Bus Split Distribution Factors

The linearised DC loadflow is a fast way to calculate the load flow in transmission networks, and it is often performed by calculating the PTDF. However, until now changes in topology due to busbar splitting cannot be dealt with in an efficient way when using the PTDF. In this study, we introduce the Bus Split Distribution Factors (BSDF) which enable an efficient way to compute the effects of busbar splitting on the DC load flow. The derivation of the BSDF formulas is based on modelling the busbar coupler as a branch with vanishing reactance and by using transformed LODF formulas. In times of the energy transition the BSDF approach might be especially helpful since optimal grid topology control is still a largely unexploited form of flexibility for system operators due to the complex combinatorial nature of grid topology reconfiguration. With the BSDF approach much faster screening of topological remedial actions (including substation reconfigurations) in congestion management applications is possible.


Bus Split Distribution Factors
Joost van Dijk , Jan Viebahn , Bastiaan Cijsouw , and Jasper van Casteren Abstract-The linearised DC loadflow is a fast way to calculate the load flow in transmission networks, and it is often performed by calculating the PTDF.However, until now changes in topology due to busbar splitting cannot be dealt with in an efficient way when using the PTDF.In this study, we introduce the Bus Split Distribution Factors (BSDF) which enable an efficient way to compute the effects of busbar splitting on the DC load flow.The derivation of the BSDF formulas is based on modelling the busbar coupler as a branch with vanishing reactance and by using transformed LODF formulas.In times of the energy transition the BSDF approach might be especially helpful since optimal grid topology control is still a largely unexploited form of flexibility for system operators due to the complex combinatorial nature of grid topology reconfiguration.With the BSDF approach much faster screening of topological remedial actions (including substation reconfigurations) in congestion management applications is possible.Index Terms-PTDF, busbar splitting, DC loadflow, BSDF, LODF.

I. INTRODUCTION
T HE European electricity system and associated energy market systems are transforming at a rapid pace due to the decarbonisation of the electricity system to meet climate goals and the facilitation of cross border energy markets.While in the past the evolution of the power system was mostly related to grid infrastructure and capacity expansion, it now becomes a matter of greater grid management and optimization over existing infrastructure.As a consequence, power system operation is becoming increasingly complex, resulting in a growing need for new forms of flexibility [1].In particular, dynamic grid topology reconfiguration is an interesting option for system operators since it is a cost-efficient and flexible solution for congestion management that uses existing infrastructure [2], [3].But it is still beyond the state-of the-art to optimally control the grid topology at scale due to the problem's nonlinear and discrete combinatorial nature leading to a large search/optimization space [4], [5].
For a fixed grid topology, the linearised DC loadflow is a fast way to calculate the loadflow and is used for many purposes.The DC power flow calculations are often performed by calculating the Power Transfer Distribution Factors (PTDF) [6], [7], [8], [9], which can be given as a |E| × |V | matrix 1 , P T DF , where |E| is the number of branches and |V | the number of nodes.The P T DF is slow to calculate, but once it is calculated for a given network topology, the power flow p L can be calculated quickly for every possible nodal power injection vector p N , by matrix multiplication: p L = P T DF • p N .
However, if the topology changes, for instance if a line trips or a switch is opened, the P T DF needs to be recalculated (which is expensive to compute due to matrix inversion).In case one line trips, there is a fast way to calculate the new P T DF post from the old P T DF pre by using the so-called Line Outage Distribution Factors (LODF) [9], [10], [11].In case several lines trip one can use the Multiple Outage Distribution Factors (MODF) [8], [12], [13], [14].Similarly, phase shifters and HVDC lines can be modelled by using the P SDF and DCDF matrices, respectively [6].
The purpose of this paper is to show that the effects of substation reconfiguration via bus splitting can also be calculated in a fast way by updating the PTDF in a linear fashion.Corresponding formulas are not available yet.We will present a novel closed formula similar to the case of line outages.More precisely, we introduce the Bus Split Distribution Factors (BSDF) which are given as a vector bsdf ∈ R |E| and represent the effect of busbar splitting on the load flow of the network.Given the PTDF of the busbar coupler as the vector ptdf bbc , the new PTDF after busbar splitting is simply given by The BSDF can also be used to calculate the new flows (c represents the busbar coupler): For the proposed method no switches need to be modelled.The BSDF calculation is based on a simple fraction, similar to the calculation of the LODF.It gives a novel and clean way to calculate the effects of a busbar splitting on the DC loadflow.
A final note on the difference between line outages and busbar splitting.On the one hand, taking a branch out of service and opening a busbar coupler are similar operations.In both cases the connection between two nodes is broken and the power flow needs to be redistributed across all other branches.On the other hand, the two operations are very different since a line outage is about effectively removing an element from the network, whereas bus splitting is about effectively creating a new node in the network.More formally, the two operations are complementary in that branch switching works on the branch dimension whereas node splitting works on the node dimension of the branches-by-nodes PTDF matrix.
The paper is structured as follows: A transformed version of the LODF formula is derived in Section II.Subsequently, in Section III the PTDF of an idealized busbar coupler is constructed.Finally, in Section IV the BSDF formula is presented and proofed.Subsequently, in Section V the BSDF approach is benchmarked against other approaches for computing the effects of busbar splitting.The paper ends with a summary and discussion in Section VI.We also note that in Appendix relevant notation is introduced (A), necessary prerequisites are summarized (B), and related work is discussed (C).

