Fast Contingency Analysis and Control for Overload Mitigation in Integrated High Voltage AC and Multi-Terminal HVDC Grids

The increasing integration of renewable energy sources calls for an extension of transmission capacity for transporting power towards load centers. Given the involved distances, the integration of high voltage DC connections into AC transmission grids becomes a reality. The interest in such AC-DC transmission grids has been further driven by the potential of multi-terminal HVDC grids interfaced via voltage sourced converters. Their capability of power flow control can contribute to overload mitigation, especially also in case of contingencies. In security analysis of AC grids, the application of linear sensitivity methods for very fast analysis of various contingencies has become established. In this article such contingency analysis is developed for integrated AC-DC transmission grids. Power transfer distribution and line outage distribution factors are reformulated to yield the novel AC-DC power transfer compensation and line outage compensation factors. Those account for the fact that voltage sourced converters allow for controlling power flows. A further innovation is the optimization for the fast identification of AC-DC voltage sourced converter operating point adjustments to mitigate system-wide overloads. Accuracy, principal functionality, robustness, and computational efficiency of the proposed methodology are demonstrated. Scenarios were studied on a modified IEEE 39-bus system and a hypothetical future German power grid.


Fast Contingency Analysis and Control for Overload
Mitigation in Integrated High Voltage AC and Multi-Terminal HVDC Grids Kai Strunz , Samuel Schilling , Maren Kuschke , and Anna Czerwinska Abstract-The increasing integration of renewable energy sources calls for an extension of transmission capacity for transporting power towards load centers.Given the involved distances, the integration of high voltage DC connections into AC transmission grids becomes a reality.The interest in such AC-DC transmission grids has been further driven by the potential of multi-terminal HVDC grids interfaced via voltage sourced converters.Their capability of power flow control can contribute to overload mitigation, especially also in case of contingencies.In security analysis of AC grids, the application of linear sensitivity methods for very fast analysis of various contingencies has become established.In this article such contingency analysis is developed for integrated AC-DC transmission grids.Power transfer distribution and line outage distribution factors are reformulated to yield the novel AC-DC power transfer compensation and line outage compensation factors.Those account for the fact that voltage sourced converters allow for controlling power flows.A further innovation is the optimization for the fast identification of AC-DC voltage sourced converter operating point adjustments to mitigate system-wide overloads.Accuracy, principal functionality, robustness, and computational efficiency of the proposed methodology are demonstrated.Scenarios were studied on a modified IEEE 39-bus system and a hypothetical future German power grid.
Index Terms-Congestion management, contingency analysis, HVDC control, HVDC transmission, integrated AC-DC power system, interior-point method, linear power flow calculation, linear sensitivity method, multi-terminal HVDC system, power system security, voltage sourced converter.

I. INTRODUCTION
B Y 2021, the global wind power capacity has increased to 837 GW, whereas 93 GW were installed in 2021 alone [1].As wind farms are typically located where wind power potential is significant, grid extension is among the solutions considered to bridge the distance to the load centers.To achieve higher public acceptance for grid extension, the usage of underground power cables is given serious consideration in a number of countries, as for example in Germany.Here, power transmission by high voltage DC (HVDC) underground cable is seen as a promising option as no reactive power compensation is needed [2], [3].The interest in HVDC power transmission has been stimulated further by the capabilities of multi-terminal DC structures based on the voltage sourced converter (VSC) [4], [5], [6], [7], [8], [9].The integration of VSC-based DC structures enables the grid operator to modify the power flow [10], [11], [12].
Yet, the operation of an integrated AC-DC grid also brings new challenges.A key challenge is the system-wide security analysis.To address this point, inspiration can be taken from the security analysis of existing AC power systems.According to [13], the issue of power system security covers the following three major functions: system monitoring, contingency analysis, and security-constrained optimal power flow.System monitoring provides the system operator with online information on the current state of the power grid [14], [15].Security-constrained optimal power flow calculation determines redispatch measures to counteract critical grid conditions [16], [17], [18], [19].Contingency analysis is aimed at the quick evaluation of a large number of scenarios in order to identify potentially critical outages.As explained in [13] and [20], linear power flow calculation is considered as an appropriate basis for contingency analysis of AC transmission grids when the main focus is on approximate but very fast algorithms.
With the increasing interest in integrated AC and multiterminal DC transmission grids, the according extension of linear power flow algorithms has attracted attention [21].Linear power flow analysis lends itself to the formulation of linear sensitivity methods, which facilitate computationally efficient contingency analysis.In [12], sensitivity factors are based on the first-order Taylor series approximation, yet the power flow calculation itself remains iterative.A linear and non-iterative application of sensitivity factors for contingency analysis was demonstrated in [22].
In this work, linear sensitivity methods are found to be useful in the analysis of contingencies, too.But beyond a contingency analysis that considers settings of controllers to be constant as for example in [22], the proposed methodology also includes a contingency control technique.The latter is capable of adjusting control settings of integrated AC-DC power systems with the aim to mitigate contingencies.The focus here is on the setting of VSC controls, as those are accessible by Transmission System Operators.To this effect, there are three complementary contributions.First, novel power transfer compensation and line outage compensation factors for integrated AC and multi-terminal DC transmission grids are developed.Those factors define the sensitivity of the line power flow to a controlled change of power injection by an AC-DC converter in the wake of a contingency.Second, a control strategy of the AC-DC converters with the objective to mitigate a detected contingency-triggered overload is formulated as an optimization problem, and the solution process is shown.Third, the performance of the contingency analysis and control is thoroughly evaluated in terms of accuracy and robustness, and the computational efficiency is verified.
The remainder of this paper is structured as follows.The linear calculation of power flows in an AC-DC grid is considered in Section II.In Section III, the linear sensitivity factors for AC-DC grids considering the opportunity of controlling VSC operating points are developed.In Section IV, the optimization problem to mitigate the detected overload is formulated, based on the previously defined sensitivity factors.The accuracy, functionality, and robustness of the proposed method are evaluated in Section V.In Section VI, the computational efficiency when applied to a realistic large-scale power system is confirmed by a case study of a hypothetical future German AC-DC power grid.In Section VII, conclusions are drawn.

