Stability of multi-terminal HVDC systems and control methods

Abstract

A multi-terminal HVDC system comprises a plurality of HVDC converters and a plurality of transfer branches. The dynamic performance of the multi-terminal HVDC system is characterised and optimised to

Description

The present invention relates to the field of power transmission and, more specifically, the present invention concerns improvements to methods of predicting and ensuring stability in High Voltage Direct Current (HVDC) converters within multi-terminal HVDC system, and multi-terminal HVDC systems implementing such controls. The present invention finds particular applications in multi-vendor multi-terminal HVDC systems. 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 (https: / / www.nationalgrideso.com / future-energy / the-pathway-203Q-holistic-network-design) 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) 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, 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 concepts of control and control system in multi-vendor multi-terminal HVDC systems (and is incorporated herein by reference), aiming to provide vendor-agnostic control of MT-HVDC systems that might enable widespread adoption of MVMT HVDC systems. Unpublished application number GB2307888.4 (also incorporated herein by reference) is concerned with the distinct aim of providing accurate assessment of the power transfer capability of asymmetrical bipolar HVDC systems and to provide a means for controlling asymmetrical bipolar HVDC systems and of a bipolar HVDC grid that might enable widespread adoption of MVMT HVDC systems. An outstanding issue however is how to avoid instability for example to avoid oscillations of voltages and currents in response to perturbations at HVDC terminals. Approaches based on passivity of impedance in frequency domain have never been implemented due to hardly achievable requirements resulting from converter control delays and the like, and compromising requirements to allow negative conductance at certain frequencies. Approaches based on Nyquist Criteria (NC) [5]

[12] requires to be able to count the exact number of right-half-plane (RHP) zeroes to ensure stability, but when there are unstable subsystems this is difficult to fulfil, and assuming a number of RHP zeroes is difficult to prove. Further the number and effect of RHP zeros present within the subsystem is not available to the network operator without requiring the Vendor to open up elements of their proprietary control system. In short, NC is not suitable for designing an MVMT HVDC grid. The most widely used approach is time-domain simulation but this requires acquisition of a complete EMT model in time domain and simulation to evaluate performance over time. This however is a “Trial and Error” approach to control stability not compatible with defining stability within a more generic model or rigorous mathematical proof, and cannot be used to specify converters in an MVMT grid. It is also necessary to consider all possible operating scenarios (e.g. powers and voltages etc) for all HVDC terminals, which is impractical at large scale through “Trial and Error” time domain analysis. The object of at least one aspect of the present invention is to ensure stability of an HVDC grid over a range of frequencies, likewise in a vendor-agnostic manner. The invention may be implemented in a method of controlling or operating a plurality of HVDC converters within a multi-vendor multi-terminal HVDC system (which may be 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. Summary of the invention According to a first aspect of the invention there is provided a method of stabilising a multiterminal HVDC system, the multi-terminal HVDC system comprising a plurality of HVDC converters and a plurality of transfer branches, the method comprising: characterising the dynamic performance of the multi-terminal HVDC system; optimising the multi-terminal HVDC system to avoid instability caused by interactions between HVDC converters; configuring a transfer network wherein each of the plurality of transfer branches is bounded input bounded output stable; and configuring each of the plurality of HVDC converters to be bounded input bounded output stable. Preferably, characterising the dynamic performance of the multi-terminal HVDC system comprises isolating each of the converters and determining its response to a current and / or a voltage perturbation. Preferably, the response of each converter is determined for a plurality of current and / or voltage perturbations. Preferably, the responses of each converter to each current and / or voltage perturbation is aggregated in a respective converter admittance matrix. Preferably, a frequency response is obtained by applying current and / or voltage perturbations at a plurality of different frequencies. The response of all converters in an offline condition can thereby be obtained over a broad range of frequencies. Preferably, characterising the dynamic performance of the multi-terminal HVDC system comprises isolating each of the transfer branches and determining its response to a current and / or voltage perturbation. Preferably, the