Asymmetrical bipolar HVDC control methods and systems

Abstract

A method for controlling or operating High Voltage Direct Current (HVDC) converters within an asymmetrical bipolar HVDC system, asymmetrical bipolar HVDC systems implementing such controls, controllers and computer software for same. The outputs of the HVDC converters are controlled by providing each HVDC converter with a respective or individual pole converter specific DC voltage reference. A feedback signal is determined for each HVDC converter, and the converter specific feedback signals comprise a modification to the DC voltage reference for the respective converter. Determining the individual pole converter specific feedback signals comprises analysing terminal characteristics of each converter and determining an index indicative of a margin against loss of equilibrium, or modelling terminal characteristics of each converter as an equivalent circuit based on or representing the response of that HVDC converter to a modification of its voltage reference for a given operating state. Each HVDC converter can be associated with an individual pole of a bipole circuit, and the DC voltage reference for each HVDC converter can be modified with a different magnitude responsive to deactivation (as the result of a fault or otherwise) of one or more HVDC converters. Distributing power flow or unbalance to the remaining HVDC converters can compensate for the deactivated HVDC converter(s).

Description

[0001] Asymmetrical Bipolar HVDC Control Methods and Systems The present invention relates to the field of power transmission and, more specifically, the present invention concerns improvements to methods of controlling High Voltage Direct Current (HVDC) converters within an asymmetrical bipolar HVDC system, and asymmetrical bipolar HVDC systems implementing such controls. Background to the invention Mass utilisation of offshore wind energy has been identified as a fundamental enabler of the UK government’s net-zero strategy for tackling climate change [1] . Within the Holistic Network Design [2] of the United Kingdom system to 2030, a range of Multi-Terminal High Voltage Direct Current (MT-HVDC) transmission systems have been considered as an effective and economical approach to transmit bulk offshore power to the existing onshore power network [3]. These more extensive Direct Current (DC) networks which are set to emerge, in a staged manner, and are doing so coincident with an unprecedented growth of demand for HVDC worldwide and the need for Transmission System Operators (TSOs) to describe staged growth of and control paradigms for these DC networks. In this context, the need for multi-vendor (MV) MT-HVDC solutions becomes increasingly likely. Within the UK, the Caithness Moray Shetland project (C-M-S) [4] is the first example of a multi-terminal Voltage Source Converter (VSC) HVDC system outside of China, although this has delivered by a single vendor. In the scheme used by some pilot projects involving multiple vendors, the vendors may be required to disclose information relating to control design and / or to modify their designs to make them compatible with other converters. This “open” approach is not practical within the UK and other international markets as this information is usually proprietary, confidential and / or otherwise unknowable. There is therefore a need for a practical and vendor-agnostic approach to Multi-Vendor Multi-Terminal (MVMT) HVDC control which does not require disclosure or otherwise obtaining knowledge of individual converter design. The principal challenge of MVMT control interoperability is how to ensure HVDC converters of different vendor solutions can work together and maintain stability in a coupled DC network without sharing details of their internal designs, beginning at planning and procurement stage [5], and then being supported across detailed design testing, operational, and refurbishment / modification stages of the asset lifecycle of a MVMT-HVDC system. A challenge within any MT-HVDC control is that unlike an alternating current (AC) system, there is no common network parameter such as AC frequency to inherently align behaviours within that DC network, and as a result many approaches will instead employ either a single point of network control referencing a single HVDC converter and its control of the DC system, or introducing / deriving a common general variable that all converters may then reference. Both of these approaches whilst viable require a strong understanding of the HVDC converter design and behaviour referencing proprietal areas of control structure, and can introduce unwelcome dependencies upon the resilience of that control system or its communication in practical operation. By introducing constant power terminals driven by the dynamics of offshore wind or onshore TSO dispatch, a MTDC system is a non-linear system. Aligning with Lyapunov’s First method [6], stability assessment of a non-linear system includes two successive components, namely the existence of equilibrium and the sufficiency of damping. Although there has been significant research into the stability of a DC system in both aspects, there has to date been no published solution that can ensure both aspects without detailed knowledge of the internal control of the participating converters. Hence, it is presently impossible to fully support industrial application of MVMT HVDC. The applicant’s earlier unpublished application number GB2219373.4 explores some similar concepts of control and control system in multi-vendor multi-terminal HVDC systems (and is incorporated herein by reference) but distinct objectives of the present application are to provide accurate assessment of the power transfer capability of asymmetrical bipolar HVDC systems and to provide a means for controlling asymmetrical bipolar HVDC systems whether as a result of a fault which would otherwise reduce capacity and / or capability, as a result of planned outages for maintenance, or simply as an alternative mode of operation. Another objective is to provide vendor-agnostic control of a bipolar HVDC grid that might enable widespread adoption of MVMT HVDC systems. The invention may be implemented in a method of controlling or operating a plurality of HVDC converters within an asymmetrical bipolar HVDC system, a controller for an asymmetrical bipolar HVDC system, an asymmetrical bipolar HVDC system, and corresponding computer program products. Further aims and objects of the invention will become apparent from reading the following description.