II. PREPARATION
In what follows, relevant background information is given.The linear calculation of the power flows of AC and DC grids is revisited in Section II-A.In Section II-B, the linear power flow equations are formulated for integrated AC-DC transmission grids, which will later form a basis for the proposed contingency analysis.Without loss of generality, with the exception of the voltage angles, the following quantities are expressed in the per unit (p.u.) system.

A. Linear Power Flow Calculation of Independent AC and DC Transmission Grids in Normal Operation
As described for example in [13], linear power flow calculation achieves a level of accuracy that makes it practical for contingency analysis for a wide range of operating conditions in electric power transmission systems.For this purpose, power flow equations of an AC grid are linearized by assuming small differences of bus voltage phase angles, neglecting the resistances and shunt capacitances of transmission lines, and setting all voltage magnitudes to 1 p.u.The power equality constraints for each bus i under these assumptions are Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where P inj AC,i is the net real power injection into bus i determined by the difference of real generator power P gen AC,i and real load power P load AC,i ; x AC,ij is the reactance of the transmission line between AC buses i and j; θ AC,i and θ AC,j are the voltage phase angles at buses i and j.
In a DC grid with n b,DC buses, the power equality constraints for each bus i are given by where P inj DC,i is the net real power injection into bus i determined by the difference of real generator power P gen DC,i and real load power P load DC,i ; r DC,ij is the resistance of the transmission line between DC buses i and j; V DC,i and V DC,j are the voltages at buses i and j.
Equation ( 4) may be linearized by assuming [21], [23], [24].This leads to where v DC,i and v DC,j are the voltage differences between the bus voltages V DC,i and V DC,j and the voltage V DC,ref at the DC reference bus, respectively.Defining one reference bus each for the AC and DC grids, respectively, the power equality constraints of (2) and ( 5) are rewritten in matrix form where P inj AC is the vector of the power injections into all AC buses except for the AC reference bus; θ AC is the vector of voltage angles at all AC buses except for the AC reference bus; P inj DC is the vector of the power injections into all DC buses except for the DC reference bus; entries of v DC are the deviations of the bus voltages from the voltage at the DC reference bus; B AC and G DC are composed of the following elements: After building preliminary matrices according to (7a) and (7b) for all i, k ∈ {1, . . ., n b,AC } or i, k ∈ {1, . . ., n b,DC }, the rows and columns corresponding to AC and DC reference buses are removed to obtain B AC and G DC .This step is necessary to obtain matrices that are invertible.The voltage at the DC reference bus is set to 1 p.u., whereas the voltage angle at the AC reference bus is set to 0 • .For all other buses, the voltage angles at AC buses and the voltages at DC buses are then analytically expressed by where X AC and R DC are composed of elements X AC,ik and R DC,ik , and constitute inverses of B AC and G DC , respectively.After solving (8a) and (8b), the power flows in p.u. over line l connecting buses i and j are obtained by By definition, P AC,l and P DC,l are positive if power flows from bus i to bus j.

B. Linear Power Flow Calculation of Integrated AC-DC Transmission Grids in Normal Operation
For the analysis of AC-DC grids, the initially independent power flow calculations for AC and DC grids are integrated by considering power exchanges through AC-DC converters.Thus, power injections of AC-DC converters are added to the bus power injections given in (1) where i denotes an arbitrary AC bus or an arbitrary DC bus; P DCAC,k is the power injection into the AC grid part through converter k; P ACDC,k is the power injection into the DC grid part through converter k; the coupling term c AC,ik , or respectively c DC,ik , is 1 if converter k is connected to bus i, and it is 0 otherwise.
To emphasize that the operating points of converters k can be adjusted according to specified objectives or requirements of the grid operator, the following definition is introduced: where P 0 DCAC,k and P 0 ACDC,k are initial power injections into the AC grid part and the DC grid part, respectively; ΔP DCAC,k and ΔP ACDC,k are changes of the power injections into the AC grid part and into the DC grid part, respectively.
Further, the following power equality constraints apply: ΔP DCAC,k + ΔP ACDC,k + ΔP loss,k = 0 (13) where P 0 loss,k denotes the power conversion losses of converter k at initial operating point; ΔP loss,k gives the changes of power conversion loss when adapting the converter operating point.Given that in this work the focus is on contingency analysis based on the linear power flow formulation, losses of power conversion are discarded.
Without loss of generality, it is assumed that power balance in the AC grid part is guaranteed by an AC reference bus, whereas an arbitrary AC-DC converter k ensures power balance in the DC grid part.Thus, in an AC-DC grid with n conv converters, the initial power injection of converter k into the DC grid part is given by the power equality constraint: When adapting the AC-DC converter operating points, the changes of power injections are related as follows, since the losses are neglected: For the integrated AC-DC grid, the power flows over an AC or DC line l are determined by inserting (10a) or (10b) into (8a) or (8b), and then using the obtained angles or voltages in (9a) or (9b), respectively.

III. LINEAR SENSITIVITY FACTORS FOR OUTAGES AND OVERLOAD MITIGATION
Outages of generators, loads, and lines are examples for contingencies that affect the power flow.The challenge of evaluating a high number of potential outages is central to contingency analysis.The application of linear sensitivity factors for overload detection in AC transmission grids was shown to be suitable in this context [13], [25], [26].Therefore, linear sensitivity factors lay the foundation for the methodology.As such, an essential preparatory stage of the contingency analysis is the calculation of sensitivity factors that measure the impact of changes of nodal power injections and of line outages.Those two sensitivity factors are formulated in Section III-A for AC and DC grids, respectively.
All underlying mathematical models are expressed through algebraic equations, and therefore differential equations are not involved.Following a contingency, it is assumed that transient stability is given [27], [28] so that a new steady state is reached following the outage.As long as this new steady state does not show overloads, the original power flow settings for the AC-DC converters can be maintained.If those original settings entail an overload following the outage and if a modification of the settings allows for mitigating the overload, then those settings are to be modified.Consequently, an interaction between AC and DC grid parts is observed as a result of the contingency.In order to describe the sensitivity of power flows thanks to AC-DC converter control during an outage, the so-called compensation factors are developed in Section III-B.