response of each transfer branch is determined for a plurality of current and / or voltage perturbations. Preferably, the responses of each transfer branch to each current and / or voltage perturbation is aggregated in a respective branch admittance matrix. Preferably, a frequency response is obtained by applying current and / or voltage perturbations at a plurality of different frequencies. Preferably, optimising the multi-terminal HVDC system to avoid instability caused by interactions between HVDC converters comprises ensuring that the real component of each complex current gain value of the converter admittance matrix is positive. Preferably, optimising the multi-terminal HVDC system to avoid instability caused by interactions between HVDC converters comprises ensuring that the real component of each complex current gain value of the branch admittance matrix is positive. Preferably, the method comprises determining a nodal current gain matrix to be positive real. Preferably, the method comprises determining bounded input bounded output stability of the admittance and impedance of each of the transfer branches. Preferably, the method comprises determining bounded input bounded output stability of the admittance and impedance of each of the converters. Preferably, configuring the transfer network comprises isolating each of the transfer branches, applying a current perturbation to each transfer branch, and ensuring that the voltage response to the current perturbation is finite. Preferably, configuring the transfer network comprises isolating each of the transfer branches, applying a voltage perturbation to each transfer branch, and ensuring that the current response to the voltage perturbation is finite. Preferably, the method comprises ensuring that current perturbation does not lead to instability by isolating each converter and each transfer branch connected to it, applying a controllable voltage source and a controllable current source to the converter. Preferably, the orders of the controllable voltage source and the controllable current source are varied to achieve predetermined DC operating points. Preferably, a converter control is adapted or configured to avoid an infinite current response at any operating point. Preferably, the method comprises determining converter admittance as a function of operating point in a time domain test. Preferably, the time domain test comprises testing the converter with a step current source applied in parallel and a step voltage source applied in series. According to a second aspect of the invention, there is provided a method for controlling (or operating) a plurality of HVDC converters within a multi-terminal HVDC system (which may be 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 (or pole specific) feedback signals comprise a modification to the DC voltage reference for the respective converter; further comprising stabilising the multi-terminal HVDC system in accordance with the first aspect; wherein determining the respective converter ( or pole specific) feedback signals comprises analysing terminal characteristics of each converter and determining an index indicative of a margin against loss of equilibrium; or wherein determining the respective converter (or 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. 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, 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 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. According to a third aspect of the invention there is provided a controller for a multiterminal HVDC system (which may be 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 (which may be an asymmetrical bipolar 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. The multi-terminal HVDC system may be a multi-vendor multi-terminal HVDC system. 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. 1 According to a seventh aspect of the invention there is provided a data carrier signal 2 carrying the computer program of the fifth aspect. 3 4 Embodiments of the fifth to seventh aspects of the invention may comprise features of or 5 corresponding to the preferred or optional features of any other aspect of the invention or 6 vice versa. 7 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 represents, in schematic form, a scheme of measuring component admittance; Figure 2 represents, in schematic form, application of (a) a current perturbation and (b) a voltage perturbation to a transfer branch; Figure 3 represents, in schematic form, application of (a) a current perturbation and (b) a voltage perturbation to a converter; Figure 4 is a flow chart illustrating the process of designing converter control; Figure 5 illustrates in schematic form the application of a (a) current perturbation and (b) voltage perturbation and the output response at a DC node; Figure 6 illustrates in schematic form the injection of a nodal current perturbation to (a) an HVDC grid and (b) a corresponding equivalent circuit; Figure 7 illustrates in schematic form the BIBO stability of an HVDC grid, in particular (a) nodal current gain matrix, (b) BIBO stability against nodal current perturbations and (c) BIBO stability against nodal