[0002] Summary of the invention According to a first aspect of the invention, there is provided a method for controlling (or operating) a plurality of HVDC converters within an asymmetrical bipolar HVDC system, the method comprising: controlling the outputs of the plurality of HVDC converters by providing each HVDC converter with a respective (or individual pole specific) DC voltage reference;; determining a respective feedback signal for each of the plurality of HVDC converters wherein the (converter / pole specific) feedback signals comprise a modification to the DC voltage reference for the respective converter; wherein determining the respective (converter / pole specific) feedback signals comprises analysing terminal characteristics of each converter and determining an index indicative of a margin against loss of equilibrium. This methodology may involve and imply definition of both converter and network equivalent impedance and control effect across fundamental and other relevant control frequencies, accounting for contingency analysis. HVDC converter terminal characteristics may include impedance, voltage droop, power, branch current and nodal voltage, etc. Preferably, each converter is associated with an individual pole of a bipole circuit. Optionally, the method comprises modifying the DC voltage reference for each HDVC converter with a different (or pole specific) magnitude responsive to deactivation of one or more HVDC converters. Deactivation of one or more HVDC converters may be the result of a fault, or may be deliberate (e.g. for planned maintenance, or any other reason), and this may be compensated by distributing power flow (or unbalance) to the remaining converters (optionally the remaining converters of the same polarity). Put another way, responsive to deactivation of an HVDC converter corresponding to a particular polarity, the DC voltage references for the remaining HVDC converters (optionally the remaining HVDC converters of the same polarity) is modified to compensate for the deactivated HVDC converter by distributing the resulting unbalance. In a particular embodiment, the system comprises a plurality of interconnected bipole circuits, each bipole circuit comprising matching pairs of positive and negative poles each with associated HVDC converters, wherein responsive to deactivation of one of the HVDC converters, the method comprises distributing a resulting unbalance across the remaining HVDC converters (optionally the remaining converters of the same polarity). Preferably, this is achieved by independent control of the outputs of each of the HVDC converters, by virtue of said converter / pole specific feedback signals. Optionally, different magnitudes of voltage reference are applied to individual pole converter terminals. Preferably, the index quantifies the interactions amongst voltage droops between the plurality of HVDC converters. Optionally, the index indicates a stability margin against DC power transfer. Preferably, the modification to the DC voltage reference is determined to optimise steady state operation (or stability) of the multi-terminal HVDC system. Preferably, the method comprises modelling the terminal characteristics of each converter as an equivalent circuit, which may be expressed as a Norton equivalent circuit or a Thevenin equivalent circuit based on or representing the effect that HVDC converter has in response to a modification of its voltage reference for a given operating state. This allows the control method to be implemented without detailed information about the converters. Preferably, the method comprises determining HVDC system characteristics by constructing a matrix of extended impedance or extended matrix of conductance for the system. Preferably, the method comprises determining DC voltage droop characteristics for each HVDC converter. Preferably, the method comprises determining an initial value of equilibrium. Preferably, the method comprises normalising the index against loss of equilibrium. Preferably the equilibrium may be identified based on both intact and contingency states of the HVDC network, such that the state may not represent an optimised position for a given operating state, but rather may represent a state optimised in its secure operation in response to change and / or contingency in its operation. Preferably, the method comprises determining branch currents and nodal voltages within the system. The branch currents and nodal voltages may be received at a central controller configured to implement the method. Preferably, the control method may be applicable to a range of differing HVDC multi- terminal topologies- including symmetrical monopole, Full Bipole, and rigid Bipole (without earth return capability). The approach to effect characterisation of the HVDC converter accordingly may apply to each pole converter terminal individually and the equivalent measurement and characterisation of the neutral current terminal is further implied. Preferably, the asymmetrical bipolar HVDC system is a multi-vendor multi-terminal HVDC system. However, this is not a necessity - the inventive concept may optionally be applied to single-vendor multi-terminal systems. The asymmetrical bipolar HVDC system may be comprised in an HVDC transmission system. The HVDC converters may be comprised in an interface or interfaces between one or more HVDC power lines and one or more AC generation systems and / or one or more AC grid networks. The one or more AC generation systems may comprise one or more offshore wind turbines or farms. The one or more AC grid networks may comprise one or more onshore AC grid networks. According to a second aspect of the invention, there is provided a method for controlling (or operating) a plurality of HVDC converters within an asymmetrical bipolar HVDC system, the method comprising: controlling the outputs of the plurality of HVDC converters by providing each HVDC converter with a respective (or pole / converter specific) DC voltage reference; determining a respective feedback signal for each of the plurality of HVDC converters wherein the (pole / converter specific) feedback signals comprise a modification to the DC voltage reference for the respective converter; wherein determining the respective (pole / converter specific) feedback signals comprises modelling terminal characteristics of each converter as an equivalent circuit based on or representing the response of that HVDC converter to a modification of its voltage reference for a given operating state. Optionally, determining the respective (converter specific) feedback signals comprises analysing terminal characteristics of each converter and determining an index indicative of a margin against loss of equilibrium. Preferably, each converter is associated with an individual pole of a bipole circuit. Optionally, the method comprises modifying the DC voltage reference for each HDVC converter with a different (or pole specific) magnitude responsive to deactivation of one or more HVDC converters. Deactivation of one or more HVDC converters may be the result of a fault, or may be deliberate (e.g. for planned maintenance, or any other reason), and this may be compensated by distributing power flow (or unbalance) to the remaining converters (optionally the remaining converters of the same polarity). Put another way, responsive to deactivation of an HVDC converter corresponding to a particular polarity, the DC voltage references for the remaining HVDC converters (optionally the remaining HVDC converters of the same polarity) is modified to compensate for the deactivated HVDC converter by distributing the resulting unbalance. In a particular embodiment, the system comprises a plurality of interconnected bipole circuits, each bipole circuit comprising matching pairs of positive and negative poles each with associated HVDC converters, wherein responsive to deactivation of one of the HVDC converters, the method comprises distributing a resulting unbalance across the remaining HVDC converters (optionally the remaining converters of the same polarity). Preferably, this is achieved by independent control of the outputs of each of the HVDC converters, by virtue of said converter / pole specific feedback signals. Optionally, different magnitudes of voltage reference are applied to individual pole converter terminals. Embodiments of the second aspect of the invention may comprise features of or corresponding to the preferred or optional features of the first aspect of the invention or vice versa. According to a third aspect of the invention there is provided a controller for an asymmetrical bipolar HVDC system, the controller adapted or configured to perform the method of the first aspect and / or the second aspect. Embodiments of the third aspect of the invention may comprise features of or corresponding to the preferred or optional features of the first and / or second aspects of the invention or vice versa. According to a fourth aspect of the invention there is provided a multi-terminal HVDC system comprising a plurality of HVDC converters and a controller according to the third aspect. Preferably, the system comprises at least one HVDC power line, a plurality of HVDC converters between the at least one HVDC power line and an offshore wind farm, and a plurality of HVDC converters between the at least one HVDC power line and an onshore AC grid. Preferably, the system comprises a plurality of interconnected bipole circuits, each pole of each bipole circuit being associated with a respective HVDC converter. The multi-terminal HVDC system may be a multi-vendor multi-terminal HVDC system. The at least one HVDC converter may comprise at least one voltage source converter. Preferably, the at least one HVDC converter comprises a nodular multilevel converter. Embodiments of the fourth aspect of the invention may comprise features of or corresponding to the preferred or optional features of the first, second and / or third aspects of the invention or vice versa. According to a fifth aspect of the invention there is provided a computer program comprising instructions which, when executed by a computer, cause the computer to carry out the method of the first or second aspect. According to a sixth aspect of the invention there is provided a computer readable medium or data carrier comprising the computer program of the fifth aspect. According to a seventh aspect of the invention there is provided a data carrier signal carrying the computer program of the fifth aspect. Embodiments of the fifth to seventh aspects of the invention may comprise features of or corresponding to the preferred or optional features of any other aspect of the invention or vice versa.