A. Linear Sensitivity Factors for Outages
In what follows, the detection of contingency-triggered overloads is described.At first, outages of generators and loads are considered, and sensitivity factors for overload detection are addressed.In addition, the formulations involve the detection of overloads as a consequence to outages of AC-DC converters.Secondly, line outages and the associated sensitivity factors measuring the changes of flows are considered.
1) Power Transfer Distribution Factor: In case of outages of generators or loads, power injections need to be shifted to ensure power balance in the grid.Power injections may be shifted to another bus or are distributed.Due to the shifting, the changed power flows may cause line overloads.The power transfer distribution factor (PTDF) measures the change of the flow on a line to a shift of power injection: sr s∈ℬ (16) where ΔP inj sr is the shift of power injection from a sending bus s to a receiving bus r; ΔP l is the change of the power flow over line l connecting bus i and bus j; ℬ is the set of buses to one of which the malfunctioning generator, load, or AC-DC converter were connected initially.By definition, ΔP inj sr is negative if the net real power injection into bus s is decreased, while the injection into bus r is increased; ΔP l is positive if the power flow from bus i to bus j increases due to the shift of power injection.
If no receiving bus is specified, then the latter becomes the reference bus by definition.As such, ( 16) is rewritten as: According to [13], the PTDF sl for AC grids is calculated by Analogously, it follows for DC grids according to Appendix A: If specific receiving buses as in ( 16) are given, then ( 18) and ( 19) are complemented by PTDF AC,lr and PTDF DC,lr as follows: PTDF DC,lsr = PTDF DC,ls − PTDF DC,lr (21) with: If the power injection is from bus s to the reference bus, the altered power flows over an AC line l or a DC line l at initial converter operating points are specified by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where P 0 AC,l and P 0 DC,l are the initial flows over the line l on the AC or DC sides; ΔP inj AC,s and ΔP inj DC,s are the shifts of power injections in the AC and DC grid parts.
Apart from considering the impact of changes of power injections due to outages of generators or loads, the PTDFs for AC and DC grids may also be applied for overload detection due to converter outages in an AC-DC grid.The outage of an AC-DC converter is modeled as simultaneous shifts of power injections in the AC and DC grid parts considering the power equality constraints of ( 12) and (13).Given the outage of a converter k, the shifts of power injections from the AC grid part to the DC grid part and vice versa are specified by ΔP ACDC,k = −P 0 ACDC,k and ΔP DCAC,k = −P 0 DCAC,k , respectively.Taking into account the power equality constraint of ( 15), power balance in the AC and DC grid parts is ensured by adapting the operating point of at least one other converter.Without loss of generality, it is assumed here that the malfunctioning converter k is connected to bus s.The changes of power injections at bus s are then given by where c AC,sk and c DC,sk are coupling terms introduced in (10a) and (10b).Inserting the above changes of power injections into (24a) and (24b), the altered power flows over an AC line l or a DC line l are obtained.
2) Line Outage Distribution Factor: Apart from shifts of power injection, overloads may also be triggered through line outages.As described in [13], the line outage distribution factor (LODF) measures the change of power flow: where P 0 m is the prefault power flow over line m; ℒ is the set of outaged lines; ΔP l again is the change of flow over line l.The calculations of the LODF for AC and of the reformulated LODF for DC grids are detailed in Appendix B.
Given the outage of a line m in the AC part or the DC part of the AC-DC grid, the altered power flow over line l at initial converter operating points is specified by where P 0 AC,m and P 0 DC,m are the prefault power flows over the line m in the AC or DC grid parts; LODF AC,lm and LODF DC,lm are the LODFs for the AC and DC grid parts.
Equations (24a) and (27a) are applied to calculate the modified power flow over an AC line in case of a generator, line, or converter outage in the AC part of the grid.On the other hand, (24b) and (27b) are applied to calculate the modified power flow over a DC line in case of a generator, line, or converter outage in the DC part of the grid.Under the assumption that the converter operating points are fixed, an outage in the AC grid part will have no influence on the DC grid part and vice versa.Such a reciprocal impact is observed if the converter operating points are variable.