II. LODF FOR SHIFTED INJECTIONS
Intuitively, the LODF is the (linearized) effect of suitable injections at the from-node and to-node of a branch such that the power flow across this branch disappears [11], [15].This section shows that the usual injections of the LODF can be shifted to other nodes than the from-node and the to-node.In Section IV, it will turn out that the BSDF can be derived using the shifted LODF formula, whereas the normal formula of the LODF diverges in the BSDF calculation.

A. Relation Between Susceptances and Flows
Before we can derive the LODF formula for shifted injections, we need the following Lemma II.1:The formula is true, independent of the slack node.This implies for every node x that the injection vector p N,x = B b :,x , which has nonzero values only on n and neighboring nodes, produces non-zero flows only on branches connected to n: :,n is a nodal injection vector having injections on the ends of the branches connected to n, then B f :,n is the resulting flow on those branches.Proof: By definition, it holds for the reduction P T DF of P T DF to the non-slack columns that P T DF • B ,b = B ,f .Let s denote the slack node.Since the column of the PTDF of the slack node is zero, for every n = s.It still needs to be shown that the same equation holds for the slack node n = s, that is, for every Note that for every branch ˜ , and every node n, This can be rewritten by extracting the terms related to the slack node Using (3), (2), and (4), we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
= − This proves the lemma.

B. Shifted LODF Formula
In this section, an outage of branch k is considered.Lemma II.1 will be used to shift the injections of the LODF (see (13)) from the from-node to the surrounding nodes.For that we need the following definitions.
Definition II.2:The nodal shifted injection vector, Consequently, the nodal correction vector, p N,corr ∈ R |V | , is given (with k = (n, m) the outaged branch) by Finally, using Lemma II.1 gives the branch correction vector, p L,corr ∈ R |E| , (with k = (n, m) the outaged branch) as Note that both the nodal shifted injection vector and the nodal correction vector are balanced, which means that the sum over the vectors is zero in both cases.
We note that by using these definitions the following relations hold, namely, (with k = (n, m) the outaged branch) and by Lemma II.1, P T DF • p N,corr = p L,corr .The relation shown in (5) can be used to rewrite the LODF leading to the following Proposition II.3:The LODF can be written as for all branches ˜ , k.If the denominator is zero, the network splits when k outages.