[0003] Brief description of the drawings Aspects and advantages of the present invention will become apparent upon reading the following detailed description and upon reference to the following drawings (like reference numerals referring to like features) in which: Figure 1 is a schematic representation of an intact, generic multi-terminal multi-vendor (MTMV) high voltage direct current network comprising full bipoles; Figure 2 is a schematic representation of the network shown in Figure 1, illustrating the immediate impact of a single, negative pole fault; Figure 3 is a schematic representation of the network shown in Figures 1 and 2, with the fault isolated a short time after the fault occurred; Figure 4 is a schematic representation of the network shown in Figures 1 to 3, showing the idealised post-fault restoration position; Figure 5 is a schematic representation of a control architecture of a MTMV HVDC network; Figure 6 is a schematic representation of an MTMV HVDC component model showing (a) nodal equivalent of each node within an MTMV HVDC network, (b) voltage dependent control and (c) voltage independent control; Figure 7 is a schematic representation of a three terminal bipolar HVDC network used for benchmarking, with a single negative pole disconnected to create an asymmetry; and Figure 8 illustrates the results of a power ramp test carried out based on the benchmark system shown in Figure 7. Detailed description of preferred embodiments As discussed in the background to the invention above it is desirable to provide methods and control concepts, and systems implementing them, which enable multi-terminal multi- vendor operation of VSC-HVDC networks. It is also desirable to enhance the applicant’s existing (application number GB2219373.4) but as yet unpublished techniques in assessing converter interactions such that the applicable scenario is expanded to bipolar HVDC networks with any kind of asymmetry in the path of power transfer. This invention will serve an enabler in planning and assessing bulk HVDC grid. As noted above, the applicant’s earlier unpublished application number GB2219373.4 explores some similar concepts of control and control system in multi-vendor multi-terminal HVDC systems, albeit with different objectives. In the present invention, a controller could physically be the same controller as discussed in the earlier application (relevant disclosure replicated below), but operating based on a different control algorithm based on different control margins, acting differently on each individual DC terminal of positive and negative poles, but using a similar approach of modification of the voltage reference of each terminal to (independently) control the HVDC converter outputs. The applicant has identified several factors limiting the functionality and preventing widespread adoption of MT HVDC systems. These include, but are not limited to, the following (at least one of which the invention and embodiments thereof seek to overcome). An MT HVDC system has no inherent operational reference which means that using conventional approaches requires that the system is enforced by a single converter which defines the overall network behaviour or requires communication and / or interaction between converters. Communication between converters introduces its own challenges. Converter design is also highly vendor-specific so control solutions which optimise the converters of one vendor may be incompatible with those of another, so approaches which might work in a single-vendor MT HVDC system (or an “open” MVMT HVDC system such as implemented in China) are generally unworkable in relatively “closed” MVMT HVDC systems. It is also impractical to implement such approaches during online operation. Furthermore, complex systems present a computational problem in that it is time consuming to compute variable cases, and engineering expertise is necessarily high on the part of the end user to implement. In response, the applicant has developed functional models and control concepts that can be implemented in control methods and systems which can facilitate interfacing in a vendor-agnostic manner with HVDC converter designs and control systems. As will be explained below, the inventive concept can be applied at any stage in the planning, specification, testing and operation of an HVDC network without requiring any knowledge of the internal design of converters (which knowledge might be proprietary, confidential or otherwise unknowable). To assess or aid in assessing control interactions in a MVMT-HVDC network, the applicant has developed a method to quantify the interactions between converters against loss-of- equilibrium. The method seeks to define the effect of converter response to a variation in terminal voltage such that conditions of stable operation may be identified and (if appropriate) directed accordingly, and this can be done without knowledge of specific converter designs. This involves representation of the encrypted control detail within a model or replica in the form of an equivalent control droop response to the voltage change, for a given operating condition. Within that data provision, this method can quantify the compatibility of multiple droop coefficients considering the impact of operating point and network metrics. Aligning with the principle of Lyapunov’s First method, this step will serve as a prerequisite and is decoupled from the assessment of small-signal before completing stability assessment for steady state. In summary, the invention (and embodiments thereof) provides methods and systems within which individual convertors can contribute to an overall de-risked voltage profile across a multi-terminal DC system without reliance upon continual updating of control across communication which respects the individual convertor control. A multi-terminal control approach is applied based on receipt of measurements of voltage and power flow within the DC system, together with other time evolving dispatch information of the DC network and current operating points of the DC system provides adjustment to individual DC voltage references to HVDC converter controllers based on an impedance-based contingency and stability assessment of the DC system. The control concept focuses on characterising the DC voltage droop characteristic of the HVDC convertor and the small signal impedance characteristic of the HVDC converter that acts upon the DC system for a given operating point rather than relying on intimate knowledge of the individual converters that delivers these and other behaviours of the convertor. By this approach the multi- terminal control is able to assess interactions between converters in a Direct Current (DC) network; and set an overall operating state of the multi-terminal DC network which is inherently stable to current and evolving operational conditions, even if not subject to further continuous updates; which serve as a course correction to that overall operating state of the multi-terminal system. Within this method, a safety margin against loss of equilibrium can be calculated with terminal behaviour of converters, i.e., without the knowledge of internal designs of converters, which are generally proprietary, confidential or otherwise unknown or unknowable. This serves to provide further protection to loss of communication or other credible disruptions in the network disruption. Hence the operating condition is not necessarily the most optimum from other measures- but is intended to be the most robust for the power flow condition and operating states of the converters present at the time. Application of the mathematical basis of control together with the associated model of this control forms a basis for evaluating, testing and defining the individual vendor approaches to its implementation in practical power system application. By counting the asymmetry between two poles in a transmission network, the model can be used to characterise a bipolar HVDC network, whose branch impedance of positive poles and negative poles of the same electrical sections are not identical, i.e. an asymmetrical power transfer path. In extreme cases of a fault, the faulty pole branch can be switched open whereas the other(s) stay closed to recover significant amount of pre- fault power exchange. With this approach, the onshore transmission system can be protected from undesirable outages caused by cascaded frequency events. The concepts described herein are not specific to any particular converter or controller can be applied to the planning, specification, testing and operating of a High Voltage DC (HVDC) network, and indeed may be extended to low-voltage, or medium voltage DC systems. As will be explained in further detail below, features of an implementation of the inventive concept may include extracting the DC components (i.e., 0 Hz) of network metrics and the terminal behaviours of the converters. The converter terminal characteristics (including constant impedance, voltage droop control, constant power, and constant current) may be linearised at 0 Hz and assumed to behave as (for example) Thevenin or Norton equivalent nodes. Optionally, switching stations can be modelled as terminals without sources. System characteristics may be extracted by constructing a matrix of extended conductance (which incorporates the model of all terminals forming a DC network of n nodes). Existence of an initial value of equilibrium to the DC network may be determined and the resulting value of the determinant can be used as an index to indicate the margin against loss of equilibrium. The index may be normalised against loss of equilibrium to generically index an HVDC network of any topology and / or control structure based on the value obtained at no-load condition. In summary, the key benefit is that the invention enables planning and assessing power transfer capability upon an asymmetrical transfer network without the need for opening up vendor’s IP of detailing converter control, and without compromising on accuracy in said assessment. Other benefits include simplicity in implementation, assessment, and characterisation. The proposed methods may provide for autonomous function with low processing demands resulting in shorter control cycles. Provision of a one-dimensional (normalised) index allows fair and explicit comparison between like and unlike / dissimilar converters, and as a reference signal is convenient and simple for vendors to react or respond to, for example to optimise the converter design and / or control responsively. Figure 1 is a schematic representation of an intact, generic multi-terminal multi-vendor (MTMV) high voltage direct current network 101, interconnecting offshore AC systems (in this example offshore windfarms 103A, 103B, 103C) and corresponding onshore AC grid entry points 105A, 105B, 105C. This example is used herein to demonstrate the inventive concept and features of a possible implementation of the control methods and systems contemplated herein. This example is a full bipole system in which each circuit A, B, C comprises two offshore HVDC terminals or converters 131,132 and two corresponding onshore HVDC terminals or converters 151,152 which are connected by an interposed electrical network of HVDC power lines 107. In each circuit A, B, C the terminals 131 and 151 are of positive polarity and the terminals 132 and 152 are of negative polarity. For the purposes of this example, the positive terminals 131 and 151 transmit 1GW at +525kV and the negative terminals 132 and 152 transmit 1GW at -525kV. Total capacity out of the offshore windfarms is therefore 6GW (being 2GW per circuit or 3GW per pole). The advantage of a bipole system of this nature is that in the event of a converter pole failure an individual circuit can be reconfigured to operate as an asymmetrical monopole which retains 50% of the transmission capacity; in the case where a bipole circuit has a capacity of 2GW, a single fault results only in a loss of 1GW on that circuit. Within the illustrated network 101 there is also provided a DC Switching Station (DCSS) that allows for the flow of electricity to be managed between the three circuits. As shown, the positive poles 131-151 are interconnected, the negative poles 132-152 are interconnected, and the ground return circuits are likewise interconnected by the DCSS. A four-pole DC circuit breaker is also provided which, in combination with the DCSS, improves the availability of residual assets following fault or outage of one of the poles, as will be explained below. Note that each terminal 131,132 comprises a modular multilevel converter (MMC) which typically controls the AC voltage and frequency of the offshore network to which it is connected, and each terminal 151,152 comprises a MMC which regulates the DC voltage of the HVDC link provided by the interposed electrical network of HVDC power lines 107. For the avoidance of doubt, these converters 131,132 and 151,152 convert, firstly, AC voltage generated by the offshore networks 103A,103B, 103C to DC voltage for transmission via HVDC power lines 107 to, secondly, subsequent conversion to AC voltage for onshore distribution (e.g., directly or indirectly to an onshore AC grid via access points 105A,). While MMC converters are the most common type of voltage source converter (VSC) for HVDC applications it will be understood that any voltage source converter (VSC) type may be employed. Figure 2 illustrates a scenario in which there is a single, negative pole fault 172 near the onshore converter station 152C. As shown this fault initially affects all of the negative poles 132A-152A, 132B-152B and 132C-152C due to the interconnectivity effected by the DCSS, resulting in an instantaneous loss of 3GW of capacity with only (effectively) three separate point-to-point (positive) monopoles 131A-151A, 131B-151B and 131C-151C in operation. However, within a short period of time (on the order of a few milliseconds) the DCCB operates to isolate the fault, as illustrated in Figure 3, such that the impact of the fault is now confined to the negative pole on circuit C. In this instant, the AC breakers associated with both converters 132C and 152C and the DC high speed switches on the corresponding ground return circuits are open. If circuits A, B and C were separate point-to-point bipoles, circuits A and B would continue (independently) with a combined capacity of 4GW, and circuit C would operate (effectively as a monopole) at a reduced capacity of 1GW. However, in the present case they are not separate (by virtue of the interposed DCSS) and we now have five active or healthy pole cables connected on the onshore side, three of which are positive and two of which are negative, resulting in a significant asymmetry. While in principle it would seem to be acceptable to operate in this manner, in reality the level of unbalance would tend to risk tripping the residual arrangement of the remaining pole circuits, potentially in a cascade. Further, significant current will start to flow in the neutral line, which is a shared path for delivering power in the circuits of both poles. This will introduce coupling dynamics between both poles in control and risk stability with growing oscillations. By way of explanation, in a bipole circuit, unbalance protection detects and responds to imbalances in current or voltage between the two poles and takes protective measures to restore balance and stable operation. Amongst other remedial actions (including alarms and notifications) an unbalance protection system can issue control commands to converter stations to correct for the unbalance, however if the unbalance cannot be corrected the protection system can trip and isolate further converters or circuits, or even the entire HDVC network to avoid damage or cascading faults. The present invention is, as noted above, concerned with a vendor-agnostic approach to Multi-Vendor Multi-Terminal (MVMT) HVDC control which does not require disclosure or otherwise obtaining knowledge of individual converter design. However, without knowledge of the proprietary, confidential or otherwise (potentially) unknowable aspects of a convertor control, it is not possible for an unbalance protection system to issue the above-mentioned control commands to correctly and reliably address the unbalance. Figure 4 illustrates the idealised post-fault restoration position, in which only the onshore converter 152C on the negative pole is outaged; the offshore converter 132C on the negative pole is switched in and the unbalance is distributed across the healthy DC converters to compensate for the enforced unbalance in circuit C. This is only achievable using asymmetrical control, i.e. the independent control of power flow on each pole of a bipole circuit, which can be used to optimise the use of the available transmission capacity by maximising power flow capability and capacity within the network while respecting network stability. In principle the more convertors participate in the voltage control of a DC network, the greater the flexibility of this control will be; but the inventive concept is enabled provided there is at least one terminal of each polarity which provides this capability whereby the overall DC network is controlled. However, the control of the DC network is subject to practical restrictions in respecting onshore and offshore AC system requirements. For the offshore AC systems providing the offshore wind farm interfaces (i.e. the respective MMCs), it is normally necessary for grid forming control to be established where the HVDC convertors define the frequency and the voltage of the offshore networks (e.g. connected windfarms), thereby representing to the DC system a fixed power flow independent of DC dynamics in its steady state operation. This needs to be accounted for in the overall solution of the DC network but does not provide further flexibility in defining that solution but rather operates as a consequence of the DC voltage as defined elsewhere within this system, within the range of DC voltage available. Onshore the connection interfaces may be operated to a specific intended power flow which does not preclude being set based on a droop power control of the voltage within the DC network, but equally would be dependent also on the operating state of the onshore AC system at the time. Direct control of the AC system, for example at the point of offtake, can be introduced but in the example above would then limit the MVMT network to one point of voltage control and an associated vulnerability to its loss. Architecture of System Control A generic control architecture 201 is illustrated in Figure 5 comprising a number n of individual converters 231. As shown, each converter station comprises inner loops of current control, or equivalent. The characteristics of lower-level design for each converter are aggregated and “masked” on the assumption that the designs will be proprietary, confidential or otherwise unknowable. The designs may include (voltage) regulators of current loops, AC voltage controls, modulation schemes, voltage balancing among sub- modules / arms / phases, main circuit, and other controls related to topology and the convertor operating principles in respect to both the AC and DC systems. At given extreme conditions of either the DC network or the associated converter operation the expression of control may be exposed to non-linearity and the objective of the supervisory control is to ensure that these are not conditions that either the network nor individual convertors are exposed to in steady state or post-steady-state operation; the later representing a range of N-1 and other contingencies informing control priority and associated control margin. The functions of supervisory control 221 are implemented by a central control unit 223 (or controller), which would for example be owned and / or operated by a transmission system operator (TSO). Conversely, the individual converters 231 may be wholly owned and independently operated by one or more third parties and may be provided by a range of vendors. As discussed above, the converter designs may therefore be proprietary, confidential or otherwise unknowable to the TSO, while the TSO desires to be able to control these converters directly it is likely this will not be possible. As such, control per se is delegated to the vendors (and more specifically to the converters) but the criteria, feedback signals, etc. are determined by / for the TSO and provided to individual converter controls for processing, e.g. by providing each converter with a DC voltage reference and by updating or providing a modification 225 to that DC voltage reference as required. Based on feedback measurements of nodal voltages and branch currents along with input orders of power dispatching and nominal voltage, the outputs of the supervisory control will update the incremental values of DC voltage reference (∆Vref) to each of the converters 231 via respective communication links. As the name implies, the purpose of the supervisory control 221 is to course-correct the operation of the overall system towards a steady state condition which is both stable and contingency robust. In this manner the concept is not dependent on the reliability of communication links, as operation for a given condition would remain stable for that operating condition and a range of credible situations ahead of any further update. Actions via the supervisory control would act to (slowly) modify voltage reference responses by the associated converter terminals to correct and drive more secure new operating conditions for the DC network, and transitions of operating state can be implemented via a forward guidance of slow changes over an extended period of time, for example achieving a ramping up or down of power flow. Nodal Model of DC Network For assessment of interoperability, an MTMV-HVDC system is modelled or otherwise represented as an electrical circuit with shunt controllable sources at selected nodes. The nodes represent the positions of converter terminals and junctions of cables within the DC network. Each branch of the circuit characterises the aggregated impedance of a corresponding point-to-point DC cable between two nodes and the connected series elements, such as DC circuit breaker, DC reactor, etc. if applicable. Each node is modelled as a Norton Equivalent Circuit 350, as Fig.6 (a) shows. The current source 351 of the Norton Equivalent Circuit 350 is controllable and represents the active control of the converter. The shunt impedance 353 represents the shunt passive elements at the node, e.g., the aggregated capacitances of MMC cells and inductances, dump resistor, etc. DC junctions (e.g., switching stations without connected converter(s)) are modelled as special nodes where the order of the controllable current source is zero (i.e., without any source). Note that in other embodiments the converter terminal characteristics may be modelled as Thevenin Equivalent Circuits, or indeed any other relevant / workable representation. During operation, the nodes of converter terminals are categorized into two types, namely voltage-dependent and voltage-independent. The model of voltage-dependent control is demonstrated in Fig.6(b). Assuming the modulation process of the HVDC converter is linearized, the S-domain admittance of converter control is modelled as the product of Kiand Regi(s) such that: where Ii, Vrefi, and Virepresent the reference voltage order, terminal voltage, and output current of active control at node i. Therefore, when it is under voltage dependent control, the order of the controllable current source of a voltage-dependent node can be expressed as: For a node with voltage-independent control, we focus on control of constant power as constant DC current control is rare and does not change nodal shunt impedance at 0 Hz. The equivalent power order is assumed to be a result of either a local converter control mechanism, e.g. autonomously balancing the connected AC network [9] or the dispatch order of operator. In both cases, the output power is determined by a one-way input to the DC system. Projecting to the control mechanism shown in Fig.6(c), at steady state, we have: Branch Model of DC Network For each DC branch, including both the positive and negative poles to give a return path, between Node i and j, the aggregated impedance ^^^(^) at steady state is written as: where ^^^(^ ≠ ^) is the aggregated resistance of the branch, including return; ^^^is the shunt resistance at the ^th node. Network Model at Steady State By interconnecting n nodal equivalents as shown in Fig.6(a) with a branch return in between any two nodes, the equilibriums of an n-node HVDC network can be characterized using Kirchhoff's law