B. Linear Sensitivity Factors for Overload Mitigation
While the application of the PTDF and the LODF is just for the overload detection, the concept of linear sensitivity factors is here further developed to consider overload mitigation in AC-DC grids.In this case, the control of the converters allows to modify the power flow and relieve contingency-triggered overloads.Measuring the change of power flow in the AC-DC grid to changes of converter operating points, the power transfer compensation factor (PTCF) and the line outage compensation factor (LOCF) are defined as follows: The PTCFs and LOCFs allow for the calculation of modified converter operating points for overload mitigation while considering outages of generators, loads, AC-DC converters, and lines.At first, the calculation of the PTCFs considering outages of generators, loads, and AC-DC converters is presented.Secondly, the calculation of the LOCFs is addressed.The calculation of the LOCFs is based on the concept of the compensated PTDF for AC grids [13].
1) Calculation of PTCF Considering Outages of Generators, Loads, and AC-DC Converters: Given the outage of a generator, a load, or an AC-DC converter connected to bus s, the altered power flows at initial converter operating points are specified by (24a) and (24b).When then adapting a converter operating point, power injections into the AC and DC grid parts are shifted, and the power flows in (24a) and (24b) are modified.Without loss of generality, the modification of the operating point of converter k connected to bus  is considered.Since the considered outages do not affect reactance or resistance matrices, the PTCF is readily given by the PTDF that refers to the same line and bus of converter connection.This PTDF is multiplied by the respective coupling term linking this bus to the converter number, as also derived for illustration below.The simultaneous change of further converter operating points to satisfy (15) is accounted for through superposition as performed in Section IV.
In contrast to (25a) and (25b), where the shifts of power injections due to a converter outage are specified by the initial converter power injections, the shifts of power injections due to a change of the converter operating points are variable.Thus, the shifts of power injections due to a change of converter operating points can be determined from a modified and more general form of (25a) and (25b): where ΔP ctl DCAC,k and ΔP ctl ACDC,k are the changes of the power injections from the DC into the AC grid part and vice versa through converter k.As a consequence of the shifts of power injections in (30a) and (30b), the power flows at initial converter operating points P AC,l and P DC,l in (24a) and (24b) are modified by adding changes of flows ΔP ctl AC,l and ΔP ctl DC,l : The calculations in (33a) and (33b) consist of two terms each.The first terms P AC,l s∈ℬ AC and P DC,l s∈ℬ DC denote the power flows over line l at initial AC-DC converter operating points while already considering the outage of a generator, a load, or an AC-DC converter connected to bus s.The second terms determine the changes of the power flows over line l when adapting the operating point of converter k.Being numerically identical to the respective PTDFs, the PTCFs in (34a) and (34b) facilitate the analysis of outages while taking into account the opportunity of changing converter operating points.This allows for the identification of modified converter operating points for overload mitigation.From ( 20) and (21), it is possible to extend (34a) and (34b) for the case that the power injection of converter k at sending bus  is balanced by an opposite injection of converter h at bus : The multiplication of the coupling terms in the equations indicates that both must be equal to one for the power shifting to take place.
Based on the calculations of (33a) and (33b), equations for the specification of modified converter operating points for overload relief are derived later in Section IV-B.
2) Calculation of LOCF Considering Line Outages: Given the outage of a line m, the altered power flow at initial converter operating points is determined by (27a) and (27b).As in Section III-B1, it is again assumed that the operating point of converter k connected to bus  is modified and that further converter operating point changes to satisfy (15) are taken into account through superposition.When adapting the converter operating points, power injections are shifted as specified by (30a) and (30b), and the power flows of (27a) and ( 27b As for (33a) and (33b), the calculations in (39a) and (39b) also comprise two terms each.The first terms P AC,l m∈ℒ AC and P DC,l m∈ℒ DC denote the power flows over line l in the AC and DC grid parts at initial converter operating points while considering the outage of line m.The second terms specify the changes of the power flows over line l as a consequence of a change of the operating point of converter k.The LOCFs introduced in (40a) and (40b) facilitate the analysis of the AC-DC power flows considering line outages and converter operating point control.
As for the power transfer compensation factors in (35a) and (35b), it is also possible to model a direct power shifting from converter k at sending bus  to converter h at receiving bus  for the line outage compensation factors.Therefore, (40a) and (40b) are extended using (20) and ( 21 Based on the calculations of (39a) and (39b), modified AC-DC converter operating points for overload mitigation can be determined.Equations for this calculation are derived later in Section IV-B.

IV. OVERLOAD MITIGATION USING COMPENSATION FACTORS
The linear sensitivity factors derived in Section III are now to be used for detection and mitigation of overloads in integrated AC-DC grids.For this purpose, Section IV culminates in the proposed algorithm for contingency analysis and overload mitigation in AC-DC grids.The algorithm benefits from the possibility to perform superposition of the effects of multiple changes of converter power injections on a line power flow by adding the respective sensitivity factors, as introduced in Section IV-A.Then, detailed equations and constraints for the calculation of modified converter operating points for overload mitigation using the introduced sensitivity factors are derived in Section IV-B.In Section IV-C, the overall proposed method for contingency analysis and overload mitigation is expressed through a flowchart.

A. Superposition of Linear Sensitivity Factors
The calculations in (33) and (39) specify the altered power flow over a line l in the AC and DC grid parts while taking into account the outage of a generator, a load, an AC-DC converter, or a line as well as the change of the operating point of converter k.However, multiple outages and changes of multiple converter operating points may also be considered.Since the presented sensitivity factors for overload detection and mitigation are linear, the effects of multiple outages and changes of multiple converter operating points on the power flow can be calculated using superposition.The power flow of an AC-DC grid considering multiple outages and changes of multiple converter operating points can be represented in matrix form as follows: where entries of the vectors P AC ∈ R n l,AC and P DC ∈ R n l,DC are line power flows at initial converter operating points considering multiple outages of generators, loads, AC-DC converters, and lines; n l,AC and n l,DC are the number of AC and DC lines, respectively; entries of the vectors ΔP ctl DCAC ∈ R n conv and ΔP ctl ACDC ∈ R n conv are the changes of power injections from the DC into the AC grid parts and vice versa through the converters; CCM AC ∈ R n l,AC ×n conv and CCM DC ∈ R n l,DC ×n conv are the newly introduced congestion compensation matrices, whose entries are specified by addition of the PTCFs of (34a) and (34b) and LOCFs of (40a) and (40b) for AC and DC grid parts, respectively, to allow for superposition of the effects of multiple changes of converter power injections on a line power flow; entry lk of CCM AC and CCM DC measures the respective change of flow over line l in the AC and DC grid parts to a change of the operating point of converter k.The calculations of (42) allow for a unified analysis of the AC-DC power flows considering multiple outages of generators, loads, converters, or lines while taking into account the opportunity of modifying multiple converter operating points.

