape. We can work on any time step, using expected fuzzy load, for instance to the peak hour.
p
t
µp(t)
Fig. 1 - Fuzzy load diagram for 7 time steps
Concerning the producers from natural resources a fuzzy trapezoidal model is used either for hydro and wind power plants. These models inference takes into account technical and legal rules, additional information about regularity of primary energy source and expert’s declarations. The figure 2 presents some of those models for mini-hydro plants and wind plants. Details of the inference procedure can be seen in [3].
The mini-hydro rated power and installed turbines’
must be known in advance for models’ construction.
Turbines without regulation are supposed to be installed in regions were a regular flow can be expected in order to allow to get economical benefits from the investment.
Turbines with regulation are less restrictive.
Wind power is far more volatile than small hydro power, due both to the intrinsic irregularity of wind and to the impossibility of storage. So, the possibility of having no power at all is high, and there is only a small possibility of generating the rated power. Typically, output values between 0.3 and 0.4 p.u. are the ones with the highest possibility.
0 0.5 1
0 0.2 0.4 0.6 0.8 1 1.2
Generated Power (p.u.)
possibi
Generated Power (p.u.)
possibi
0 0.5 1
0.0 0.2 0.4 0.6 0.8 1.0 1.2
mini-hydro with regulation mini-hydro without regulation
Generated Power (p.u.) 0
0.5 1
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Wind power generator
Fig.2 Fuzzy models for different kinds of independent producers
In the figure 2 a general example of the basic fuzzy models of independent producers is presented. A mini- hydro with regulation, a mini-hydro without regulation and the basic model for a wind power generator are.
Details of the building up of these models can be seen in [3].
If extra information about natural sources is available we can get more accurate models composing basic models with linguistic declarations like those presented in set (4):
{
most regular; very regular; regular;}
little regular; very little regular (1) The final possibility distribution results from the composition of the basic models with the appropriate function from figure 3 using the rules of fuzzy composition [4].
The models obtained this way are too detailed for mathematical calculations. To be more easily included in mathematical models the possibilities distributions are modified and approximated to trapezoidal ones.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.00 0.20 0.40 0.60 0.80 1.00
M VR R LR VL
Fig. 3 - Qualifiers of flow regularity
This approximation was obtained after composing the distribution possibility corresponding to the mini-hydro without regulation, presented in figure 2, with the flow qualification most regular (curve MR from figure 3). Low values of possibility lessen, and the resulting distribution is more "narrow", meaning that we are more certain about the true value of flows than in the base case.
Generated Power (p.u.)
possibi
0 0.5 1
0 0.2 0.4 0.6 0.8 1 1.2
Fig. 4- Trapezoidal Approximation of a modified fuzzy generation
Independent producers from traditional resources will be modelled using a similar procedure to the presented above to model loads.
3 Operational Model
3.1 Rough Allocation
The injection points are the MV network nodes, which consist on the aggregated loads on secondary substations, either public or private, and on the production from dispersed generation. These injection points were grouped into five different types as a function of available data. The groups are defined as follows:
1 - Groups the loads with least available data (minimum data). Related information will consist on the rated kVA of installed (transformer) capacity. The uncertainty related to this group will be modelled using the experts’ information and experience. This data corresponds to the minimum required one.
2 – Consists on groups of loads with information from the billing services (energy) or registered measurements, for instance the cases, namely on private secondary substations, where peak power for different periods is registered. This data can have a triple, double or single period of time basis corresponding for peak, empty and full hour average power. Once again we can define an interval or a triangular fuzzy number to model the uncertainty.
3 – Consists on loads with the level of information of group 2 plus estimated load diagrams. This data has a 24 hour time basis. The figure 1 presents a 7 period diagram and the dark and grey areas as represented define a region of uncertainty. For each time period uncertainty can be defined as represented by the trapezoidal fuzzy number.
4 – Loads having information from actual on line measurements, available from the SCADA such as power measurement or current and power factor measurement at a given point of the network. This information is introduced as deterministic.
5 – Groups the independent producers, either from conventional or renewable sources. This kind of injections will be treated as negative loads associated with legal and inference rules for this kind of producers.