˜
Substitution of (5) in the nominator and denominator of the LODF formula gives In the normal LODF formula, and thus also in (8), the network splits if and only if the denominator is 0 [12].Since b N n is non-zero, the above equations also show this for the LODF formulations of ( 6), (7).
The shifted version of the LODF of the proposition will be used in Section IV.
As an example, we consider the same network as in Section II-A, Fig. 1.For that we choose k = ˜ 4 .The LODF of branch k = ˜ 4 is given by (see (13)) The term P T DF ˜ ,n 3 − P T DF ˜ ,n 4 can be interpreted as the flows produced by the nodal injection vector e n 3 − e n 4 .Fig. 2 (upper-left) shows the nodal injection vector e n 3 − e n 4 and the resulting load flow P T DF ˜ ,n 3 − P T DF ˜ ,n 4 for the example network.Moreover, Fig. 2 shows the terms of the re-written LODF formula as given in the Proposition II.3.These are p N,shift (upper-right) and p N,corr (lower) with the resulting load flows.One can see that for the shifted injections the injection related to the from-node of k = ˜ 4 (i.e. for n 3 ) vanishes and is shifted to the neighbouring nodes (see Def. II.2).The shifted injection vector only produces different flows than P T DF :,n 3 − P T DF :,n 4 on the branches connected to n 3 .Correspondingly, the correction load flow is vanishing on branches not connected to n 3 .Intuitively, p N,shift is a vector with withdrawal at the to-side m of k, and injection shifted from the from side n of k to all the neighbors of n.And p N,corr is a correction for the flows on all branches connected to n, by Lemma II.1.

III. PTDF OF AN IDEALIZED BUSBAR COUPLER
Representing the busbar coupler in the PTDF can be done in two ways: Either the busbar coupler can be modelled as a branch with a small reactance X (and using the formulas in Appendix B), or the busbar coupler is modelled as a branch with vanishing reactance and the entries in the PTDF are based on Kirchhoff's current law ((2) of [11]).
More precisely, we assume the following situation: To model bus splitting, substation configurations need to be considered before and after the busbar coupler opens.A substation configuration for which the busbar coupler is closed is called the pre-bus split situation.Is the busbar coupler open, then the situation is called post-bus split.In this section, the busbar coupler is denoted by c, and is considered to be a branch from bus A to bus B.Moreover, only substation configurations are considered where every branch is exclusively connected to either bus A or B (i.e.not to both busbars).This ensures that the buses split when the busbar coupler opens.Now let |V | denote the number of nodes where A and B are not counted separately.Using the PTDF, the pre-bus split situation can be modelled in two ways: 1) The busbar coupler c is modeled as a branch with a reactance of X (a susceptance of 1/X), where X is small, and c is connected to the nodes A and B (which have separate columns in the PTDF).Assuming a given substation configuration and using the formulas in Appendix B results in a PTDF denoted by P T DF pre,X ∈ R (|E|+1)×(|V |+1) .
Here the additional elements are the busbar coupler and the busbar B. With the LODF the effect of opening the busbar coupler can be calculated.However, the PTDF and LODF depend on the given substation configuration (that is, they need to be re-computed for each substation configuration) 2 .Consequently, it is important to note that P T DF pre,X does not appear in the BSDF formula below but is only used in the proof thereof.
2) The busbar coupler c is modeled as a branch with vanishing reactance.Consequently, in this ideal model initially the busbars are merged into one node (that is, no additional column for busbar B and no additional row for the busbar coupler c are used in computing the initial PTDF with the formulas in Appendix B).In order to model bus splitting, the PTDF must be extended by a column for busbar B and a row for the busbar coupler c.Since the busbars A and B are effectively identical the column related to busbar A 2 This is because the from/to nodes to A and B needs to be changed for any different substation configuration.Only the flow over the busbar coupler changes much if a substation configuration changes.The PTDF on the other branches do almost not change, although for the LODF, it turns out that the changes, tiny as they may be, are significant.Since the normal LODF formula of the busbar coupler needs subtractions P T DF pre,X :,n − P T DF pre,X :,m , it turns out that the tiny differences in P T DF pre,X :,n , P T DF pre,X :,m when the configuration changes, matter in a great way for the outcome of the LODF.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.2) in [11]).In particular, the sum µ n − ptdf pre,ideal,c n must be 0 if is simply copied to create the column for busbar B. We denote the PTDF extended by the column for busbar B by P T DF pre,ideal ∈ R |E|× (|V |+1) .The essential step is the computation of the row for the busbar coupler c which is given by the following Definition III.1:The theoretical PTDF of the busbar coupler is defined to be a nodal vector ptdf pre,ideal,c ∈ R 1×(|V |+1) given by for nodes n, slack node s, and where Obviously, μ n represents the flow that enters node (or busbar) A excluding the flow that is transmitted via the busbar coupler c.In other words, if 1 MW is injected at node n and withdrawn at the slack node s, then μ n enters bus A over any branch except the busbar coupler.Consequently, Definition III.1 is simply a consequence of Kirchhoff's current law (see e.g. ( 2) in [11]).Fig. 3 illustrates Definition III.1.We note that P T DF pre,ideal ∈ R |E|×(|V |+1) is independent of the specific switch configuration in a station (i.e.how the different elements in a substation are connected to the busbars), whereas ptdf pre,ideal,c n depends on the specific substation configuration.Finally, we denote the PTDF which includes ptdf pre,ideal,c n by P T DF pre,ideal ∈ R (|E|+1)×(|V |+1) .Then P T DF pre,X converges to the ideal model P T DF pre,ideal if the reactance X converges to 0: for any ˜ .In particular, this holds for the busbar coupler since the equations in Definition III.1 also apply to the case of nonvanishing X using P T DF pre,X instead of P T DF ,pre,ideal (and interchanging limits and finite sums).

