A Power Flow Method for Radial Distribution Feeders with DER Penetration

Abstract: This paper presents a novel power flow method suitable for radial distribution feeders, which consists a modification of the simplified power flow concept known as the DistFlow method, already available in the literature. The proposed method relies upon a differentiated manipulation of power losses, which are taken into account in voltage calculations, unlike other simplified methods, where losses are totally neglected. As a result, calculation accuracy is greatly improved, in terms of node voltages, losses and overall active & reactive power flows. In addition, the proposed method is non-iterative and entirely linear, being easily implementable and fast in execution. The method is particularly suited for feeders with a high penetration of Distributed Energy Resources (DER), providing results that closely match those of a full non-linear power flow and are considerably more accurate than the traditional linearized distribution power flow methods, without any increase in computational burden. The new method is applied to a variety of case studies in the paper, to demonstrate its accuracy and effectiveness, comparing its performance with the simplified (linearized) DistFlow and a conventional non-linear power flow method.


INTRODUCTION
Load flow analysis is a fundamental tool for design and operation of distribution networks [1][2][3][4][5], whose importance is amplified by the growing attention on distribution systems, the ever increasing penetration and significance of Distributed Energy Resources (DER) [4][5][6], the development of Distribution Management Systems (DMS) [7,8] the implementation of new and advanced functionalities such as demand response and the continuously increasing load demand [9]. At the same time, the size and complexity of the systems are also increasing [2,4,9], rendering the task of analysis more difficult and cumbersome, especially if the non-linear nature of power flow is taken into account. For these reasons, fast, accurate, robust, and computationally efficient distribution power flow algorithms become all the more significant [3,7,9].
Conventional load flow techniques (e.g. Gauss Seidel, Newton Raphson, Decoupled and Fast Decoupled Load Flow [1][2][3][9][10][11][12][13][14][15][16]) are known to be unsuitable for distribution networks, in terms of efficiency, performance and robustness [1][2][3][9][10][11][12][13][14][15][16][17][18][19][20][21]. Hence, a variety of power flow algorithms  have been developed so far for radial or meshed distribution networks, different load models, varying levels of convex relaxation etc. The vast majority of these methods are either iterative or based on non-linear equations and rather complicated mathematical or numerical approaches, which renders them inefficient and cumbersome to apply. A notable exception relying on basic circuit theory and simple voltage equations is the DistFlow power flow method, initially presented in [25][26][27], which is suitable for radial distribution networks. It is based on a set of simple, recursive equations, where no trigonometric terms are used. Dropping the quadratic terms that represent losses and making certain approximations in voltage calculation, a simplified (linearized) form of DistFlow is derived. A comparison of two DistFlow algorithm variants is presented in [16] for different loading conditions and feeder R/X ratios. In [12], three algorithms (decoupled, fast decoupled & very fast decoupled distribution loadflow algorithms) have been proposed for radial distribution networks, based on the DistFlow methodology.
Although the DistFlow equations have been widely used in the literature, in their original form or with minor modifications [15,23,24,[28][29][30][31][32][33][34][35][36], they still suffer from high errors in power flows, losses and voltage calculation in networks with heavy load and/or high DER penetration, as a result of the fundamental assumption of neglecting active and reactive power losses. Inaccuracies become more pronounced when operating power factors deviate from unity (e.g. in networks where DER participate in voltage regulation), as well as in feeders with high R/X ratios or high overall impedance.
In this paper, a novel distribution power flow method is introduced, based on a modification of the standard, simplified DistFlow equations, so as to maintain active and reactive power losses and properly incorporate them in the algorithm equations. The proposed method, referred to in the following as the ModDistFlow method, is linear, non-iterative, easy to implement, fast and particularly robust, presenting enhanced accuracy in power flow, losses and voltage calculations, compared to the linearized DistFlow method. To demonstrate its performance, a variety of benchmarking scenarios are presented in the paper, addressing different loading conditions, DER penetration levels, network characteristics (R/X ratios) etc., including application to a real world MV network. In all cases, the proposed method is compared to the simplified (linearized) DistFlow and to the full non-linear power flow.
The paper is organized as follows. Section 2 describes the standard simplified DistFlow method. The proposed method is introduced in Section 3. The evaluation of the method using a simple reference network is presented in Section 4, while in Section 5 a real MV rural feeder is used as a case study, whose parameters are provided in the Appendix. Concluding remarks are summarized in Section 6.