voltage perturbations; Figure 8 illustrates in schematic form time domain test schemes for the stability of nodal admittance / impedance for (a) DC terminal with linear voltage droop control and (b) DC terminal with non-linear power control; Figure 9 represents the transfer admittance to formulate generalised nodal admittance for a bipolar HVDC showing (a) positive pole and (b) negative pole; Figure 10 illustrates in schematic form a three-terminal HVDC grid; Figure 11 illustrates the control strategies of (a) voltage controlled pole ends and (b) power controlled pole ends of the HVDC grid illustrated in Figure 10; Figure 12 illustrates models of control responses (a) current loop response, (b) power response and (c) damping control, corresponding to the control strategies illustrated in Figure 11; Figure 13 shows graphs of time domain ramp tests of a DC grid for (a) a balanced network and (b) an unbalanced network; Figure 14 are phasor diagrams of nodal admittances for (a) a balanced network and (b) an unbalanced network; and Figure 15 are phasor diagrams of nodal current gains for (a) a balanced network and (b) an unbalanced network. Detailed description of preferred embodiments As intimated in the background to the invention above, the control and design of HVDC networks are expected to be covered by vendor agnostic specifications by Transmission System Operators (TSOs) for HVDC converters and other related infrastructure. It is critical that control interactions between converters will not lead to instability once they are inter-connected. However, each vendor typically has a unique converter design, including and especially converter control, which are kept secret. In any case, TSOs are focused upon the operation of the HVDC system and are not expected to be involved in the internal design of converters and their internal controls. Thus, the Applicant has developed a universally applicable approach to multi-terminal control that characterises and where appropriate then specifies the measurable performances with reference to the terminal behaviour of every disconnected converter [2]-within the DC network operated by the TSO, echoing similar philosophies of operation which have operated upon the AC system for several decades. Unlike the AC system which is dominated by classical physical concepts of synchronized power generation and consumption across network characteristics, in relation to an inherently coupled power frequency a DC system is control dominated by the characteristics of the convertor system responses to the resultant voltages across the DC system under operation. Therefore, if vendor secrecy is to be respected and a practical control solution reached to DC network operation, an enabling methodology to quantify the stability of an HVDC grid, formed by interconnected “black-boxes” across a transfer network, must be developed. It is also desirable to enhance the applicant’s existing (application numbers GB2219373.4 and GB2307888.4) but as yet unpublished techniques in assessing converter interactions and serve as an enabler in planning and assessing bulk HVDC grid. As noted above, the applicant’s earlier unpublished application numbers GB2219373.4 and GB2307888.4 explore concepts of control and control systems in multi-vendor multi-terminal HVDC systems, 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. 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). In summary, the invention (and embodiments thereof) provides methods and systems within which the terminal behaviour of HVDC converters is specified to ensure that there are no oscillations caused by control interactions after interconnecting them via a transmission network. 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 characterising the dynamic performance of an HVDC grid (for example in an aggregated matrix format), optimising system design to avoid instability caused by interactions between converters, design the transfer network ensuring that the impedance and admittance of every transfer branch is Bounded Input Bounded Output (BIBO) stable, similarly design the converter impedance / admittance for BIBO stability. These features and variants thereof are expanded upon below. With reference to Figure 1, to characterise the dynamic performance of an HVDC grid, the admittance of every individual component (or model thereof) of the HVDC grid is defined as a record of responses to a small current perturbation. The components may include shunt components, typically converter, and transfer components (branch) between the poles of shunt components, e.g. a transmission line connected in series with DC reactance. Applying a perturbation current of sinusoidal wave as ip = / pSin (rot) (i) The voltage response to the current perturbation is measured as: vr = l^sin {cot + 0) (ii) where Ip and Vp are the amplitudes to the perturbation and response, respectively; to is the frequency in rad / sec; t is the time in sec, and 0 is the shift of phase angle of voltage response with reference to the perturbation. With one frequency value of to at steady state, the admittance of the measured component y at the specific value of frequency co can be calculated as a complex number: *The above definition of converter admittance can be measured according to Fig. 1 and Equ. (i)(ii)(iii) in the assessment of compliance. The array of admittance y{jco) over broad frequencies can be obtained by repeating the process of Fig. 1 with (i)(ii)(iii) over a range of frequencies, for example,. 