IV. BUS SPLIT DISTRIBUTION FACTORS
The BSDF models the opening of a busbar coupler.Consequently, the derivation of the BSDF proceeds in two steps.First, the PTDF is extended by a node (representing the second busbar) and a branch (representing the busbar coupler), as presented in Section III.Second, the busbar coupler is opened using the rewritten LODF formula (see Proposition II.3).In this section the BSDF will be defined and derived by combining these two steps.
However, we first briefly reexamine unsuccessful ways of modeling busbar splitting.The obvious approach is to model the busbar coupler as a branch and use the standard LODF formula (13).But using P T DF pre,ideal in (13) leads to ) which is equal to 0 0 (noting that the columns related to A and B are identical except for the row ptdf pre,ideal,c where the difference is 1).
An alternative approach is to use ( 6) from Proposition II.3.However, the limit of the nodal correction factors Fortunately, the alternative formula (7) of Proposition II.3 resolves the issue, and can always be calculated.Hence, we can now state the main results of this study given by Definition IV.1:The Bus Split Distribution Factors (BSDF) vector is a branch vector bsdf ∈ R |E| defined for a branch ˜ by bsdf ˜ := (P T DF pre,ideal • p N,shift Theorem IV.2:If the denominator of the BSDF vector is nonzero, the following equation holds: .
The denominator is zero if and only if the network splits into two components due to bus splitting.When specific flows p L,pre ∈ R |E|+1 are given, then it holds Note that the shifted injection vector, p N,shift ∈ R |V |+1 , and the branch flow correction term, p L,corr ∈ R |E| , as defined in Definition II.2, take the following specific forms Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
In particular, terms related to the busbar coupler cancel each other out.Again, the intuition behind the shifted injection vector is that 1 MW is injected at the nodes connected to A (via a branch) and withdrawn from B (note that the specific switching states in a substation matter here).As an example, the description of Fig. 2 in Section II-B still essentially applies.The proof of Theorem IV.2 is as follows Proof: First note that with P T DF pre,X the standard LODF formula (13) does not diverge.That is, for P T DF pre,X all LODF formulas can be used.Hence, the approach is to use P T DF pre,X in the LODF formula (7) and to show that in the limit X → 0 formula (7) converges to the BSDF formula given in Definition IV.1.For that note that p N,shift and p L,corr are independent of X.Consequently, applying (9) to II.3 (7) it follows The results immediately follow from same properties of the LODF.
For illustration we provide another example which explicitly includes the busbar coupler.
Example IV.3: Consider the example network in Fig. 4.Here every branch has a susceptance of 1.The substation considered has two busbars n 3 and n 4 .The slack node is n 4 .The PTDF before busbar splitting without the busbar coupler, P T DF ,pre,ideal , is given by (using the formulas provided in Section B-A) • The calculated theoretical PTDF of the busbar coupler P T DF pre,ideal c,: is given by (see Definition III.1) • n 1 n 2 n 3 n 4 n 5 n 6 n 7 24 18 30 0 12 24 18 .
The PTDF after busbar splitting, P T DF post , is given by (also using the formulas provided in Section B-A) The BSDF vector is given by (see Definition IV.1) The required equality (see Theorem IV.2) indeed holds.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

A. More Than Two Busbars
If there are more than two busbars at a substation, the BSDF can still be used to calculate the effect of busbar splitting by applying it iteratively.Starting with the PTDF of the pre-ideal case, first use the BSDF to split busbar A from the other busbars, then to split busbar B from the remaining busbars, and continue like this until all busbars are split.The only difference from the two-busbar system is that the BSDF and the PTDF update has to be done multiple times.The order in which the busbars are split does not matter.

