Complete System(i.e. Power Flow Eq.)

Secondary System

Results (X) Index (X)

Full Monte CarloSimulation

Full Monte Carlo Simulation

Indexing of therandom input

~10.000 NRmodel calls

~100.000 NR model calls(3 quantiles of interest)

Random input variable X

Transformation

~350 million NRmodel calls

Figure 1: CDF of the nodal voltage varying the DER’s active power

Figure 2: Exemplary flowchart of the proposed method

Common Rank Approximation – A new method to

speed up probabilistic calculations in distribution grid

planning

Marco Lindner, Rolf Witzmann

Associate Professorship Power Transmission Systems

Technische Universität München

Germany

marco.lindner@tum.de

Abstract—Probabilistic considerations of uncertainties in

distribution grid planning is one of the keystones to avoid

oversizing of grids and equipment due to unreal worst-case

assumptions. However, the computational costs to integrate

uncertainties in calculation processes reach from worth

mentioning to unfeasible. In this paper, a new method is

introduced, which consists of a simplification and a rank

comparison process and yields high accuracy as well as low

computational costs. Various DER control strategies and

nonlinear network equipment can be integrated and time

savings of up to a factor of 9000 can be reached.

Index Terms—Power distribution, Power systems analysis

computing, Power systems planning, Renewable Energy

Sources, Uncertainty

I. INTRODUCTION

Uncertainties in current or future states of distribution grids are mostly unconsidered in today’s planning processes. In case of additional decentralized energy resources (DER), the grid is designed for the worst case instead of real scenarios. This often leads to an exponential increase of the costs, even though the scenario is most likely not to occur. To consider DER uncertainties, their random positions and rated powers have to be analyzed.

The most basic and straightforward way to integrate them is to compute a significant amount of simulations varying the random variables. This is known as Monte Carlo Simulation (MCS). Probability functions are retrieved and specific quantiles are evaluated for decision making. Different authors recommend 10.000 variations of the random vector to obtain stochastically acceptable results [1], [2]. Fig. 1 shows an exemplary cumulative distribution function (CDF) of the voltage at the feed-in node with 10.000 probabilistic variations of the feed-in power. The simulation took about 30 seconds. The most common quantiles used for numerical comparisons (median, 5%, 95%) are marked in green and red. If the intention is to compare yearly values, like energy losses or yearly reactive energy (VAR-hours), 10.000 time series

simulations have to be done. Assuming a time step of 15 minutes, the corresponding computational effort rises to an infeasible amount of ~350 million NR model calls with a computing time of about 122 days.

In the last decades, different methods have been proposed to reduce the simulation effort by analytical as well as statistical means. These reach from classic convolution methods [3], [4] to advanced concepts like moments and cumulants [5], [6]. All of these have in common, that they are trying to construct the complete output distribution function. However, for comparison and further processing of the results, only specific quantiles of interests are stated. The motivation behind the proposed method is, whether it is possible to only calculate the quantiles of interest instead of the complete CDF. For this purpose, the variation of the input variable with regard to its possible impact on the output is assessed and indexed (Fig. 2).

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Figure 3: Required correlation between Z and Y

II. GENERAL IDEA

Let X be the uncertainty vector of both system A with the output Y and system B with the output vector Z. System B (secondary system) is a mathematically simplified version of system A with a significantly reduced computational burden. When thinking of successfully indexing the influence of X on the output Y using the simplified system B, it can be stated that sequences of Y and Z have to be identical.

rank rank

𝑋 = [123 ] ∶ 𝑌(𝑋) = [

164 ]

132

𝑍(𝑋) = [102011 ]

132

The similarity in sequences can be numerically evaluated using Spearman’s Rank Correlation Coefficient (SRCC), which is defined as Pearson’s Correlation Coefficient between two ranked variables. When yielding the described property, vector Z can be used to determine probabilistic samples out of X, which will generate certain quantiles of Y. Because of this property, the proposed method is named Common Rank Approximation (CRA).

A. Mathematical Requirement

The mathematical requirement to be hold by Y and Z can be stated as the monotonic increase of Z over Y. Analytically, the first partial derivative of Z with respect to Y should be of positive sign according to (1).

𝜕𝑍(𝑋)

𝜕𝑌(𝑋)> 0

𝜕𝑍(𝑋)

𝜕𝑌(𝑋)∙𝜕𝑋

𝜕𝑋=𝜕𝑍(𝑋)

𝜕𝑋∙ [𝜕𝑌(𝑋)

𝜕𝑋]

−1

Four examples of a valid correlation between Z and Y are shown in Fig. 3. Note, that possible correlations are diverse, not only linear, which is a major advantage of the CRA-method. Focusing on the ranks of Y and Z, system B does not need to approximate the real output Y nor be any physical representation of system A, which allows it to be simplified as much as possible.