B. Calculation of New Converter Operating Points for Overload Mitigation
Overloads on lines can be mitigated if it is possible to transmit the excessive power over alternative lines of the grid.Given the power flow calculations of (42a) and (42b), and considering the limits of power injections of AC-DC converters, an inequality system can be defined as follows: ⎡ where denotes elementwise smaller or equal; entries of P 0 DCAC and P 0 ACDC are the initial converter power injections from the DC into the AC grid parts and vice versa; entries of P lim AC and P lim DC are the transfer capacities of AC and DC lines; entries of P lim ACDC and P lim DCAC are the limits of the real power injections of AC-DC converters.The changes of converter power injections ΔP ctl DCAC and ΔP ctl ACDC are related by the power equality constraints of ( 13) and (15).Neglecting converter losses, those power equality constraints are rewritten in matrix form as where σ 1×n conv is a vector of ones of dimension 1 × n conv : To enable a unified analysis of the inequality system of (43) and the equality constraints of ( 44) and ( 45), the equations are combined and rewritten into the standard form Ax b.Therefore, the two inequality systems of (43) are reformulated as where denotes elementwise larger or equal; I n conv is an identity matrix of dimension n conv × n conv ; 0 n conv is a zero matrix of dimension n conv × n conv ; 0 1×n conv is a zero matrix of dimension 1 × n conv .The inequality system of (47) is multiplied by −1 and combined again with inequality system (48), leading to where CCM and ΔP lim are defined in (50) and (51).Congestion management through adaptation of converter operating points is possible if the inequality system of (49) has a feasible solution.Feasible solutions can be determined by applying an interior-point method to the inequality system [29].Given a feasible solution of (49), new VSC operating points for overload mitigation are specified.

C. Flowchart of Process
Based on the foundations established in Sections III-A to IV-B, a process for contingency analysis and overload compensation was developed and formulated in the flowchart of Fig. 1.After importing the grid data and initializing the outage counter ν, sensitivity factors PTDF and LODF of the AC and DC grid parts are calculated.The altered AC-DC power flow is specified considering an outage of a generator, a load, a converter, or a line.Overloads are detected by analyzing if the power flow exceeds the transfer capacity of the line.
If overloads are detected, the outage scenario is saved, and CCM as well as ΔP lim of inequality system (49) are calculated using (50) and (51), respectively.Depending on the considered outage, the entries of CCM are determined by using (34a) and (34b) or (40a) and (40b).The entries of ΔP lim are specified by using the limits of real power transfers and by (24a) and (24b) or (27a) and (27b).
Overloads can be relieved by adapting converter operating points if the inequality system of (49) has a feasible solution.A solution of (49) can be determined by applying an interiorpoint method.If a feasible solution exists, then the new converter operating points for overload mitigation are saved.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.If there is no feasible solution, then an alarm message is displayed [13].The alarm message indicates that the considered outage would lead to congestion of transmission lines and that the congestion cannot be relieved by modifying AC-DC converter operating points.
To deal with situations in which the modification of AC-DC converter operating points does not yield a solution, the process may be extended.In this context, security-constrained optimal power flow (SCOPF) calculation as a function of system security may be applied to determine appropriate redispatch measures.

V. VALIDATION
The performance indicators considered here are the functionality, accuracy, and robustness.The validation of the proposed method comprises three stages.In Section V-A, the process of overload detection is addressed.The functionality in terms of detected overloads and the accuracy of the power flow calculation based on linear sensitivity factors compared to the nonlinear power flow calculation are analyzed.In Section V-B, the assessment of functionality and accuracy is extended towards the novel compensation factors and their application for overload mitigation.In the third stage in Section V-C, the robustness of the method towards parameter uncertainty is addressed.The robustness is tested by analyzing the impact of stochastic variations in network parameters on the linear sensitivity factors.Fig. 2. Version of IEEE 39-bus system modified from [31] through DC grid.
The considered test system is the IEEE 39-bus system augmented with five-terminal six-bus DC grid as depicted in Fig. 2 and modified from [30], [31].The dataset of this AC-DC 39+6bus system is included in Appendix C. All program code was implemented in Matlab R2020b and executed on a personal computer with a processor Intel(R) Core(TM) i7-7700K and 32 GB RAM.

A. Overload Detection
In what follows, the functionality and accuracy of the overload detection process is evaluated.First, the results of the proposed AC-DC contingency analysis and overload detection are demonstrated.Then, the accuracy of these results is verified.For this purpose, the relative deviation in branch power flows and currents calculated with the proposed linear AC-DC power flow calculus and a nonlinear AC-DC power flow calculus as in [32] is assessed.
The rated power of each AC-DC converter is 1000 MVA.The initial power exchanges from DC to AC grid parts are specified for converters C1 to C4, while converter C5 ensures power balance, which is accomplished using (14)  According to Fig. 1, single outages of AC transmission lines, AC generators, and DC transmission lines were considered.For each possible scenario, the power flows were calculated by (24) or (27), respectively.The power flows were then compared with the limits of real power flows specified in Tables VI, VII, and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.I.
To verify the accuracy of the proposed method, an outage in the DC grid part was considered, and line D2-D6 was taken as an example.The results of the linear and nonlinear power flow calculations for the real power flows are shown in Table II.The corresponding results for the root-mean-square (RMS) values of the currents in the AC grid part as well as the magnitudes of the currents in the DC grid part are considered in Table III.
Since the method's scope is centered on the evaluation of highly loaded or overloaded lines, only branches with a loading of at least 70% are considered.In Table II, the maximum of the magnitudes of the relative deviations between the results of the linear and nonlinear power flow calculations is 2.4%.The results are in accordance with the accuracy of linear power flow calculation of AC transmission grids, given at around 5% [13].Since the linear power flow calculus is primarily formulated for the computation of real power flows, the results for the latter are generally more accurate than for current flows.This is also observed in Table III, where the recorded results for AC RMS values and DC magnitudes reveal magnitudes of deviations of up to 4%, respectively.Both lower and higher deviations were observed for other contingencies.Thus, for the detection of the violation of a flow limit, a margin should be considered to adjust for the relative deviation compared with nonlinear power flow calculus.In this context, it is recommended to modify the actual flow limit by 10% for the purpose of detecting overloads based on linear calculations.Then, the accuracies are such that both power flows or current flows may serve as a basis for comparisons regarding branch flow limits.In the flowchart of Fig. 1, limits for real power flows are considered.
In accordance with the made assumption that AC voltage magnitudes are at 1 p.u., linear power flow calculus is as such not suitable to evaluate voltage magnitudes in AC grids [13].The results obtained in the context of this validation confirm that the assumption is plausible.During all studies of normal operation and contingencies performed for the AC-DC 39+6-bus system in Fig. 2, there was only one contingency where the magnitude of the voltage of a single bus fell out of the range from 0.9 p.u. to 1.1 p.u., which is the acceptable range as specified by the European Network of Transmission System Operators for Electricity (ENTSO-E) [33].This specific voltage drop occurred at bus A15 during the outage of transmission line A15-A16.During this outage, bus A15 is connected to the remaining part of the AC-DC system by just one other transmission line A14-A15.Also, bus A15 is a bus with a relatively high load in terms of both active and reactive power, and the bus is not directly linked to any bus with controlled voltage close by.When such or similar situations are observed, the nonlinear power flow calculus provides a basis for further analysis of the AC voltages.