V. BENCHMARKING OF COMPUTATIONAL PERFORMANCE
In this section we benchmark the BSDF approach with other methods for bus splitting previously described in the literature.More precisely, we contrast the BSDF approach with (i) an approach for updating the Phase Angle Distribution Factors (PADF) [16], [17], and with (ii) a post-compensation method [16], [17] which we term LU-factor approach in the following.Both approaches are used to compute the DC voltage angles and are summarized in detail in Appendix C.
In order to perform the computations we consider 4 different power networks of considerable size, namely, the IEEE case300 network [18], the MATPOWER case3120sp network [18], a TenneT network of the Netherlands, and a (TenneT) high voltage network of Europe.Table I shows the number of nodes (1st column) and branches (2nd column) of each network.Moreover, for each network a specific node with a specific configuration is chosen for bus splitting.Table I also shows the number of branches connected to busbar A (3rd column) and busbar B (4th column) for the corresponding configuration.Finally, the calculations are performed by considering either all branches of the network or by restricting the network to 100 monitored branches [19], which speeds up calculations considerably.Since the PADF and LU-factor approaches are based on a node-bynode matrix this leads to a corresponding reduction in nodes which is shown in the 5th column of Table I.However, a drawback of the LU-factor approach is that the L and U matrices themselves cannot be reduced in size (only the resulting nodal susceptance matrix can be reduced).
All methods are implemented in Python using NumPy [20] and SciPy [21].For calculating the PTDF and PADF the sparse structure of the network is exploited by using SciPy's sparse linear solver.For the LU-factor approach SciPy's splu function is used which employs the underlying SuperLU library [22] 3 .The benchmarking was done using an Intel i5-10210U CPU at 1.60 GHz.The run times of the various methods are tested using Python's timeit.
Tables II, III, IV, and V show the compute times for the three different bus splitting approaches for different calculations types.In Fig. 5 the values of each table are plotted in a separate panel.First, Table II (upper-left panel in Fig. 5) shows the computation time of calculating the base matrix.The LU-factor approach is significantly faster than the PTDF or PADF approaches.
Second, Table III (upper-right panel in Fig. 5) shows the computation time of calculating the change matrices for a single bus split (e.g. the BSDF vector).Again the LU-factor approach is fastest.However, in this case all three approaches exhibit the same order of magnitude.
Third, Table IV (lower-left panel in Fig. 5) shows the computation time of updating the load flow using pre-computed change  matrices.Here the BSDF approach turns out to be one order of magnitude faster (for larger networks) than the other two approaches.We note that in this case the PADF and LU-factor calculations are identical.Consequently, the BSDF approach is advantageous for applications in which the load flow related to plenty of injection scenarios needs to be computed after a topology reconfiguration.
Finally, Table V (lower-right panel in Fig. 5) shows the computation time for the BSDF and PADF approaches 4 of updating the pre-computed base matrices using the change matrices.In other words, comparing Table V with Table II indicates how many consecutive bus splits can be computed before recomputing the base matrix is faster.Both approaches exhibit the same order of magnitude with the BSDF approach being faster in most network cases.