[0010] for voltage as: ^^^= ^ where G is an ^ × ^ symmetrical matrix of network conductance (i.e., network admittance at 0 Hz). ^^is the ^–dimension vector of nodal voltage; its ^th (0 < ^ ≤ ^) dimension ^^represents the voltage measurement of ^th node. Similarly, ^ is the ^–dimension vector of nodal current injection from converters. Representing the element of G at ith (0<i≤n) row and jth (0<j≤n) column with Gijgives: The ith component of ^^ and ith component of ^ Considering the control in Fig.5 and Fig.6, the vector of nodal current injection can also be expressed as: where ^ is an ^ × ^ diagonal matrix of droop gain (supervisory droop control?). Its diagonal element at ith row ^^(0 < ^ ≤ ^) represents the voltage droop coefficient of the ^th node; if the node is not droop controlled, then ^^^= 0.^^^^^^^^^^^^ is an n-dimension vector ofthe reference value of droop control (coefficient?), whose ith (0<i≤n) component is ^^^^^(the voltage of the ith node); ^^^ is an n×n diagonal matrix, whose diagonal element ^^^^^ (0<i≤n) is^^;^^^^^^^^^^^^is an n-dimension vector of the arbitrary power injection at all the nodes, whose ith (0<i≤n) component ^^^^^represents the expected power injection at the ith node. ^^^^^is either determined by a dispatching signal from supervisory control function, or an embedded function of the converter at ^th node. Substituting, one can define a function: Treating vector ^^as a dependent variable driven by the independent variable^^^^^^^^^^^^ , thenthe corresponding function of solution to ^^, ^^= ^(^^^^^^^^^^^^), is implicitly defined by the function If a converter node is designed to operate either as voltage- independent or voltage- dependent exclusive, then this assumption can be mathematically expressed as ^^^^^^^= 0 Assessing Existence of Equilibrium Based on the definition in (9), for a practical design of HVDC network, the function of ^^^^,^^^^^^^^^^^^^ is a linear combination of polynomial functions and reciprocal functions; hence it is continuously differentiable and analytic