THE DISTFLOW AND SIMDISTFLOW POWER FLOW METHODS
The widely used DistFlow simplified power flow method for radial distribution networks is based on the work of Baran and Wu, [25][26][27]. Its primary equations are outlined in the following, incorporating modifications introduced since its introduction.
With reference to Figure 1, V 0 represents the voltage at feeder departure. For MV feeders this typically is the voltage of the substation MV busbars, regulated by the On-Load Tap Changer (OLTC) of the HV/MV transformer and voltage regulator relay, and therefore it is assumed to be known. Lines are represented by their series impedance z j =r j +jx j , ignoring shunt capacitance. All loads are considered constant power. S j,s is the "sending" complex power flow of branch j, i.e. the left-hand side complex power flow entering the branch at the side of node j, while S j,r is the "receiving" complex power flow of branch j, i.e. the right-hand side complex power flow leaving the branch, at the side of node j+1.
Given the voltage of node j and the power flow at the left-hand side of branch j ("sending" power), the voltage of node j+1 and the sending power of branch j+1 can then be calculated based on the following equations: Using V j as the reference complex vector, Eqs. (1) and (2) for branch j yield the following real equations: Eqs. (3.i)-(3.iii) are known as the forward DistFlow branch equations [27], employing only quantities at the left end of each branch (sending).
Assuming that node voltages are close to nominal, a reasonable approximation when node voltage deviations are small [32][33][34] leads to the following Eq.
Rewriting Eq. (4) for node j+1 and subtracting from Eq. (4) yields: Eq. (5) can be substituted in Eq. (3.iii), while the non-linear terms in Eqs. (3.i)-(3.iii) represent active and reactive power losses and can be neglected. Based on these assumptions, the following simplified DistFlow equations [27,33] (hereinafter referred to as the SimDistFlow method) are derived: The Eqs (6.i)-(6.v) are solved using the following boundary conditions for node 0 at feeder departure and node n at feeder end: A slightly modified version of these equations is presented in [31,32,34], where V 0 is used instead of V nom .

THE PROPOSED MODDISTFLOW METHOD
As mentioned in the previous section, SimDistFlow employs quantities at the left end of each branch (sending), which are not readily obtainable from node loads, unless active and reactive losses are ignored. In the method proposed in this paper, this fundamental shortcoming of SimDistFlow is overcome using the power flows at the right end of each branch (receiving) and incorporating power losses in the voltage calculations in a novel and intuitive manner.
With reference to Figure 2 and using V j+1 as the reference complex vector: Eq. (8) is differentiated from Eq. (2) in the selection of the reference complex vector, which now is V j+1 instead of V j , facilitating implementation of the method.
Eq. (5) can then be employed to simplify Eq. (9.iii) as follows: where the assumption V j+1 ≈V nom has been used in the denominator, as it is common in the literature, in order to transform a hard non-linear term to a linear one. The impact of this assumption on the accuracy of calculations is only moderate, if an effective voltage regulation policy is implemented and node voltages deviate reasonably from nominal.
The quadratic term in Eq. (10), often ignored in the literature, can be expressed in terms of the losses on branch j: Eq. (11) implies that active and reactive losses on branch j effectively cause the same voltage drop as an equivalent load would produce on a branch having half the impedance of the actual branch j. Hence, the effect of losses on voltage drop calculations can be accounted for by introducing a fictitious node at the middle of each branch, where a virtual load is connected whose complex power is equal to the active and reactive losses of the branch (Figure 3).
Based on the above, the final set of equations of the proposed method, referred to as the ModDistFlow method, is the following. Quantities involved and notation used are clarified in Figure 3: Eqs. (12) show that losses are properly accounted for in the equilibrium of active and reactive powers for each node, as well as in voltage calculations. Notably, the fictitious nodes in Figure 3 offer a physical explanation for the effect of losses, but they are not necessary for the implementation of the ModDistFlow method.
At the end of the feeder: ii. At feeder departure (bus 0 in Figure 2 and Figure 3), corresponding to the substation MV busbars, the voltage is determined by the OLTC regulator and therefore it is considered to be known: The equations of the proposed power flow method are solved in a mixed forward and backward manner: The proposed method is directly applicable to networks including DER, simply replacing p load_j and q load_j with the net active and reactive power demands at each node (local consumption minus local generation).