0.1 rad / sec, 0.2 rad / sec, ... 3000 rad / sec. The grid admittance matrix Yex{jco') can be obtained by: Ygx(j^) Ynet(j(f)) Ycon(j<xY) 0^) where yex,i7( / w) refers to i th row and j th column element of admittance matrix Yex(Ja)y, Ynet(ja)) refers to the admittance matrix of transmission network, and Ycon(ja)) the converter matrix. For the avoidance of doubt, when used in sequence numbers or subscripts, i and j are integers between 1 and n (inclusive), where n is the number of HVDC nodes. Ynet^ = {ynet.ijM) can be obtained by: (ynet,ii j^iV . . ( V I ynet,ij = ~y ij V * / ) where yij(jM) is the admittance array measured across the transfer branch of the HVDC grid between ith and jth node over the interested range of frequencies. Note that this can be generalised for designing bipolar (asymmetrical or symmetrical) HVDC grid by replacing equation (v) above with (27) and (28) described under the heading “Generalisation” further below. YconCj^) = {ycon^d^} can be obtained by: (ycon,u(j^ =yt(J^ ( } (yCOn,ij0") =0 Vl where yt(ja)) refers to the admittance array measured across the converter at ith terminal over interested range of frequencies. The grid impedance matrix is calculated as the inverse of admittance matrix as: ZeX(J^ = [Zex.ijQ'^} = Yex(jMYr (vii) Optimising the system design to avoid instability caused by interactions between converters is to ensure the “domination of nodal impedance” (discussed in detail further below). Here the real component of the complex current gain values, Ze^iiO^yexjiC / 6*). extracted from the aggregated matrices (iv) (vii) above are positive. This fulfilled by complying with following condition at all interested frequencies and at all DC nodes: Re[zexii(ja))yexM(Ja))] >0 (i = 1,2,..., n) (viii) This ensures control interactions between any two converters will not lead to instability. See discussion further below under the headings “Sufficient Condition for Stability” and “Generalisation”. With reference to Fig. 2, the next objective is to ensure the impedance and admittance of every transfer branch is Bounded-Input-Bounded-Output (BIBO) stable, i.e. it produces finite outputs responsive to finite inputs. This step ensures that any single transfer branch will not lead to instability. Firstly, every transfer branch of the HVDC grid is isolated. Secondly, it is ensured that every transfer branch complies with both of the following conditions: • When applying a current perturbation ip to the transfer branch between ith and jth node of the HVDC grid, as shown in Fig. 2(a), the voltage response vr to a perturbation step response remains finite. • When applying a voltage perturbation vp to the transfer branch between ith and jth node of the HVDC grid, as shown in Fig. 2(b), the current response ir to a perturbation step response remains finite. As this stage does not involve components supplied by multiple vendors it can be fulfilled by design based on corresponding time-domain simulations, passivity, or NC, whichever is convenient to a party. With reference to Figure 3, the converter impedance / admittance is then designed to ensure the Laplace(s)-domain nodal admittance yex,u(s) and nodal impedance —of all transfer branches are BIBO stable. This step ensures that any DC node, typically a converter, will not lead to instability by interacting with its connected transfer branches. To ensure that current perturbation will not lead to instability: • Short all the shunt branches except for the one at ith DC node. The remaining effective part of the circuit becomes interconnections of the shunt branch, i.e. ith converter (represented by admittance yj and all the transfer branches (represented by admittance of y^y^, ■■■.ym) connected to it. Applying a controllable voltage source V and controllable current source I as Fig. 3(a) shows. • Manipulate the orders of the controllable voltage source V and controllable current source I to make sure the interested DC operating points of (pi0,vi0) is reached. -see discussion further below at "A. Test of (D-lll) nodal impedance / admittance stability”. • Design the converter control such that any step perturbation of current ip shall not lead to an infinite response of vr at any interested operating point. To ensure that voltage perturbation will not lead to instability: • Short all the shunt branches expect for the one at ith DC node. The remaining effective part of the circuit becomes interconnections of the shunt branch, i.e. ith converter (represented by admittance yj and all the transfer branches (represented by admittance of y^.y^, ...,ym) connected to it. Applying a controllable voltage source V and controllable current source I as Fig. 3(b) shows. • Manipulate the orders of the controllable voltage source V and controllable current source I to make sure the interested DC operating points of (pi0,vi0) is reached. -likewise see discussion further below at "A. Test