[0011] with respect of all elements of ^^and^^^^^^^^^^^^ .By applying the theorem of implicit function

[0012] to the above, there must be a unique function of ^(^^^^^^^^^^^^) in the neighbourhood of (or an open set that contains) an operating point (^^^^^ , ^^^^^^^^^^^^^^^), as long as the following two conditions both stand: Condition (I): One equilibrium does exist at the operating point of (^^^^^ , ^^^^^^^^^^^^^^^ ).Condition (II): The Jacobi matrix of ^^^^,^^^^^^^^^^^^ ^ with respect of ^^is not singular at theequilibrium. That is: By complying with Condition (II), the domain of the above-mentioned implicit function ^(^^^^^^^^^^^^) is expandable by iteratively repeating the following steps: 1) selecting one new equilibrium of on the boundary of the neighbourhood to satisfy Condition (I); 2) then satisfying Condition (II) based on the new equilibrium again. By iteratively expanding the domain, the operating point can be expanded throughout a set of manifolds that consistently complies with the above and Condition (II); hence the existence of equilibrium is ensured throughout this aggregated manifold. This process is aligned with the principle of analytic continuation

[0011] . Substituting yields: |^ + ^ − ^^^^| ≠ 0 where ^^^^ is an ^ × ^ diagonal matrix, whose diagonal element on ith row, ^^^^^^(0 < ^ ≤ ^) is^^^^^^^^. Define ^^^as the extended conductance matrix with: ^^ ^^^= ^ + ^ − ^^^^ For a practical HVDC network, the operating points of power must be bounded. To ensure the condition (i.e., the existence of equilibrium) consistently stands within a bounded space of power vector defined by: ^ = ^^^^^^,^^^^^,⋯,^^^^^^^^^^^≤ ^^^^^≤ ^^^^^^ any one of the following two conditions should exclusively stand for all eligible operating points (^V^^^^ , ^^^^^^^^^^^^^^^ ) that satisfies the condition of ^^^^^^^^^^^^^^^ ∈ P:Condition 1 in which the determinant of ^^^is positive: |^^^|> 0 (A) Condition 2 in which the determinant of ^^^is negative: |^^^|< 0 (B) And therefore, the operational margin against loss-of-equilibrium can be defined as|^^^|, whose greater value indicate a better margin. Considering the extended conductance matrix is real and symmetrical by its mathematical definition, Condition 1 (A) can be replaced by a sufficient (enhanced) condition that the extended conductance matrix is positive definite as it will guarantee its leading principal minor of all orders, which include its determinant, are all positive

[0013] ,

[0014] , which is expressed as Condition 3 in which ^^^is positive definite: or equivalently ensure the lowest eigen value of ^^^is positive [13, 14] Similarly, Condition 2 (15) can be replaced by an enhanced condition of consistent negative definiteness of the extended conductance ^^^throughout the space of P as Condition 4 in which ^^^is negative definite: or equivalently ensure the highest eigen value of ^^^is negative as: ^^^^(^^^)< 0 (D) For the enhanced conditions, the operational margin against loss-of-equilibrium can be expressed by ^^^^(^^^)or ^^^^(^^^)when the extended conductance matrix, ^^^, is positive definite or negative definite, respectively. Initial Equilibrium and Refined Assessment Criteria The existence of equilibrium in an HVDC system can be guaranteed in every cycle of control by iteratively complying with Conditions (I) and (II) and securing a pre-defined margin with (A), (B), (C), (D). As a start of the iteration, an initial equilibrium of operating point must be guaranteed for Condition (I). This initial equilibrium can be practically selected as the condition of a no-load condition. Mathematically, such condition is represented by assigning 0 to all the components of^^^^^^^^^^^^in the above. Therefore, the initial operating point to the operating voltage vector ^^=^^^^^^^^^^ is solvablefrom the above as:^^^^^^^^^^ = −(^ + ^)^^^^^^^^^^^^^^^as long as the following condition stands

[0014] : In an HVDC system, the probability of not satisfying the above is practically zero. By assigning the nominal value of DC voltage ^^(>0) to every component of the voltage reference vector^^^^^^^^^^^^ , every component of initial voltage ^^^^^^^^^^shall stay at the nominal value and therefore the initial equilibrium is secured to satisfy Condition (I). At this initial condition, there is where ^^is defined as a diagonal matrix of self-conductance of the HVDC network and its diagonal elements ^^^^will be defined as the reciprocal of shunt (self) resistance 1 ^^^^= ^ ∈ [0, +∞) ^^^^^^Substituting as appropriate, one can obtain the elements of the mutual conductance matrix^^ as Considering the mutual conductance matrix ^^is a Laplace matrix by its definition in the above and the positivity of all branch resistance, ∈ (0, +∞], ^^must be semi- positive definite

[0015] . Thus, for any non-zero n-dimension real vector ^, the quadratic form of ^^shall comply with [13, 14]: ^^^^^≥ 0 (25) As a practical design of droop coefficient must be greater or equal to zero, the diagonal elements of the diagonal matrix K must be greater or equal to zero; considering the above, so must ^^. Therefore, there is: ^^^ ^^^ ≥ 0 ^^^^≥ 0 Summing up the inequations of (25) and (26) on both sides of the operators, one can write the following to prove the semi-positive definiteness of the extended conductance matrix ^^^at the initial energization condition as ^(^^+^^+ ^)^^= ^^^^^^^^^≥ 0 Since the initial condition must be practically included in the expected manifold set of operating power, ^, Conditions 2 and 4 above are therefore ruled out from practical scenarios of HVDC operation, as a semi-positive definite matrix ^^^^^^can neither have a negative determinant nor eigen value