EVALUATION OF THE PROPOSED POWER FLOW METHOD IN A SIMPLIFIED DISTRIBUTION NETWORK
This section demonstrates the enhanced accuracy characteristics of the proposed ModDistFlow method, as compared to SimDistFlow, using the simplified MV feeder of Figure 4 as a case study. Feeder characteristics are summarized in Appendix (see Table  3).
The analysis encompasses different loading conditions, DER penetration levels, feeder R/X ratios and DER power factor regulation alternatives. To evaluate the accuracy of the methods, the full nonlinear power flow on the feeder (hereinafter referred to as the Full Power Flow) is used as a benchmark. The Full Power Flow has been performed using the commercial load flow software PSS-Adept.

Implementation for Different Feeder Loading Conditions and DER Penetration Levels
Two case studies are analyzed, corresponding to different loading conditions of the feeder:

•
High feeder load: 9 MW • Low feeder load: 3 MW Figure 5 shows the voltage profile along the feeder for two extreme scenarios (high load conditions and no DER connected and light load conditions and 9 MW DER connected), as calculated by ModDistFlow, SimDistFlow and the Full Power Flow, as well as the relative voltage error of the two former methods compared to the latter for both scenarios.
ModDistFlow clearly proves substantially more accurate than SimDistFlow, especially at distant nodes where the standard SimDistFlow exhibits highest errors. Overall, the results obtained by ModDistFlow essentially coincide with those of the accurate Full Power Flow. In all cases, DER units operate at full output power with unity power factor. With either method, calculation error is minimized when DER output matches the load of the feeder, as power flows on feeder branches then reach a minimum, whereas highest errors are noted when DER and load levels differ significantly. Again, the ModDistFlow method achieves superior accuracy (error reduced by at least one order of magnitude).

Besides improvements in voltage calculation,
ModDistFlow permits a more accurate evaluation of active and reactive power losses along the feeder, as demonstrated in Table 1 and 2. At high load conditions, the proposed method proves to be 10 times more accurate, with the absolute error remaining below 1%. At low feeder loads, ModDistFlow remains more accurate but by a lower margin.