III. APPLICATION IN MODERN DISTRIBUTION GRID

PLANNING

As already mentioned, today’s planning processes do mostly not include uncertainties. In the context of extensive integration of DER, the amount of plants as well as their rated powers and positions in the grid are of topmost importance when it comes to the determination of nodal voltages, losses, reactive power needs, grid reinforcements and many more.

A. Scenario

Considering the planning phase of a low voltage distribution grid with an unknown future distribution of PV-plants, the uncertainty vector X of length n becomes

𝑋 = [𝑃𝑟,𝐷𝐸𝑅,1, 𝑃𝑟,𝐷𝐸𝑅,2, 𝑃𝑟,𝐷𝐸𝑅,𝑖 , … , 𝑃𝑟,𝐷𝐸𝑅,𝑛],

with 𝑃𝑟,𝐷𝐸𝑅,𝑖 representing the rated DER power at the node ‘i’.

The capacity to be installed is fixed according to an extension plan. Note, that X changes for each probabilistic variation.

B. Defining an objective quantity

A full Monte Carlo Simulation results in CDFs of all quantities available in the system. These quantities react differently on changes in X. To correctly model the secondary system B, an objective quantity has to be chosen and analyzed. As an example, the yearly reactive energy (VAR-hours) exchanged at the slack bus between low and medium voltage as well as the energy losses in the system are chosen in this paper. These values are usually concerned for an economic assessment and are traditionally evaluated using time series simulations with a high computational burden. However, any quantity can be defined as the objective quantity, keeping in mind that some details of the secondary system might be addressed more carefully.

C. Modeling the secondary system

To model the secondary system B, the classic power flow (PF) equations connecting powers and voltages are simplified. As shown in (3), the system is divided in two parts.

𝐵(𝑋) = 𝐵𝑝𝑎𝑠𝑠𝑖𝑣𝑒(𝑋) + 𝐵𝑎𝑐𝑡𝑖𝑣𝑒(𝑋)

The first part considers passive network equipment like lines and transformers, which are also included in a standard NR-Jacobian. The second part represents active network equipment bearing mainly nonlinear behavior, like DER with voltage dependent reactive power dispatch, on load tap changers (OLTC), static synchronous series compensators (SSSC), STATCOMS, etc.

1) Passive network equipment The passive part of the secondary system B is obtained in

(4) by linearizing the original system A around X0 using the first term of its Taylor expansion.

𝐵𝑝𝑎𝑠𝑠𝑖𝑣𝑒(𝑋) =𝜕𝐴(𝑋)

𝜕𝑋|𝑋=𝑋0

∙ ∆𝑋

In convolution methods, the systems obligatory linearization is done around the expected values of the random variables (i.e. X̅) and the resulting Jacobian is used for all following calculations. This is only valid if the deviation of the random variable from the mean is not too much [1], [6]. Having rated power as well as positions of the DER as inputs, this requirement cannot be hold. Therefore, the Jacobian is evaluated using the nominal voltage vector with X0 = [0] (i.e. no DER input). This is equivalent to a constant current power flow or stopping NR after the first iteration.

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Figure 4: Z-Y-graph of the power losses

Being an associated error, computed values like power flows, losses and reactive power needs in the network will deviate from the real solution. In case of power infeed only, they will be consistently higher and deviate linearly according to the impedance between nodes and the slack. Due to its linear correlation, this does not violate the requirement in (1). For an exemplary visualization, a Y/Z-graph of the line power losses similar to Fig. 3 is drawn in Fig. 4. The ranks of the probabilistic samples X are equal within Z and Y, implying system B to be a valid simplification of system A.

1) Active network equipment Active network equipment is mainly bearing nonlinear

behavior unable to be included in a fixed Jacobian. Integrating local voltage control through reactive power dispatch, on load tap changers, STATCOMs or SSSCs into conventional MCS is done by manipulating the Jacobian or implementing cascaded loops around the NR process. To incorporate the active equipment’s functionality into the CRA process, a second step has to be done, which bases on the linearized load flow results of the previous sub-chapter.

Using the results of the linearized secondary system B, operating points of the active network equipment using their original equations can be determined in (3). As an illustration, a first estimation of the reactive power dispatch of an inverter with active Q(U) is derived using the calculated, slightly overestimated voltage at the respective node, without regarding closed loop effects. Although the reactive power dispatch is overestimated, the results of the CRA-method are only slightly distorted.

𝐵𝑎𝑐𝑡𝑖𝑣𝑒(𝑋) = 𝐴𝑎𝑐𝑡𝑖𝑣𝑒 (𝜕𝐴(𝑋)

𝜕𝑋|𝑋=𝑋0

∙ ∆𝑋)

a) Active DER / Compensators

In case of local voltage control by DER or other distributed compensators, their behavior has to be included in Bactive. Exemplarily, two operating modes are analyzed in this paper, whose principal characteristics are sketched in Fig. 5.