[0013] ,

[0014] . As a result, the assessment criteria to existence of equilibrium can be reduced to Condition 1 or 3. Normalisation of Assessment Margin Although Condition 1 or 3 can serve as a margin against loss-of-equilibrium, the actual implication of the value may vary significantly with variable structures of circuit or control. One consequence is that it is difficult for an operator to interpret the electrical implication of the resultant margin. This makes it difficult to set up a generic standard in specifying the requirement of an interoperable HVDC system for TSOs. Considering this, two types of normalised index, namely CX-index-I (corresponding to Condition 1) and CX-index-II (corresponding to Condition 3), are created to generically index an HVDC network of any topology and control structure as below: CX-Index I (in which the determinant of ^^^is positive): CX-Index II (in which ^^^is positive definite or the lowest eigen value of ^^^is positive): Either index can adequately indicate the operating condition against loss-of-equilibrium. A value close to 1 indicates a closer operating condition to a risk-free no-load status, whereas a value more towards 0 indicates higher risk of loss-of-equilibrium (voltage collapse). To obtain either index, the inputs are droop-coefficients K, network metrics G, (expected or measured) power injections^^^^^^^^^^^^, and measurements of terminal voltage ^^. As required for interoperability, the information of internal design of converters are not required. Inclusion of Supervisory Control The extended matrix of conductance Gexcan be further expanded by including another component matrix of supervisory droop control, Ksas follows: ^^^= ^ + ^ + ^^− ^^^^ where Ks is an n×n matrix. Each of its elements KSij defines the weight of the voltage measurement at the jth node (0<j≤n) towards a global reference value in regulating the voltage at ith node (0<i≤n). This component is dispatched by a supervisory control function (or supervisory control functions) based on the feedback of nodal voltage ^^via communications between converters and a central control unit (e.g., comprised in the supervisory control, see Fig.5). The impact of the communications will be reflected by its non-diagonal elements. Since the inclusion of supervisory control does not change the nature of the methodology, ^^is assumed to be a zero matrix for simplicity. Impedance of Asymmetric Transfer Network The foregoing does not include an explicit model of transfer impedance and associated assessment of power transfer capability in a bipolar HVDC network, so there is now described a generic approach to modelling a bipolar HVDC network which aims to characterise the asymmetry of transfer path as well as shunt elements for stability assessment. By way of practical demonstration, an analytical indicator is proposed to assess the security margin against loss-of-equilibrium in an asymmetric transfer network. Considering a bipolar DC network of ^ bipolar nodes, we may define the vector of nodal voltage as: ^^= [^^^, ^^^, ^^^, ^^^, ⋯ , ^^^, ^^^]^ and nodal shunt current as: where ^^^, ^^^(^ = 1,2, ⋯ , ^) represents the voltages of the positive pole and negative pole at ^th (^ = 1,2, ⋯ , nodes, respectively; whereas, ^^^, ^^^the shunt current at ^th node. The resistance of positive pole, negative pole, and neutral return between nodes ^ and (^ = 1,2, ⋯ , ^ ≠ ^) are defined as ^^^^, ^^^^, and ^^^^, respectively, and the transfer currents flowing from ^th node to ^th node carried by the branch of positive pole, negative pole, and neutral return are defined as ^^^^, ^^^^, and ^^^^, respectively. Applying Kirchhoff's current and voltage laws between bipolar Nodes i and j at steady state, one can write: Solving from (1),(2) where Applying Kirchhoff's current laws to ^th bipolar nodes and then the entire network yields: ^^^×^^^^= ^ (6)where ^^^×^^is the 2^ × 2^ matrix of transfer conductance (network admittance at 0Hz). Substituting (3),(4),(5) into (6) yields: Importantly, the definition in (4) and (7) does not require symmetry of transfer impedance between poles. When the branch section between Bipolar Nodes ^ and ^ is subject to a monopole operation, after isolating a pole-to-ground fault on its negative pole for instance, the corresponding transfer admittance shall be replaced by: 0 ^^^ ^^ ^ (8) 0 To complete a typical multi-terminal HVDC system, two types of shunt branches are considered, namely voltage(droop)-controlled branch and power-controlled branch. For the voltage droop type at ^th node, the shunt current at 0 Hz is expressed as: where ^^^^^^, ^^^^^^are the voltage reference values as ^th node for positive pole and negative poles, respectively; and ^^^, the droop coefficients for positive pole and negative pole. For power-controlled type at ^th node, the shunt current at 0 Hz is expressed as: where ^^^and ^^^are the powers injected to the DC grid at positive and negative poles at ^th node, respectively. Differentiating (9),(10) with respect of nodal voltages, the gradient of shunt currents with respect of nodal voltages is a diagonal matrix of 2n dimensions as: whose diagonal elements are: (^th node is voltage controlled) (12) (^th node is power controlled) (13) Assessment of Equilibrium We may reorganise (6) and define: Define the Jacobian Matrix as the gradient of ^ with respect of ^^as

[0012] Substituting (14) into (15) yields ^ = ^^^×^^− ∇^(^^) (16) Since the cable resistance at the fundamental frequency, i.e.0 Hz, are positive values, it can be inferred that the matrix of transfer conductance is semi-positive definite

[0018]

[0012] , i.e. ^ and therefore it can be inferred from (11) that: Excluding the condition of|^||^^^,^^^ = 0 from real-world application and thus an initial equilibrium can be mathematically secured when all voltages are identical at nominal value with no-load current. From this initial equilibrium, a normalized security margin against loss-of-equilibrium can be obtained by applying the theorem of implicit function as

[0018] where a resultant value close to ^^^^^^= 1 indicates a risk-free no-load condition and ^^^^^^= 0 indicates the loss-of-equilibrium. According to (4)(7)(12)(13), |^|is a diagonally dominated symmetrical real matrix with positive diagonal elements when all constant power orders are zero at 0 Hz. Thus, is semi-positive definite such that