The first phase of the process consist on the construction of the fuzzy models as suggested in previous section. The second phase develops into 3 steps. The first one consists on determining all independent trees (sub- trees with a root measurement) to be studied separately.
Each sub-tree will be treated as a group 4 load when concerning mother-tree.
For each sub-tree and time schedule (for on-line studies the time schedule will be the hour of the study), we start the process, as already mentioned, by allocating, successively, loads with less uncertainty. We begin calculating the average power per group 2 and 3, Proot.ave2 and Proot.ave3, based on the record of energy consumption.
Next we estimate maximum fuzzy power based on (P~root.peak ) maximum power for each load at the study time hour. In that case of loads with unknown peak this will be estimated using a factor introduced by the operator, kload.rated., and used as default.
[ ]
∑
∑
+
=
peak unknown
peak known
loads
with loads all
rated . load rated . load with
all
peak . load peak
. root
P ~ . k
P ~ P ~
(2)
In the next step the root ratio is evaluated. This ratio will result from the defuzzyfication of the fuzzy value obtained in (3). For the purpose we apply the method presented in [5] by Lee and Li.
peak root
ave root ave
root PE ratio
root
P
P k P
.
3 . 2 . .
.
~
~
~ = ~ +
(3)After obtaining the deterministic value for the root.ratio.PE we are able to calculate estimated power for loads 2 and 3. Next step will be to calculate the average power for loads 1 after subtracting the estimated values for loads 2 and 3 using (3).
This process will be iterative till we attain convergence that is; the sum of all allocated loads equals the root measurement. This must be followed for all the roots (sub-trees with valid measure) we can find in the network. Note that independent producers, injection from group 5, will be dealt with using the described procedure applied to the related models presented in section 3, and considering the power flow of opposite sign.
∑
=
1 class
loads . .
. 1
.
. ~
~
all
peak load PE
ratio root ave
root
k P
P
(4) The reactive allocation will follow the same steps.
The main difference consists on the much less amount of available data. In this case, when there is no information about the power factor, it will be estimated based on typical load power factor and on legal rules imposed to the consumers.
3.2 Robustness and Severity Concept
As loads and dispersed generation are estimated in advance by possibility distributions (fuzzy injection in nodes), constraints of maximum branch flows and
maximum node's voltage drop must be met. This leads to the introduction of the concept of robustness [6,8] that is illustrated in figure 5.
0.2 0.4 0.6 0.8 α
0 1
I(A) 0
0.2 0.4 0.6 0.8 1
0 I(A) 0.2 0.4 0.6 0.8 1
I(A) β=0.22 β=0
α
Imax=100 120 Imax=100 140
α
β=0.22 0.2
0.4 0.6 0.8 α
Fig. 5 - Illustration of robustness and severity A given network element is robust regarding a specific constraint if the constraint holds true for every possible value of the uncertain variables and constants. In that case β=1. Otherwise, if some instances of those quantities lead to violation of the constraint, βelement equals the maximum possibility value for which the constraint is not violated. In figure 5 the element of left is not robust while the elements of right have a robustness equal to 0.22.
A global index, βConfiguration, to evaluate network configuration concentrates this robustness information, allowing the planner to get information about plans that face the risk of not being able to accommodate load demands.
element element ion
Configurat
min β
β =
(5)From an operational point of view this index allows the planner, that must establish in advance the guidelines for operation, to have an overall network information and proceed, in the second phase, with a supervised search for robust configurations.
1.3 Adjustment Considering Losses
After having estimated the injections, for the configuration under analysis, as described, we are able to include system losses that were omitted in the rough allocation.
Fig. 6 - Network reduction.
Once again and for operational reasons, (note that this process should give results in real time), we introduce a simplified procedure. Suppose we know the estimated (aggregated) load on node j from figure 5. That’s the situation after estimating the injections on all nodes. The objective of any distribution network will be the operation under minimum voltage drops. This allows us to state, without introducing severe errors, that node voltage level will assume a value around 1 p.u. If we know the legal limits we can state that: it is possible that node voltage is not less than 0.95 p.u. and not more than 1.05 p.u.. This linguistic declaration can be modelled as presented in figure 7.
We can also state that, at this point, voltage argument is not very important for estimation purposes. Distribution networks, on account of the operation conditions, present very small angles for the voltage phase.