Figure 5: Qualitative characteristics of cosφ(P) (l.) and Q(U) (r.)

a) Open loop control: cosφ(P) b) Closed loop control: Q(U)

In case a, the reactive power dispatch depends on the ratio of active power feed-in over rated power. If comparing probabilistic variations of scenarios with a fixed DER capacity (e.g. integration of 100 kW PV power in total) and a given irradiation profile, the sum of the reactive power output and the yearly reactive energy stay the same. From the CRA methods point of view, an offset would be added with no relevant change in the output vectors sequence (Z). Hence, the term does not need to be considered in the CRA method.

In case b, the reactive power contributions of DER depend on the nodal voltages they are connected to. As mentioned in (5), the output of the secondary system B’s passive part is inserted in the original system Aactive to determine operation points of the DER. The calculated values of DER reactive power dispatch are overestimated, primarily because of the missing closed loop effect on the reactance in the grid and secondly because of the current’s missing dependency on the voltage. Nevertheless, this “first guess” of the reactive power contribution of active DER is valid to be used within the CRA-method, since its overestimation is of systematic nature.

b) OLTC / SSSC

In case of an OLTC installed at the distribution transformer, its nonlinear influence on the voltage is simulated by manipulating the resulting voltage vector (i.e. adjusting the transformer busbar voltage). In principle, any type of characteristics can be implemented within this approach, but the high sensitivity of the transformers voltage drop on reactive power flows has to be regarded carefully. For simplicity, a step less control of the busbar maintaining nominal voltage is considered in this paper. Local linear regulators like SSSCs are modeled likewise by manipulating the voltage vector at the corresponding node.

D. Further reduction of the computational effort

In case of time series simulations, a major reduction of the computational effort can be done by considering the regional character of low- and medium voltage grids.

1) Reduction of yearly profiles to a single point in time Due to their local closeness, all photovoltaic DER are

exposed to almost the same irradiation profile. Cloud-drifts and other short-time phenomena are smoothed by the 15-minute moving average. If the implemented profile is the same for all photovoltaic DER, the impact of each probabilistic variation of X on the yearly reactive energy as well as the energy losses and maximum nodal voltages in the grid should correlate with each variations maximum infeed scenario. Using CRA, it is possible to assess yearly quantities of probabilistic simulations by a simply calculation of one significant out of 35040 points in time (equal to feed-in factors) and compare the differences. If only passive grid elements are installed, any chosen feed-in factor would produces identical results. However, low feed-in factors are not able to represent network equipment inheriting nonlinear behavior depending on voltage levels like Q(U) or P(U), since activating/disturbing voltage levels will not be reached.

)

𝑃

𝑃 𝑎

) 𝑖𝑛

cap.

) 𝑖𝑛

ind.

QMax

Q

U

-QMax

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By choosing the maximum feed-in factor (i.e. 1), the equipment’s maximum impact is comprised.

Summarized: instead of a time series simulation with 35040 NR calculations, a significant scenario is defined for each probabilistic variation of X. Its output value (i.e. reactive power flow LV/MV) is linked to the desired yearly value of the original system (i.e. yearly reactive energy LV/MV) using the CRA method.

2) Consideration of loads Loads are modeled as annual mean values of the

corresponding load profiles at each node. This accounts for the partial self-sufficiency of customers as well as for reduced power losses and reactive power demand of lines. Reducing yearly load profiles to one single value is a major source of errors if profiles are highly diverse.

IV. EXEMPLARY RESULTS

The proposed CRA-method has been validated in various scenarios against detailed results of full MCS. The following scenarios are chosen to generate representative results:

A. Case A: small village LV grid

LV-Grid: Small village with a radial grid structure, 36 households (probabilistic load profiles [7]) and a mix of cables and overhead lines.

DER: A total PV capacity of 100 kW peak with unknown counts of inverters, positions and rated powers is installed.

Technologies: The following 5 scenarios are considered:

1. No active DER, OLTC (0 / OLTC) 2. Q(U), passive transformer (Q(U) / 0) 3. Q(U), OLTC (Q(U) / OLTC) 4. Cosφ(P), passive transformer (cosphi(P) / 0)

5. Cosφ(P), OLTC (cosphi(P)/ OLTC)

At first, 100 variations of X have been generated. Full MCS of these variations in all scenarios was done with a total duration of ~110 hours on a single Intel i5. Subsequently, the same variations were fed into the CRA-method to calculate three quantiles of interest (median, 5%, 95%). Objective quantities have been set to yearly reactive energy at the slack (VAR-hours), the active energy fed back at the slack, the energy losses in the grid and the maximum/mean nodal voltages during the year. For simplicity, grid voltages results are represented by the voltages at the electrically most remote node (#44, ~400 m of overhead lines). The computational effort of the CRA method was about 2 seconds, whereas the successive full MCS of the selected quantiles took about 3.3 hours. The reduction of computational burden equals to 97%.