of (D-lll) nodal impedance / admittance stability”. • Design the converter control such that any step perturbation of current vp shall not lead to an infinite response of ip at any operating point. As with the preceding stage, this stage does not involve components supplied by multiple vendors so it can be fulfilled by design based on corresponding time-domain simulations, passivity, or NC, whichever is convenient to a party. The above is summarised in Figure 4, being a flowchart showing the process of specifying HVDC converters. The benefits of the design approach set out generally above and in more detail below are many and varied. It provides sufficient conditions for stable operation of an HVDC grid and there is no reliance on unprovable assumptions. When compared with Nyquist Criteria (NC) or derived generalised NC approaches, there is no need to count the number of RHP zeroes for sub-systems consisting of multiple converters. When compared with passivity of impedance approaches, this approach is achievable despite of presence of converters under constant power inversion operation, and control delays. For time domain simulations, mandatory simulations are advantageously reduced to a subsystem involving a single converter when designing the converter impedance / admittance. For the first time, it is possible to define and evaluate network stability within the Multi Terminal Control to which is vendor agnostic in nature. Specifically, the invention offers a way to quantify the design of an HVDC grid to avoid oscillations caused by control interactions. There follows fuller detail of an implementation of the above approach culminating in numerical illustrations which demonstrate the effectiveness of the underpinning concepts and achievability of stability criteria. Defining Stability of an MVMT-HVDC Grid Linearised model of an HVDC grid Without requiring (or even knowing) the internal design of any converter and applying nodal analysis on every DC terminal, the dynamic behavior of a monopole HVDC grid can be represented by an equation in s-domain [8]: (1) where V(s) is the vector of voltage at all DC terminals of interest; and Yex(s) the admittance matrix of the HVDC grid. The grid admittance matrix can be decomposed into two components as [8]: Ye^)=Ynet&-YC0^ (2) where Ynet is the admittance matrix of the transfer network and Ycon the admittance matrix of converters under active control. When communication based centralized control mechanisms, i.e. multi-terminal control, are not in place, Ycon is a diagonal matrix as: K-on (s) = diag{yt(s)} (3) where yt is the admittance introduced by / th ( / = 1,2,---, n) converter in open circuit with the convention of voltage source (positive power flows into the DC grid). The elements of the network matrix, Y^ts) = {ynet,y(s)}. can be expressed as: IYnet.ij GO = -Yij (s) G * j) where yij(s)\^j is the (mutual) admittance of the transfer network between / th and / th DC terminals. Herein, “? and “7 are integral numbers between 1 and the number of converter terminals n when used as in subscript or in sequential terms. Note that the frequency response of the elements of Ynet(ja)) and Ycon(JoY) can be obtained through frequency scan over every disconnected converter and transfer branch. With reference to Figs. 5 (a) and (b), the perturbation vector of nodal current injection / p(s) and perturbation vector of terminal voltage Vp (s) can be defined as: W = [vpl(s),vp2(s),---,vp^^ where vpi(s) and ip,(s) represent the perturbation voltage and current at the / th terminal, respectively. Vectors of nodal current and (converter) terminal voltage in response to the perturbations can be defined as: (W) = Kl(s),Vr2(s), ( Ir(s) [Xl ($), lr2 (5) / "’ <^ri (X), "' >irn (5)] where vri(s) and iri(s) represent the response terminal voltage and response nodal current injection at the / th terminal, respectively. When a perturbation vector of terminal voltage Vp (s) , is applied to the grid (and assuming there is no structural change in the system), the response of nodal current flowing into all positive nodes of DC terminals contributed by the rest of the system, Ir(s), shall comply with Kirchhoffs current law again with nodal current analysis

[14] as: Yex(s)Vp(s) = Ir(s) (7) Vice versa, when a perturbation vector of nodal current injection / p(s), is applied to the same grid, the response of (converter) terminal voltage, Vr (s), shall again comply with: Zex(s)Ip(s) = K-ts) (8) When the impact of control delays and travelling wave are assumed negligible in calculating grid impedance and Yex is non-singular in frequency domain, this can be expressed as: = {zeXilJ(s)} « XzX)1 = {yex,ij(5)} 1 (9) provided the grid is linearizable around an equilibrium guaranteed by appropriate control strategy over DC power flow [2], Based on (7) and (8), the input / output dynamics of an HVDC grid against perturbations of terminal voltage and nodal current injection can be represented by Multi-Input-Multi-Output (MIMO) transfer functions, e.g. Yex(s) for the transfer functions from input of l^,(s) to output of Ir(s), and Zex(s) for input of Ip(s) to output of K-(s), respectively. BIBO Stability of MVMT-HVDC Grid Once an equilibrium