Additional Investigations
The effectiveness of the ModDistFlow method increases when applied to MV feeders with increased R/X ratios (conductors of lower cross section or      Figure 7 (mean error for all 5 nodes), as a function of the total DER capacity on the feeder and for different loading levels, assuming R/X ratios of 1.0 and 1.5 (instead of 0.63 for the base case feeder). Higher R/X ratios lead to increased voltage calculation errors using the standard SimDistFlow method, as well as to greater accuracy enhancement by applying ModDistFlow. Similar remarks hold for the estimation of power losses on the feeder. Gains in accuracy by applying the proposed method, presented in Figure 7 become more significant at high R/X ratios, where the level of losses is increased. It is obvious in the figures on the left there is a similar monotony for several load levels along the aggregate DER capacity for voltage and power losses calculation. The same happens also in the figures on the right where a higher R/X ratio is applied.
Comparing the left to the right figures, it is evident that the accuracy by implementing the ModDistFlow is much higher (almost 1 time and 0,7 time in voltage and power losses calculation respectively).
In all previous analysis, DER units were assumed to operate at unity power factor (zero reactive output power). In Figure 8, indicative results for voltage and power losses calculation are provided when the power factor of DER units varies between 0.9 inductive (VAr consumption) to 1, for a reference case study of low feeder load conditions. In the same figure, similar results are presented for 0.95 capacitive (VAr production) and unity power factor operation, for a reference case study of high feeder load conditions; capacitive power factor operation may not be realistic, but it has been included only for the sake of completeness of the analysis. As it is observed, node voltages and losses calculation accuracy is enhanced by applying the ModDistFlow method in all cases.

IMPLEMENTATION IN A REAL MV NETWORK
To assess the performance of the ModDistFlow method in more realistic conditions, a real world MV feeder with substantial RES penetration is examined,  located on Rhodes island, Greece. Its main characteristics are given in the Appendix (see Table 4).
The feeder is shown in Figure 9 while load and DER distribution along the feeder is shown in Figure 10. Figure 11 depicts the calculated voltage profile and respective error using ModDistFlow, SimDistFlow and the Full Power Flow methods for operation of the feeder at minimum load and maximum DER output. Both distribution power flow methods prove sufficiently accurate, with an average absolute error around 0.1-0.2%, with ModDistFlow exhibiting again improved accuracy characteristics. As for power losses, ModDistFlow presents an error of 7.8%, compared to 10.1% for the SimDistFlow method.
The differences between the two methods are not spectacular in this case, as the installed DER capacity on the feeder is relatively low at 5 MW, hence power flows and therefore losses are not significant enough to play a crucial role in voltage calculations, as in other cases examined in the previous section. At higher DER penetration levels errors increase, as well as the improvement achieved by the proposed ModDistFlow method. It is reasonable to state that if the power flow is very low then there is almost no margin for presenting inaccuracy by implementing any method. But what increases the merit of ModDistFlow is that the power flows in the future are expected to be higher taking into account the increase of both energy consumption as well as energy generation at least    Figure 9: Real-world case study MV distribution feeder. during some hours of a day. This is indicatively shown for voltage calculations in Figure 12, where DER capacities are increased by 50% at all nodes. In this case, the accuracy in losses calculation improves by 4%.

CONCLUSION
In this paper, a new distribution power flow method is proposed (ModDistFlow), derived from the simplified equations of the SimDistFlow method after suitable modifications. The proposed method does not neglect active and reactive power losses, as is the case with other simplified methods commonly used in the literature, and incorporates them in the voltage calculations in a novel and intuitive manner which substantially enhances accuracy in voltage, power flows and losses calculations. Further, the ModDistFlow method is non-iterative and therefore   simple to implement and computationally efficient, while it is particularly suited for networks with high DER penetration levels.
The proposed method has been compared to the widely used SimDistFlow algorithm, already available in the literature, using both simplified and real world MV feeders as case studies and addressing a variety of loading, DER penetration conditions and feeder R/X ratios. ModDistFlow proved to be more accurate than SimDistFlow, the differences being magnified for heavily loaded feeders, high DER penetration levels and DER inductive power factor operation. On the other hand, when feeder load and DER output powers were relatively balanced all methods performed comparably, as the power flows on the network were reduced and therefore the effect of neglecting losses was not significant.
The substantial gains achieved by the proposed method in the calculation of voltages, power flows and losses, besides their self-evident value, eventually permit a more dependable evaluation of the DER hosting capacity of the network, more accurate estimation of HV/MV transformer losses and evaluation of reactive power compensation requirements at HV/MV substation level, as well as a more reliable calculation of the total active and reactive powers at the interface point with the HV system.