1) Reactive energy, active energy and energy losses Fig. 6 shows the energetic results. The blue line represents

each variation of X in the full MCS, whereas the green and red markers stand for the three quantiles approximated by the CRA-method. Green marks the median, while the 5% and 95% quantiles are circled in red. Spearman’s rank correlation coefficients are noted in the northwest corner of each graph. The SRCC comparing Y and Z is always greater than 99%. Corresponding errors with respect to the full MCS are shown

in Fig. 7. The error of the CRA method is less than 1% in any case. These results are replicable for all investigated rural and suburban grids (also MV grids) as well as different penetrations of DER (total PV capacity of 10 kW to 600 kW).

2) Nodal voltages The voltage results of the most remote node #44 are

presented exemplarily in Fig. 8. The SRCCs are higher than 99% and the errors are less than 0.03%.

Figure 7: Example case A – CRA Errors

Figure 6: Yearly reactive energy - Full MCS vs. CRA

Figure 8: Yearly voltages at Node #44 – Full MCS vs. CRA

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3) Comparison between reduced MCS and CRA Running MCS with 10.000 or more variations does not

promise gaining precise results but approximates the stochastic behavior closely. If a reduction of the number of variations in MCS is suggested (e.g. to 100 or 1000 variations), the results are possibly less accurate than having an approximation method like CRA running 10.000 variations. The results of a direct comparison are illustrated in Fig. 9, clearly revealing the error made by reducing MCS variations.

Fig. 10 shows, that using CRA with 1000 variations involves less errors than running a reduced number of MCS. The CRAs additional computational burden is only marginal.

B. Case B: rural LV grid

Analogous to example case A, a rural LV-grid with a radial structure, 8 households (probabilistic load profiles [7]) and a high share of overhead lines with small diameter is examined in case B. Fig. 11 gives a summary of the accuracy. The SRCCs are above 99.2% and the error is negligibly small.

V. PROPOSED METHOD

Refining Fig. 2, the method proposed in this paper consists of three steps depictured in Fig. 12. After modelling the simplified secondary system B, the CRA method is fed with a high number of probabilistic variations of X (i.e. 10.000). The quantiles of interest are selected and the corresponding variations of X are computed individually using the original system A. Time savings reach from a factor of ~4 (CRA vs. single NR for one point in time) to a factor of ~9000 (CRA vs. full MCS yearly simulation). Various DER control strategies and nonlinear equipment can be integrated to a certain extend.

VI. REMARKS AND PROSPECTS

The proper functioning of CRA in DER environments with different feed-in profiles is anticipated. Its evaluation and demonstration is subject to further work.

REFERENCES

[1] R. N. Allan, a. M. L. Da Silva, and R. C. Burchett, “Evaluation Methods

and Accuracy in Probabilistic Load Flow Solutions,” IEEE Trans. Power

Appar. Syst., 1981. [2] A. Meliopoulos and G. Cokkinides, “A new probabilistic power flow

analysis method,” Power Syst. IEEE, 1990.

[3] M. Gödde, F. Potratz, and A. Schnettler, “Analysis of a convolution method for the assessment of distribution grids within probabilistic

power flow calculations,” in CIRED Workshop, 2014.

[4] P. Chen, Z. Chen, and B. Bak-Jensen, “Probabilistic load flow: A review,” 3rd Int. Conf. Deregul. Restruct. Power Technol. DRPT 2008,

no. April, pp. 1586–1591, 2008.

[5] M. Fan, V. Vittal, G. Heydt, and R. Ayyanar, “Probabilistic power flow studies for transmission systems with photovoltaic generation using

cumulants,” Power Syst. IEEE Trans., vol. 27, 2012.

[6] C. Viggiano, “Comparison of probabilistic and deterministic methods in distribution grid planning,” Technische Universität München, 2015.

[7] M. Wagler and R. Witzmann, “Open loop operational strategies of a

virtual power plant and their impacts on the distribution grid,” in 23rd International Conference on Electricity Distribution, 2015.

Figure 9: Yearly reactive energy - CRA vs. MCS100 vs. MCS1000

Figure 10: Example case A - CRA vs. MCS Errors

Figure 11: Example case B – CRA Errors

Secondary System

CRA

Select quantilesof interest

CRA Calculation

~2 min(10.000 model calls)

Original System

Y

Original System(i.e. Full MCS)

Full MCS(quantils of interest only)

e.g. 3 quantiles

e.g. 3 quantiles

X

Figure 12: Exemplary flowchart of the proposed method

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