[0014] , is ^th eigen value of the Jacobian matrix ^. Excluding the condition of a singular matrix of transfer conductance, the alternative security margin against loss-of-equilibrium can be obtained as above: As an mathematical alternative, the circuit equation (6) can also be expressed in an equivalent coordinate with linear transformation, namely asymmetrical transformation, as where the transformation matrix is defined as and thus the conductance matrix ^^^^^×^^, vector of nodal voltage^^^^^^^^^, and vector of shuntcurrents^^^^^^^^in the new coordinate from can be defined as Therefore the vector of nodal voltage in the new coordinate frame, namely CD coordinate, is expressed as and nodal shunt current as: where ^^^, ^^^(^ = 1,2, ⋯ , represents the voltages of the balanced (differential) mode and residual (common) mode at ^th (^ = 1,2, ⋯ , nodes, respectively; whereas, ^^^, ^^^the shunt current at ^th node. In an asymmetrical HVDC network, the neutral currents of all sections are determined by residual (common) -mode currents. Substituting (22)(24) into (21), the system equation (6) can be alternatively represented in the CD coordinate as ^^^^^×^^^ ^^^= ^^^(27) By substituting (24) to replace the conductance matrix ^^^×^^, vectors of voltage ^^and current ^ of the original reference frame, the effect of the methodology in (14-20) can be repeated for an asymmetrical HVDC network. Case Study To verify the methodology, a benchmark system was developed for case studies and is illustrated in Fig.7 which shows a 3-terminal bipolar HVDC network 461 is illustrated with each terminal comprising a pair of Half-Bridge Multi-Modular-Converters (HB-MMC) 463, 465, 467. Terminal 1463 and Terminal 2465 are voltage (droop) controlled terminals, and Terminal 3 is assigned as a constant power control terminal. Note that on Terminal 2 the negative pole P2- is disconnected to create an asymmetry analogous to the fault situation illustrated in Figures 1 through 4. Power flow simulation was carried out with MATLAB / Simulink. For every pole of Terminals 1 and 2, the voltage reference and droop coefficient are set at 525 kV and 16.4 A / kV, respectively. The resistance of all inter-station conductance are 5 Ω To test power transfer against loss of equilibrium, a ramp of constant power is applied to both poles of Terminal 3 at a rate of -1000 MW / s (load). The result is illustrated in Figure 8 and described below. After opening the transmission line between Terminals 1 and 2 at the negative pole, when the ramp load of both poles at Terminal 3 starts at 0.5 s, the power is accommodated by the Terminals 1 and 2. Since the system is asymmetrical, the voltage trajectories at both poles are not overlapping each other at every terminal. As the load increases, all voltages, except for the disconnected pole P2-, drop as expected. Simultaneously, both equilibrium margins, IndCX1 and IndCX2 drop from 100% at no-load condition towards 0. Once pole power of Terminal 3 reaches approximately -1050 MW per pole, both indexes reach zero simultaneously and then the system collapses, after which all quantities oscillate chaotically. This result shows that the above-described model of an asymmetrical network accurately reflects real system behaviour and the assessment approach proposed can effectively indicate the existence of equilibrium in this bipolar HVDC system. Conclusions Simulations confirm that for planning and testing the interoperability of a multi-vendor HVDC network, the above-described model and proposed assessment approach, which enable and underpin the inventive concept, is ready for use. It can be directly programmed into a software module of test environment for functional simulation tests. For operating a real-world HVDC network, the inventive concept can be programmed into a central control device, for example as an embedded software package of notch function. The invention provides a MVMT HVDC control paradigm not reliant upon proprietary, confidential or otherwise (potentially) unknowable aspects of a convertor control, verified by simulation under relatively extreme conditions. By definition and solution of the DC network condition based upon the representation of the effect of the convertor control, and a definition of margin in that operation relevant to steady state stability, an overall, vendor- agnostic solution may be obtained in alignment to the TSO priorities for intended operation of a DC network. This therefore allows both the MVMT HVDC control and the associated convertor contributions to it to be described, designed, specified, tested, and deployed with clear definition of individual vendor role and performance. This vendor-agnostic and unified concept of control enables multi-vendor interoperability within a multi-terminal multi- vendor HVDC network. With reference to the simulation proof outlined above, the proposed indices, namely CX- Index I and CX-Index II, effectively quantify the interactions among converter droop co- efficient present, the DC network metrics and operating points within a HVDC network. Either index can indicate the stability margin against DC power transfer, or equivalently loss-of-equilibrium, with a scalar between 0 and 1. Defining a control index for each converter provides a number of advantages, not least simplicity in implementation which may for example be programmed as an autonomous function, and reduced demand on computing power means that shorter control cycles can be accommodated resulting in system-wide improvements. Converter-specific normalised indexes can be communicated to the converters as a feedback signal, shifting the burden for optimising the system onto the vendors providing and / or operating the converters but maintaining control on the TSO side as to how the optimised system should be defined (e.g., prioritising stable operation). The TSO does not need knowledge, detailed or otherwise, of the converter design or more complex aspects such as frequency response. The provision of control index also addresses the issue with existing systems (or attempts to implement them) in that a multi-terminal HVDC system has no inherent or common reference, and would require communication between what might be incompatible resources operated by different vendors, each with their own approach to optimisation, and known systems in which a single, common feedback signal is shared amongst all converter stations based on an overall voltage level in the DC transmission network. Although this indexing methodology is derived from static behaviour of an HVDC system, real time EMT simulations show that it provides good accuracy in an online assessment form one steady state to a new steady state operation across a transition of a significant power ramp. In contrast with prior art approaches, the inventive approach rather seeks to quantify the “effect” of the HVDC converter upon the DC system within a multi-terminal control scheme such that a stable operating point across converters can be both identified and instructed that is then resilient across a range of contingencies, including subsequent loss of the multi-terminal control for a given period of time. This effect may be quantified without need to identify sensitive areas of proprietal control structure within the converter itself and will be different across different operating points and control priorities. Across these, the converter in its effect - as seen from the DC system - will either represent a droop- controlled response to a DC voltage or a constant power terminal - providing connection to offshore wind (for example). Other behaviours such as grid forming controls of various applications upon the AC system may be represented by these two above DC-side representative behaviours within a given control tolerance provided / captured within the multi-terminal control. While the models described herein are based on linear circuit analysis at the fundamental frequency, i.e.0 Hz, it is envisaged that the proposed structure of conductance matrix can be expanded to broader frequencies for small-signal studies in frequency domain as well. The admittance matrix at any interested frequency can be obtained by replacing the branch resistance defined in (4) with the corresponding impedance at the frequency and then substituting the updated admittance in (4) into (7). Reversely, communication based supervisory control can also be designed for transmission system operators by modifying the off-diagonal elements of the matrix in (15) and informing updates of dispatched orders of Vrefat each pole. Also described herein is the first application of asymmetric derivation. However, it can be immediately expanded to three further applications: i) where full bipole converters are situated in close electrical proximity to one- another and operating in parallel, individual pole balancing controls may interact in managing small levels of imbalance under normal operation across the converters leading to hunting and undesired oscillation without additional supervisory correction; ii) where, following the fault / outage, it is possible to switch back into service unaffected elements of a multi-terminal bipole system, this methodology allows management of the inherent asymmetry present in this arrangement; and iii) where the multi-terminal bipole arrangement includes a rigid bipole, this methodology allows the rigid bipole voltage to be balanced to enable its ongoing symmetrical operation within that larger asymmetrically operating network. All features of this methodology have been deliberately constructed to be agnostic to individual vendor approaches to pole balancing such that the behaviour of that control is accounted for and course-corrected as relevant. The use of the control strategies described above within a multi-terminal Bipole HVDC system has the potential for further innovative application if used together with i) the underpinning development of the derivation of a sequence transform for an unbalanced DC system into unbalanced, balanced and residual return current formulations, ii) an ability within a MVMT control to express the network impedance relative to this transform and to dispatch the voltage reference of the two poles of the Bipole convertor control to different reference voltages, to control these new DC sequence components to new power flow conditions, whilst respecting the limitations of the individual convertors to do so iii) to control the voltages at nodes other than the convertor terminal themselves under DC operation, for example at a DC switching station within that network. These above features address a concern that would otherwise exist within extensive DC networks populated by Bipole HVDC arrangements that their reliability and availability under fault / outage would reduce to 50% of the available capacity provided by full bipole arrangements (those where a ground return current cable is provided between terminals), with rigid bipoles (those without a ground return cable) subject to unbalance needing to be switched out of service. Benefits of the approach are significant in nature and include, but are not limited to- A) Enabling a rigid bipole to remain in operation even if it is connected to other arrangements which have become asymettric due to a pole connection loss. This is achieved by the control distributing the unbalance across available full bipole terminals to achieve a power flow condition that results in equal and opposite voltages being respected at the point of interface of the rigid bipole route. This allows bipole routes over long distance to remain in service despite asymmetry of powerflow within the overall network without requiring a 3rdground return cable cost. B) Allows a full bipole DC network to maximise its availability following a fault- for example a 6 terminal network can continue to function with 5 terminals all operating together sharing power flow. C) Allows DC networks to continue to operate when elements of ground return cable have been lost. D) Allows individual full bipole controls operating in close electrical proximity to each other within a DC network to avoid “hunting” the control of a stable overall voltage profile when each voltage reference of each pole of that bipole is being influenced by the action of a MTMV control E) A “future-proofing” of a DC network control to accommodate DC circuit breakers that would transiently open in response to DC faults upon a pole to create these arrangements. F) Can support measurement and control of the ground return network present, which in nominally operating at a low voltage and current can be problematic to measure with accuracy. Throughout the specification, unless the context demands otherwise, the terms 'comprise' or 'include', or variations such as 'comprises' or 'comprising', 'includes' or 'including' will be understood to imply the inclusion of a stated integer or group of integers, but not the exclusion of any other integer or group of integers. The foregoing description of the invention has been presented for the purposes of illustration and description and is not intended to be exhaustive or to limit the invention to the precise form disclosed. The described embodiments were chosen and described in order to best explain the principles of the invention and its practical application to thereby enable others skilled in the art to best utilise the invention in various embodiments and with various modifications as are suited to the particular use contemplated. Therefore, further modifications or improvements may be incorporated without departing from the scope of the invention as defined by the appended claims. For example, the inventive concept is described with reference to offshore wind connections, but it will be understood that it is equally applicable to other DC grids more generally. Also, as noted above the inventive concept is not limited to application in high voltage DC applications; the principles are also suitable for low- and medium-voltage DC applications. Furthermore, the inventive concept is not limited to MMC converters but may include two- and three-level converters, hybrid converters, and any combination of different converter types as may be appropriate.