VI. SUMMARY AND DISCUSSION
In this study, the Bus Split Distribution Factors (BSDF) are defined and derived.The BSDF enable a fast and clean way to model the effects of bus splitting on the power flow.Given a PTDF matrix (which is expensive to compute due to matrix inversion), for each substation reconfiguration the PTDF can quickly be updated solely via linear operations (similar to the LODF for branch outages).This enables, for example, much faster screening of topological remedial actions (including substation reconfigurations) in congestion management applications [2], [3], [4], [5].
The derivation of the BSDF formulas is based on modelling the busbar coupler as a branch with vanishing reactance and by using transformed LODF formulas.The problem is that the standard LODF formula diverges for opening the busbar coupler (i.e. a branch with effectively vanishing reactance).The trick is to transform the LODF formula by shifting the injections (and again correcting for the shift).
Considering 4 different power networks of considerable size, the BSDF approach is benchmarked against other methods for bus splitting which are based on updating the DC voltage angles [16], [17].It turns out that the BSDF approach outperforms the other approaches by almost one order of magnitude in terms of DC load flow update.
This work can be extended in several directions.Obvious next steps include generalizing the BSDF to work with multiple substations in one go, similar to the MODF which extend the LODF [8], [12], [13], [14].Also interesting is finding a BSDF formula that can handle multiple busbars at the same substation simultaneously.It may also be possible to find a BSDF formula for the case that the network splits, similar to the outage of bridges described in section 4.2 of [24].
Moreover, the work can be extended by combining different forms of distribution factors in one framework.For example, modeling branch outages and adding branches have long been known [15].Similarly, the complement to the BSDF are the Bus Merge Distribution Factors (BMDF) (called a Line Closing Distribution Factor (LCDF) in [25], see (15) in that paper).By combining busbar splitting, busbar merging, branch outage, and branch addtion into a single framework any kind of topological change to the network can be calculated in a fast way.
Finally, the presented work can also be extended in the following way.The PTDF matrix is closely related to the Moore-Penrose pseudo-inverse of the weighted graph Laplacian (i.e.B b in our case) and the matrix of the effective resistance between two nodes (see [8], [11], [24], [26]).Consequenlty, there should be BSDF-like formula for the pseudoinverse and the matrix of effective resistances to allow for node splitting.

APPENDIX A NOTATION
Consider a network represented by a directed weigthed graph G = (V, E) with V the set of nodes and the set of E ⊆ V × V branches, and weights vector b L ∈ R |E| .The weights vector will also be called the branch susceptance vector.The network has |V | nodes and |E| branches.We denote nodes by n, m, s, whereas branches are denoted by ˜ , k, c.Each branch ˜ has a direction ˜ = (n, m) ∈ E, with n being the from-node and m being the to-node.The nodal vector of phase angles is given by ϕ N ∈ R |V | .The vector of nodal power injections is written as p N ∈ R |V | , and the vector of branch flows as p L ∈ R |E| .
The sets of branches from n (S F n ), toward n (S T n ), with arbitrary direction (S n ), or parallel branches (S n,m ) are defined by The branch-by-node connectivity matrix C ∈ R |E|×|V | is given by and the branch-by-node matrix for n = m 0 else .

APPENDIX B PREREQUISITES
Here we introduce well-known formulas necessary for understanding the paper.

A. Power Transfer Distribution Factors
In Appendix A the branch-by-node matrix B f ∈ R |E|×|V | and the node-by-node matrix B b ∈ R |V |×|V | where introduced.
These matrices are related to the nodal power injections and the power flow over the branches by If the matrix B b would be invertible, then the power flow on the branches could be related to the power injected at the nodes by where B f • (B b ) −1 would be the PTDF matrix.However, B b is not invertible, and a slack node s must be selected and removed to be able to perform inversion.Then B b restricted to the non-slack nodes is an invertible matrix.Denote the the restriction of B f to non-slack columns and B b to non-slack rows and columns by The PTDF is a matrix, P T DF ∈ R |E|×|V | , calculated as follows [6], [7], [8], [9], [15], [27]

P T DF
Intuitively, P T DF ˜ ,n is the (linearized) effect on a branch ˜ of injecting 1 MW to node n and extracting 1 MW from the slack node.

B. Line Outage Distribution Factors
The Line Outage Distribution Factors (LODF) represent the effects of a branch outage.If k = (n, m) is the outaged branch, then the LODF vector, lodf k ∈ R |E| , can be given as (see [9], [10], [11]), This formula assumes that the denominator is non-zero.The denominator is 0 if and only if the network splits after the branch outages [12].The LODF vector is used to calculate the postoutage PTDF by Alternatively, the LODF vector can be used to update branch flows The simultaneous outage of multiple branches can be calculated using the Multiple Outage Distribution Factors (MODF).The MODF is a generalization of the LODF [8], [12], [13], [14].
The matrix inversion lemma [28], [29] describes how the inverse of the updated matrix relates to the inverse of the original matrix: with Q ∈ R r×r given by Bus splits and merges can be described in this framework by using r = 2 and the following matrices [16] Next we describe two strategies for updating the voltage angles induced by bus splitting.