B. Overload Mitigation
Having confirmed the functionality and accuracy of the method for overload detection, a similar assessment is conducted for the proposed overload mitigation.For each outage scenario in which too high a real power flow over a branch was detected, the converter operating points were optimized by solving (49).As indicated in Table I, in 18 out of 26 scenarios a feasible solution was found, so that the overload can be mitigated.The results of the proposed method were also confirmed by nonlinear power flow analysis that was performed for comparison.For the cases where overloads could not be relieved by just controlling converter operating points, the results were reconfirmed by an exhaustive search.
The optimization of converter operating points to mitigate overloads is based on the novel compensation factors.The accuracy of the power flow calculation using these factors was evaluated again taking the nonlinear power flow calculation as a reference.For the sake of illustration, the results for one outage Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE IV REAL POWER FLOWS OVER BRANCHES WITH AT LEAST 70% LOADING AT OPTIMIZED CONVERTER OPERATING POINTS AND OUTAGE OF LINE D2-D6
scenario, the outage of line D2-D6, are presented in detail.At the initial converter operating points (52), the line D2-D3 was overloaded, for example.After solving the optimization problem (49), the following adjusted converter operating points given in MW were found: The optimization of the AC-DC converter operating points led to modified branch power flows as shown in Table IV for linear and nonlinear power flow calculus.As given in the last row of this table, the absolute value of the power flow over the line D2-D3 is decreased, such that the power flow does not violate the limit.This result confirms the functionality of the proposed optimization method for overload mitigation.
For the considered outage, the magnitudes of the relative deviations between the results of the linear and nonlinear power flow calculations are listed in Table IV.In general, the observations resulting from the analysis of the accuracy of the deviations made in Section V-A also apply to overload mitigation.

C. Robustness Towards Parameter Uncertainty
When calculating linear sensitivity factors, the grid parameters, such as admittances of transmission lines, are typically considered as given.In reality, however, those parameters are subject to a certain degree of uncertainty.For evaluating the robustness of the developed method, the impact of parameter uncertainty on the linear sensitivity factors is analyzed using Monte-Carlo simulation as a stochastic method.
For each AC and DC branch admittance, a normal distribution with the mean equal to the admittance given in the grid data set and the standard deviation equal to 10% of the admittance was considered.Then, the Power Transfer Distribution Factors were calculated according to (18) and (19), whereas each branch admittance was randomly sampled from the corresponding normal distribution.In accordance with guidelines from [34], 7500 samples were taken.As such, a set of 7500 values was obtained for P T DF AC,ls and P T DF DC,ls .The mean of the 7500 P T DF AC,ls and P T DF DC,ls for each s and l generated in the stochastic process was eventually compared to the values of the P T DF AC,ls and P T DF DC,ls calculated with given admittances The second and third columns of Table V give the maximal magnitudes of the absolute and relative deviations between the values of each linear sensitivity factor obtained with and without considering the parameter uncertainty, respectively.To calculate a relative deviation with respect to a value of a linear sensitivity factor calculated without parameter uncertainty, a deviation with an absolute value smaller than 10 −12 was rounded to 0. In all cases considered, the relative deviations remained lower than 1%.
To further examine the extent to which parameter uncertainty affects the linear sensitivity factors, the analysis was repeated with a larger range of uncertainty.For each AC and DC branch admittance, 7500 samples from a normal distribution with the mean equal to the admittance given by the grid data and the standard deviation equal to 25% of the admittance were taken.Then, the same approach was conducted to calculate the deviations in PTDF, LODF, PTCF, and LOCF values obtained with and without considering parameter uncertainty.The maximal magnitudes of the absolute and relative deviations under these conditions are listed in the fourth and fifth columns of Table V.As the relative deviations do not exceed 5%, the simulation results confirm that the method remains accurate even if the network parameters are subject to significant uncertainties.

VI. APPLICATION
In the following, the benefits in terms of reduced computational effort when using the developed linear contingency analysis and control are confirmed by applying the method for a realistic large-scale integrated AC-DC grid.The test system represents a hypothetical scenario of the future German transmission system and is characterized by high wind power generation in the northern part of the country, while several important load centers are located in the southern part.Thus, power transport over long distances is necessary, and the multi-terminal overlay DC grid depicted in Fig. 3 is introduced [2].
The DC grid consists of nine AC-DC converter stations with a maximal capacity of 6 GVA, and twelve DC transmission Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.corridors arranged in north-south and east-west corridors, each with a capacity of 10 GW in this study.The AC grid model comprises 194 buses, 75 generators, and 398 transmission lines.The conducted case study considered single outages of generators as well as of AC and DC transmission lines.In total, 485 scenarios were evaluated.
For all scenarios, the algorithm of Fig. 1 is used.Regarding the outages of AC generators, in 42 out of 75 cases an overload was detected, yet in 41 cases it could be mitigated by the contingency control through optimization of the converter operating points.Regarding the AC transmission line outages, overloads occurred in 180 out of 398 scenarios, subsequent overload reliefs were achieved in 164 cases.Finally, outages in the DC grid part led to three identified overloads, out of which one was alleviated after the optimization of converter operating points.The results are visualized in Fig. 4.While overloads were detected in a substantial share of scenarios, thanks to the developed optimized control of AC-DC converters 91.6% of them could be mitigated.
As such, a secure operation of the system could be maintained after 466 out of 485 possible outages.
Using Matlab R2020b and the personal computer mentioned in Section V, the whole process took around 20 seconds.A nonlinear power flow calculation considering single AC line outages, single AC generator outages, and single DC line outages was also conducted.It took around 92 minutes.This outcome proves the applicability and the practical benefits of the proposed linear contingency analysis for realistic large-scale AC-DC power systems.