0 0.2 0.4 0.6 0.8 1
V(p.u) )
α
0.95 1.0 1.05
Fig. 7 - Fuzzy uncertainty related to voltage level With this premises we are able to estimate load losses for each branch. The procedure of evaluating losses will be applied from root to leaves, separately for active and reactive losses, using equation (6 and 7).
(
2 2)
2
~ ~
~ 1 .
~ r P x Q
U P
j chij
lossesbran
= ⋅ + ⋅
(6)
(
2 2)
2
~ . ~
~ 1
~ x P r Q
Q U
j chij
lossesbran
= ⋅ + ⋅
(7) As long as losses are calculated, estimated injections will be adjusted following a top down process for each sub-tree. Nevertheless we cannot be sure that, after this approximated procedure, fuzzy flow will be coherent with data and some adjustments must be made. A corrective procedure will follow. As the sum of loads and losses will not be equal to the measure value for the sub-tree in analysis, remaining or lacking power will be redistributed in a proportional ratio to the allocated loads.
1.4 Fuzzy Load Flow
The traditional structure of the primary distribution system, although meshed, is almost always radially operated. Although the load flow algorithms applied to interconnected systems can be applied, we applied a radial flow algorithm more suitable for on-line answers claimed by SCADA systems. This load flow can be solved by a simple iterative method were some simplifications can be applied. To accomplish the load flow studies we supported on the method presented in [9] applied to triphasic balanced network and extrapolated it to be dealt with using fuzzy flows.. This fuzzy load flow for radial networks will be applied iteratively till we reach the convergence limits imposed by the operator.
1.5 Inclusion of Loops
As the flexibility and complexity associated to distribution systems grow, available tools must cope with stronger standards of exigency. Systems must be prepared to face the new electricity market rules where quality and security must not be forgotten. The tendency is that utilities sometimes need to operate the network using loops. This situation turns the network configuration into a weekly meshed one.
i j
As the rough allocation is independent from configuration, yet for this case we propose the presented procedure. For the corrective process to include losses we propose that radial independent network trees should be studied the way suggested and the remaining looped parts analysed with traditional fuzzy power flow.
Results for this case are not yet available.
4 Case Study
The effectiveness of the proposed methodology is illustrated through a small example. The case study consists on a small radial network with 20 independent loads and 9 nodes. The network configuration is presented in figure 8 and node injections and group is presented in Table I.
TABLE I – Load Data
Load Rated power (p.u) Type Available data for the hour of study (p.u.)
load01 1,6 1 …
load02 0,5 1 …
load03 0,8 1 …
load04 1,6 1 …
load06 0,9 5 …
load07 0,08 1 …
load08 0,25 2 0,2
load09 0,5 3 0,12
load10 1,6 1 …
load11 0,07 1 …
load12 0,03 1 ..
load13 0,8 1 ..
load14 0,5 3 0,12
load15 0,8 1 ..
load16 0,25 1 ..
load17 0,02 3 0,02
load18 0,9 2 0,08
load19 0,08 1 …
load20 0,25 2 0,2
load21 0,5 3 0,12
We also have as data a valid measure on node 0 equal to 6+j2 p.u. From the loads we have information about the group they belong and the rated power as presented in table I. We will use the fuzzy proposed models to capture uncertainty.
G1 groups loads 1 and 2, G2 load 3 and 4, G3 load 7 and 8, G5 load 13 to 20 and G7 load21 tan φ is estimated 0.4.
G1 G7
G7 5
G2 G3
G4 G5
G6
0
1
2 3
4
6 8
7 9
Fig 8 – Network configuration
Loads are represented by triangular possibility distributions obtained from nominal given power. For purposes of simplification we considered a triangular membership function with left and right values respectively equidistant from central value.
The tables II, III and IV group the results obtained with the proposed procedure respectively for the fuzzy loads, line fuzzy flows and losses.
Table II presents the fuzzy injections, for the case study network. This fuzzy injections were obtained using the methodology presented in section 2.