A. Phase Angle Distribution Factors
The first approach directly calculates the updated inverse matrix (B b ) −1 by using the matrix inversion lemma.The (B b ) −1 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.for every branch ˜ = (n, m) and node x.

B. LU Decomposition
Another approach to update flows is by calculating LU factors of B b and using those to solve the linear system.More precisely, the LU decomposition is a decomposition B b = LU with L ∈ R n×n a lower triangular matrix and U ∈ R n×n an upper-triangular matrix.A solution ϕ N to ϕ N = (B b ) −1 p N can be found by first solving ψ = L −1 p N by using forward substitution operations, and then solving ϕ N = U −1 ψ by using backward substitution operations [19].Highly optimized libraries are available for finding the LU decomposition of sparse matrices and using it to solve linear systems of equations [22].
More specifically, the technique described in [17], [29] uses LU factors to solve part of the matrix inversion lemma (15).Specifically, the equation can be solved in various ways.Solving X = (B b ) −1 M and Z = M T X is useful when having an earlier calculated solution ϕ N .Then the delta voltage angle after the network change is given by Δϕ N = −XQM T ϕ N , and ϕ N,new = ϕ N + Δϕ N is the solution to (14).This technique is called post-compensation.

Fig. 1 .
Fig. 1.Example network to demonstrate Lemma II.1.In the left panel, the susceptances are shown.In the right panel, the power flow according to Lemma II.1 for n = n 3 is shown.It is crucial to note the PTDF is dependent on a chosen slack node, but the flows in this diagram are not.

Fig. 1
Fig. 1 shows a simple example.The left panel shows the susceptances of each branch.The right panel shows the nodal injections B b :,n 3 and the resulting flows B f :,n 3 .There is only non-zero flow on branches connected to n 3 .The susceptances of the red branches give the flows.In other words, if B b :,n is a nodal injection vector having injections on the ends of the branches connected to n, then B f :,n is the resulting flow on those branches.Proof: By definition, it holds for the reduction P T DF of P T DF to the non-slack columns that P T DF • B ,b = B ,f .Let s denote the slack node.Since the column of the PTDF of the slack node is zero,

Fig. 2 .
Fig.2.All panels show the branch flow in the network under the nodal injections given inside the circles.The green circles give nodes with positive injections and the red circles give nodes with negative injections.

Fig. 4 .
Fig. 4. network before (left panel) and after (right panel) bus split.The numbers inside the circles are nodal injections.The nodes are colored green if the nodal injection is positive, and red if the nodal injection is negative.The dashed line indicates a busbar coupler.

Fig. 5 .
Fig. 5. Comparison of the average CPU time based on Tables II-V.
More specifically, one can formulate the DC loadflow via the inverse of the nodal susceptance matrix, B b ∈ R (|V |−1)×(|V |−1) (prime indicates the removal of the slack node): find the voltage angles ϕN ∈ R |V |−1 that satisfy B b ϕ N = p N for a nodal power vector p N ∈ R |V |−1 (see(10)).When the voltage angle ϕ N is known, the branch flow on a branch ˜ = (n, m) ∈ E is given by pL ˜ = b L ˜ (ϕ N n − ϕ N m ) (see (11)).Considering modifications of B b of the following form [29] ΔB b = M • δB • M T where M ∈ R (|V |−1)×r , δB ∈ R r×r , for some integer r, the equation for the new voltage angle ϕ N,new ∈ R |V |−1 becomes B b + ΔB b ϕ N,new = p N .
: M ∈ R (|V |−1)×2and δB ∈ R 2×2 are given by M x matrix is sometimes called the Phase Angle Distribution Factors (PADF) matrix.The PADF is related to the PTDF byP T DF ˜ ,x = b L ˜ • (P ADF x,n − P ADF x,m )

TABLE IV COMPARISON
OF CPU TIME FOR THE UPDATE/CALCULATION OF THE LOADFLOW AFTER A BUS SPLIT TABLE V COMPARISON OF CPU TIME FOR THE UPDATE/CALCULATION OF THE BASE MATRICES AFTER A BUS SPLIT