VII. CONCLUSION
A methodology for fast contingency analysis and control of integrated AC-DC transmission grids was developed.Inspired by the contingency analysis of AC transmission grids, the proposed contingency analysis and control of integrated AC-DC transmission grids is also based on linear sensitivity methods, allowing for the efficient evaluation of a large number of scenarios.To readily address contingency-triggered overloads, the newly developed algorithm enables the mitigation of the detected overloads by making use of the power flow control capability of the AC-DC voltage sourced converters without referring to a redispatch of power generation.
As part of the development, three key contributions were made.First, sensitivity factors as known from contingency analysis of AC power transmission systems were redeveloped for the analysis of integrated AC-DC grids.An important role is attributed to the line outage compensation factor LOCF.This factor gives the sensitivity measuring a change of power flow on the studied line to a change of power injection by an AC-DC converter while there is a simultaneous outage of another specified line.The factor is formulated in two variants depending on its application for AC or DC grid parts.
Second, a method of line overload mitigation was developed to integrate congestion management with the contingency analysis of AC-DC grids.This contingency control makes use of the sensitivity factors and an interior point method to optimize the operating points of the AC-DC voltage sourced converters.The objective of the optimization is to reach a system-wide overload relief of lines during a contingency.Third, the claims were substantiated through validation and application studies, covering accuracy, robustness, and efficiency.Those studies covered the IEEE 39-bus system augmented with a five-terminal DC grid, and a large-scale AC grid model of Germany with an integrated overlay DC grid.The contingency analysis and control functions demonstrated a high level of accuracy.All contingencies were shown to be detected.In the thorough accuracy analysis based on the modified IEEE AC-DC 39+6-bus system, magnitudes of relative deviations in power flows of heavily loaded branches between the proposed linear and the nonlinear solution were in the same range that is observed when AC grids alone are considered.Impressive was also the robustness of the proposed sensitivity factors toward parameter uncertainty.For standard deviations of all branch admittances as large as 25%, the changes of the power transfer distribution factors did not exceed 5% for both AC and DC grid parts, respectively.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
For a large-scale future AC-DC grid covering Germany, the contingency analysis and integrated optimized control of the AC-DC converters for overload relief was shown to be fast in the system-wide mitigation of line congestion as triggered by contingencies in AC-DC grids.On a personal computer, the whole process implemented as a code of Matlab R2020b took just 20 seconds, while the nonlinear counterpart needed 92 minutes.Thus, the proposed method also lends itself to online application in practice.At the same time, the accuracy of the AC-DC linear power flow calculation results is consistent with the one reported for linear contingency analysis of just AC power transmission grids.As such, the proposed methodology of contingency analysis and control was proven to be efficient and valuable in contributing to power system security analysis of modern grids integrating AC and multi-terminal DC power transmission.
APPENDIX A CALCULATION OF PTDF FOR DC GRIDS From (9b), the change of flow over DC line l to a change of power injection at bus s and opposite change at bus r is determined by: The DC voltage differences v DC,i and v DC,j are obtained from (8b):  −1) .(56) Thus, the derivatives of the DC bus voltages with respect to bus power injections are given by: For the case where r is the reference bus, the entries R DC,ir and R DC,jr are omitted, leading to (19).

APPENDIX B CALCULATION OF LODF FOR AC AND DC GRIDS
The LODF for AC grids is a function of PTDFs.The LODF measuring the change of flow over line l due to the outage of line m is given by [13]: where buses s and r were originally connected by line m.In the special case where the outage of line m leads to islanding, then PTDF AC,msr = 1, and (62) is not defined.To handle this issue, the LODF may be set to zero [13].
The derivation of a formula for the case of the DC grid is inspired from the procedure that led to (62).Fig. 5(a) shows DC line m, which is connected to the remainder of the grid through fictitious breakers at bus s and bus r, respectively.The outage of line m can be modeled by the opening of the fictitious breakers as depicted in Fig. 5(b).Alternatively, the outage may be modeled without changing the network topology as illustrated in Fig. 5(c).For this purpose, power P DC,m is injected into bus s and − P DC,m is injected into bus r so that P DC,m flows over DC line m.Thanks to those injections, there are no power flows over the breakers even if they remain closed -effectively representing the outage of line m as far as the remainder of the grid is concerned.The modified flow P DC,m is specified by applying the PTDF: . (68)

APPENDIX C PARAMETERS OF MODIFIED IEEE 39-BUS SYSTEM
The parameters of the IEEE AC-DC 39+6-bus system are modified from [30], [31].The base apparent power is set to 1000 MVA, the base voltage for the AC grid part is 345 kV, and the base voltage for the DC grid part is 645 kV.The series reactances, the series resistances, the shunt susceptances, and the limits for real power flows of AC transmission lines are given in Table VI.The series reactances, the series resistances, the tap ratios, and the limits for real power flows of transformers are summarized in Table VII.The series resistances and the flow limits of DC transmission lines are given in Table VIII.The real and reactive power injections at AC buses, and the set voltage magnitudes for the considered grid operating points are presented in Table IX.
The applied converter station models are governed by (10) to (13) in Section II-B.The real power injections from the converters into the AC and DC grid buses occur according to (10).The initial operating point was selected such that the real power injections of converters C1 to C4 are controlled to specified values, given by the first four entries of (52).The real power injection of C5 into the AC grid ensures power balance in the DC grid part with the DC bus voltage of C5 controlled to a set value.Thus, for this type of control, D5 acts as a slack bus in the DC grid part with a bus voltage at 1 p.u.For the linear power Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE IX PARAMETERS OF AC BUSES IN AC-DC 39+6-BUS SYSTEM
flow calculus, the power balance in the DC grid part is maintained via (14).For consistency with the predefined setpoints (52), power injections at buses A7 and A31 are modified from Table IX.Alternatively, the setpoints (52) may be modified instead to match the specified power injections.Beyond that, A39 serves as a slack bus with adapted power injection.
Since the converters considered are realized through modular multilevel converter technology, filters are not needed since sinusoidal waveforms are closely approximated.Voltages inside the converters can be calculated by considering relevant transformer and arm reactances, which are chosen at 0.16 p.u. and 0.13 p.u., respectively, for a base apparent power of 1000 MVA and a base voltage of 345 kV.Reactive power injections into the AC grid part are set to zero, and those settings are only relevant for the nonlinear power flow calculus.