TABELA II – Load results
Node P(p.u) Q(p.u.)
load01 1,25000 1,25000 1,25000 0,50000 0,50000 0,50000 load02 0,08039 0,09475 0,11318 0,01576 0,01897 0,02219 load03 0,58643 0,60777 0,66988 0,11496 0,12313 0,13131 load04 1,25000 1,25000 1,25000 0,50000 0,50000 0,50000 load06 0,69470 0,69460 0,69480 0,01363 0,01362 0,01364 load07 0,09924 0,09925 0,09925 0,01946 0,01945 0,01947 load08 0,15274 0,16332 0,17417 0,01838 0,02219 0,03202 load09 0,08039 0,09475 0,11318 0,01576 0,01897 0,02219 load10 1,25000 1,25000 1,25000 0,50000 0,50000 0,50000 load11 0,02010 0,06664 0,11318 0,00394 0,01300 0,02219 load12 0,02346 0,02513 0,02680 0,00460 0,00493 0,00525 load13 0,05864 0,06282 0,06699 0,01150 0,01231 0,01313 load14 0,08039 0,09475 0,11318 0,01576 0,01897 0,02219 load15 0,58643 0,60777 0,66988 0,11496 0,12313 0,13131 load16 0,14333 0,16332 0,17417 0,01838 0,02219 0,03202 load17 0,09378 0,10348 0,11318 0,01838 0,02029 0,02219 load18 0,69470 0,69460 0,69480 0,01362 0,01363 0,01364 load19 0,09924 0,09925 0,09925 0,01945 0,01946 0,01947 load20 0,15274 0,16332 0,17417 0,01838 0,02219 0,03202 load21 0,08039 0,09475 0,11318 0,01576 0,01897 0,02219 Table III presents the fuzzy flows on the lines and also the robustness evaluated for all the lines. As it can be seen in line 1-2 power capacity limits are surpassed. The operator should or the system should purpose another robust configuration.
TABLE III - Fuzzy results for the allocation procedure
line P(p.u.) Q(p.u.) Lim ββββ
45 1,330 1,344 1,368 0,515 0,518 0,522 2 1 46 1,836 1,857 1,919 0,614 0,623 0,631 2 1 48 0,251 0,262 0,273 0,037 0,041 0,051 2 1 13 1,373 1,436 1,503 0,524 0,536 0,549 2 1 12 1,850 1,926 2,038 0,218 0,239 0,272 2 0,5 47 0,694 0,694 0,694 0,013 0,013 0,013 2 1 49 0,080 0,094 0,113 0,015 0,018 0,022 2 1
TABLE IV - Fuzzy results for the losses Losses(p.u.)
line p(p.u.).10-3 q(p.u.) 10-3
45 4,19000 1,68000
46 4,19000 1,68000
48 0,00015 0,00006
13 0,00473 0,00189
12 0,00168 0,00068
47 0,00000 0,00000
49 0,00006 0,00006
Table IV presents the results for line losses allocated.
5 Conclusions
Today the automatic information systems provide the tools for analysis with a considerable amount of information. Aside with this jump in quality engineers must be able to develop calculation tools that take advantage of the quality of available information to enhance the profits we expect from the results.
This approach allows obtaining the estimated loads with reference to a certain instant. At the same time it characterises the present topology in terms of robustness taking into account the uncertainty that affects the estimation and allowing the operator to deal with risk in an objective way.
As a summary we can say that the data for injections in the existent distribution analysis tools is in general based on common sense and experience, due to the lack of precise information. This contribution models the uncertainty of non-probabilistic nature using possibilistic membership function together with inference rules. This procedure will be adequate to feed system's data either for on line or planning studies and allow the planner to be able to take his decisions taking into account uncertainty including risk and robustness analysis.
6 References
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Contributions, Subject area: 6 (Study and planning of Supply Systems), Birmingham, Junho 1997.
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[5] S.-J. Chen, C.-L. Hwang, Fuzzy Multiple Attribute Decision Making - Methods and Applications, LNEMS 375, Springer-Verlag, 1992.
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[7] Fidalgo J. N., Matos M. A., Ponce de Leão M. T.,
“Assessing Error Bars in Distribution Load Curve Estimation”, Proceedings of the 7th International Conference on Artificial Neural Networks, ICANN´97, Lausanne, 1997
[8] Matos, M. A., Ponce de Leão, M. T, “Electric
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