[0004] References [1] H. Government. "Net Zero Strategy: Build Back Greener." Department for Business, Energy & Industrial Strategy. https: / / www.gov.uk / government / publications / net- zero-strategy. [2] N. ESO. "The Pathway to 2030 Holistic Network Design." National Grid ESO. https: / / www.nationalgrideso.com / future-energy / the-pathway-2030-holistic-network-design. [3] X. Chen et al., "Integrating wind farm to the grid using hybrid multiterminal HVDC technology," IEEE Transactions on Industry Applications, vol.47, no.2, pp.965-972, 2010. [4] T. N. H. Centre. "Caithness-Moray HVDC Project During Delivery." The National HVDC Centre. https: / / www.hvdccentre.com / our-projects / caithness-moray / . [5] A. Shetgaonkar, L. Liu, A. Lekić, M. Popov, and P. Palensky, "Model predictive control and protection of MMC-based MTDC power systems," International Journal of Electrical Power & Energy Systems, vol.146, p.108710, 2023. [6] M. S. M. Saat, S. K. Nguang, and A. Nasiri, Analysis and synthesis of polynomial discrete-time systems: an SOS approach. Butterworth-Heinemann, 2017. [7] J. Beerten, D. Van Hertem, and R. Belmans, "VSC MTDC systems with a distributed DC voltage control-A power flow approach," in 2011 IEEE Trondheim PowerTech, 2011: IEEE, pp.1-6. [8] S. Dong, Y. Chi, and Y. Li, "Active voltage feedback control for hybrid multiterminal HVDC system adopting improved synchronverters," IEEE Transactions on Power Delivery, vol.31, no.2, pp.445-455, 2015. [9] D. B. Rathnayake et al., "Grid forming inverter modeling, control, and applications," IEEE Access, 2021.

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[0018] Dong Chen, Benjamin Marshall, Colin Foote, Suresh Rangasamy, and Shangen Tian, “Towards HVDC Interoperability - Assessing Existence of Equilibrium with Reference to Converter Terminal Behavior”, The 19th International Conference on AC and DC Power Transmission, Glasgow, 2023, in press.

Claims

Claims 1. A method for controlling a plurality of HVDC converters within an asymmetrical bipolar HVDC system, the method comprising: controlling the outputs of the plurality of HVDC converters by providing each HVDC converter with a respective (or individual pole converter specific) DC voltage reference; determining a respective feedback signal for each of the plurality of HVDC converters wherein the feedback signals comprise a modification to the DC voltage reference for the respective converter; wherein determining the respective feedback signals comprises analysing terminal characteristics of each converter and determining an index indicative of a margin against loss of equilibrium.

2. The method of claim 1, wherein each converter is associated with an individual pole of a bipole circuit.

3. The method of claim 1 or claim 2, comprising modifying the DC voltage reference for each HDVC converter with a different magnitude responsive to deactivation of one or more HVDC converters.

4. The method of any preceding claim, wherein one or more HVDC converters are deactivated as the result of a fault or deactivated deliberately.

5. The method of claim 3 or claim 4, comprising compensating for the one or more deactivated HVDC converters by distributing power flow or unbalance to the remaining converters.

6. The method of any preceding claim, wherein responsive to deactivation of an HVDC converter corresponding to a particular polarity, the DC voltage references for the remaining HVDC converters is modified to compensate for the deactivated HVDC converter by distributing the resulting unbalance.

7. The method of any preceding claim, wherein the system comprises a plurality of interconnected bipole circuits, each bipole circuit comprising matching pairs of positive and negative poles each with associated HVDC converters, whereinresponsive to deactivation of one of the HVDC converters, the method comprises distributing a resulting unbalance across the remaining HVDC converters.

8. The method of any of claims 5 to 7, wherein the power flow or unbalance is distributed between the remaining HVDC converters of the same polarity as the deactivated HVDC converter.

9. The method of any of claims 5 to 8, wherein the power flow or unbalance is distributed by independent control of the outputs of each of the HVDC converters, by virtue of said converter specific feedback signals.

10. The method of any preceding claim, wherein different magnitudes of voltage reference are applied to individual pole converter terminals.

11. The method of any preceding claim, wherein the index quantifies the interactions amongst voltage droops between the plurality of HVDC converters.

12. The method of any preceding claim, wherein the index indicates a stability margin against DC power transfer.

13. The method of any preceding claim, wherein the modification to the DC voltage reference is determined to optimise steady state operation of the multi-terminal HVDC system.

14. The method of any preceding claim, comprising modelling the terminal characteristics of each converter as an equivalent circuit based on the effect that HVDC converter has in response to a modification of its voltage reference for a given operating state, which may be expressed as a Norton equivalent circuit or a Thevenin equivalent circuit.

15. The method of any preceding claim, comprising determining HVDC system characteristics by constructing a matrix of extended impedance or extended matrix of conductance for the system.

16. The method of any preceding claim, comprising determining DC voltage droop characteristics for each HVDC converter.

17. The method of any preceding claim, comprising determining an initial value of equilibrium and normalising the index against loss of equilibrium.

18. The method of any preceding claim, comprising determining branch currents and nodal voltages within the system and receiving same at a central controller configured to implement the method.

19. The method of any preceding claim, wherein the asymmetrical bipolar multi- terminal HVDC system is a multi-vendor multi-terminal HVDC system.

20. The method of any preceding claim, wherein the asymmetrical bipolar multi- terminal HVDC system is comprised in an HVDC transmission system, and wherein the HVDC converters may be comprised in an interface or interfaces between one or more HVDC power lines and one or more AC generation systems and / or one or more AC grid networks, wherein the one or more AC generation systems may comprise one or more offshore wind turbines or farms, and wherein the one or more AC grid networks may comprise one or more onshore AC grid networks.

21. A method for controlling a plurality of HVDC converters within an asymmetrical bipolar HVDC system, the method comprising: controlling the outputs of the plurality of HVDC converters by providing each HVDC converter with a respective pole specific DC voltage reference; determining a respective feedback signal for each of the plurality of HVDC converters wherein the pole specific feedback signal comprises a modification to the pole specific DC voltage reference for the respective converter; wherein determining the respective pole specific feedback signals comprises modelling terminal characteristics of each converter as an equivalent circuit based on or representing the response of that HVDC converter to a modification of its voltage reference for a given operating state.

22. A controller for an asymmetrical bipolar multi-terminal HVDC system, the controller adapted or configured to perform the method of any preceding claim.

23. A multi-terminal HVDC system comprising a plurality of HVDC converters and a controller according to claim 22.

24. The system of claim 23, comprising at least one HVDC power line, a plurality of HVDC converters between the at least one HVDC power line and an offshore wind farm, and a plurality of HVDC converters between the at least one HVDC power line and an onshore AC grid.

25. The system of claim 23 or claim 24, comprising a plurality of interconnected bipole circuits, each pole of each bipole circuit being associated with a respective HVDC converter.

26. The system of any of claims 23 to 25, wherein the multi-terminal HVDC system may be a multi-vendor multi-terminal HVDC system.

27. A computer program comprising instructions which, when executed by a computer, cause the computer to carry out the method of any of claims 1 to 21.

28. A computer readable medium or data carrier comprising the computer program of claim 27.

29. A data carrier signal carrying the computer program of claim 27.

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