P
AC + CCM AC ΔP ctl DCAC P DC + CCM DC ΔP ctl ACDC P 0 DCAC + ΔP ctl DCAC Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 1 .
Fig. 1.Flowchart of contingency analysis and overload mitigation based on linear sensitivity methods.

Fig. 5 .
Fig. 5. Model of DC line outage; (a) before outage of line m; (b) outage of line m; (c) model of outage of line m for contingency analysis with no power flow through fictitious breakers.Modified from [13].

P 1 − 1 r
DC,m = P 0 DC,m + PTDF DC,msr P DC,m .(63)Assumingthat PTDF DC,msr = 1, (63) is rewritten as:P DC,m = 1 PTDF DC,msr P 0 DC,m .(64)The outcome of (64) is used to calculate the power flow over any line l in the remainder of the grid:P DC,l = P 0 DC,l + PTDF DC,lsr P DC,m .(65)Inserting (64) into (65), the power flow over line l can be expressed as follows:P DC,l = P 0 DC,l + LODF DC,lm P 0 DC,m(66)with the newly formulated LODF for DC grids given by:LODF DC,lm = PTDF DC,lsr 1 − PTDF DC,msr .(67)For special cases, the same applies as stated for (62).The final expression of the LODF for DC grids is obtained by inserting (61) into (67):LODF DC,lm = DC,ij ((R DC,is − R DC,js ) − (R DC,ir − R DC,jr ))1 − 1 r DC,sr ((R DC,ss − R DC,rs ) − (R DC,sr − R DC,rr )) LOCF lmk Sensitivity of power flow change on line l to change of injection by VSC k during outage of line m.LODF lm Sensitivity of power flow change on line l to outage of line m.Voltage phase angle at bus i. σ 1×n conv Vector of ones of dimension 1 × n conv .0 n conv Matrix of zeros of dimension n conv × n conv .0 1×n conv Vector of zeros of dimension 1 × n conv .
conv Identity matrix of dimension n conv × n conv .lkh Sensitivity of power flow change on line l to shift of injection from VSC k to VSC h.PTDF lsr Sensitivity of power flow change on line l to shift of injection from bus s to bus r.
l s∈ℬ AC = P 0 AC,l + ΔP ctl AC,l + P T DF AC,ls ΔP inj AC,s (31a) Since the changes of flows ΔP ctl AC,l and ΔP ctl DC,l result from the shifts of power injections ΔP inj AC, and ΔP inj DC, in the AC and DC grid parts, they can be specified taking into account the PTDFs for AC and DC grids introduced in Section III-A: PTCF DC,lk ΔP ctl ACDC,k (33b) where the newly introduced sensitivity factors PTCF AC,lk and PTCF DC,lk for congestion management considering outages of generators, loads, or AC-DC converters are specified by PTCF AC,lk = PTDF AC,l c AC,k DC = PTDF DC,l ΔP inj DC, .(32b)Including(30a)and(30b) in (32a) and (32b) and inserting the result together with (24a) and (24b) into (31a) and (31b) yield the modified power flows considering the control of converter k:P ctl AC,l s∈ℬ AC = P AC,l s∈ℬ AC + PTCF AC,lk ΔP ctl DCAC,k(33a)P ctl DC,l s∈ℬ DC = P DC,l s∈ℬ DC + (34a) PTCF DC,lk = PTDF DC,l c DC,k .( Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.pointsPAC,land P DC,l of (27a), (27b)P ctl AC,l m∈ℒ AC = P AC,l m∈ℒ AC + LOCF AC,lmk ΔP ctl m∈ℒ DC = P DC,l m∈ℒ DC + LOCF DC,lmk ΔP ctl LOCF AC,lmk = (P TDF AC,l +LODF AC,lm P TDF AC,m )c AC,k (40a) LOCF DC,lmk = (P TDF DC,l +LODF DC,lm P TDF DC,m )c DC,k .

TABLE I OUTAGE
SCENARIOS WITH CONTINGENCY-TRIGGERED OVERLOADS AND IDENTIFICATION IF OVERLOADS CAN BE MITIGATED TABLE II REAL POWER FLOWS OVER BRANCHES WITH AT LEAST 70% LOADING AT INITIAL CONVERTER OPERATING POINTS AND OUTAGE OF LINE D2-D6

TABLE III CURRENT
FLOWS OVER BRANCHES WITH AT LEAST 70% LOADING AT INITIAL CONVERTER OPERATING POINTS AND OUTAGE OF LINE D2-D6 VIII of Appendix C. Overloads were detected in 26 scenarios, as listed in Table

TABLE V MAXIMAL
MAGNITUDES OF DEVIATION BETWEEN VALUES OF SENSITIVITY FACTORS OBTAINED WITH AND WITHOUT PARAMETER UNCERTAINTY without parameter uncertainty.The same process was applied to the other linear sensitivity factors: LODF, PTCF, and LOCF.

TABLE VI PARAMETERS
OF AC TRANSMISSION LINES IN AC-DC 39+6-BUS SYSTEM TABLE VII PARAMETERS OF TRANSFORMERS IN AC-DC 39+6-BUS SYSTEM TABLE VIII PARAMETERS OF DC TRANSMISSION LINES IN AC-DC 39+6-BUS SYSTEM