is secured within a pre-defined domain of operating point [2], a stable HVDC grid should ideally settle at the equilibrium after a small disturbance. Such stability can be assessed by checking if all accumulative quantities, e.g. capacitive voltages, inductive currents, register of integral regulators of converter, etc., converge to the equilibrium after the small disturbance, which is aligned with Lyapunov stability [6], However, since the information of converter design is unknowable or unknown (to the TSO or the other vendors at least), the state space model to determine Lyapunov stability will not be available. Thus and as a compromise, the responses of terminal voltages Vr and shunt converter current injection to any perturbations of terminal voltage Vp and currents Ip are checked to confirm that they are finite over time, which is aligned with the “Bounded-Input-Bounded-Output” (BIBO) stability [6], Once the system is linearized as “small-signal model” around an equilibrium, the principle of superposition applies. Thus, the dynamics against the perturbations of voltage and current can be analyzed independently in this MIMO system. Referring to transfer matrix, this approach seeks BIBO stability for the outputs introduced by current perturbations through grid impedance Zex(s) / P(s), voltage perturbations through grid admittance Tex(s)^,(s), and the resultant converter current / against both perturbations, shown in Fig. 5, as: (^ = (1^),1^),-^^ Vi(s) — tri(s) tpi(s) = tij(s) where i^fs) refers to the current of transfer branch from Terminal i to Terminal j. Sufficient Condition for Stability BIBO Stability Against Perturbation of Nodal Current To facilitate the construction of a sufficient and practical condition in determining internal stability of a HVDC grid, a new set of transfer functions (matrix) is introduced in this section to exclude the impact of interactions in stability. Given an HVDC grid is linearized at an equilibrium [2][6], the small-signal dynamics can be characterized by a linear circuit in Laplace (s) domain. Therein, a nodal current perturbation ipj(s) is exclusively injected to an n - terminal HVDC grid at j th terminal, which is demonstrated in Fig. 6(a). As shown, the response voltage vector at all terminals are: ^rj2 (^), <Vrjt (s), •", Vrjn (s)] (11) where represents voltage response at / th terminal excited exclusively by tpj(sy, referring to the definition of the linearized impedance matrix (8), these SISO voltage responses can be written as: (12) Applying Kirchhoff’s current law, the nodal current at the point of injection shall comply with: (13) where irjjj(s') is the shunt current response of the local converter to the perturbation iPj(sy and lrjij(sy is the response of transfer current flowing into the local Node / from Node I, which are defined as: ( irjjjt5) = Vrjjt^yjC5) ( / rjij'Cs) ^rij OOlT / i Substituting (14) into (13) yields: ipj(s) O’) trjl,j O’) ^rj2,j COirjij ($) ■" ^rjn,J (15) where irjj,jXs) and irji,j'(.s) are the response currents (to the single perturbation current) flowing out of, and into the positive node of Terminal j, contributed by voltages of the local / th terminals and a remote / th terminal, respectively. By substituting (14) into (13) and comparing with (13), the response nodal currents in response to the perturbation (current) can be rewritten as: irjjj'W = vrjj(s)[yj(s) - (16) And irjij (A) (17) Preserving the response of terminal voltages, and then applying theorem of substitution

[14] and superposition

[14] to the linearized circuit represented by Fig. 6(a), the circuit of an HVDC grid can be decomposed into n isolated equivalent circuits, with one (the top middle) corresponding to (16) and the rest of (n-1) circuits corresponding to (17), which is shown in Fig. 6(b). In those circuits, the current responses are preserved by the controllable current sources and the response terminal voltages preserved by the voltage across the current sources. Applying the circuit decomposition shown in Fig. 6(b) to every terminal, a new n x n transfer function H(s) (Matrix) of “nodal current gain” can be constructed as follows: 1) refer to (16) to set the output of diagonal elements at / th column in response to the inputs of all the nodal current perturbations / p; 2) refer to (17) to set the output of off-diagonal element at / th row and / th column in response to the inputs of all the nodal current perturbations Ip. As is shown in Fig. 7(a), the output of every element (in response to one current perturbation signal) is represented by the order of the current controlled source in the corresponding block. When the impedance matrix Zex and admittance matrix Yex are obtained by measurement [8], the axillary transfer matrix of current gain, H(s), is a function of the elements of grid impedance matrix {Zj / s)} and admittance matrix {yjt (s)} as: H(s) {zei-.iy (-^yexji (-^)} (18) Referring to the circuit analysis in Fig. 6 and Kirchhoff’s Current Law, the sum of the response current at / th column shall equal to the injected current perturbation irii, which is demonstrated in Fig. 7(a); hence the terminology of matrix of nodal current gain. When the following condition, i.e. positive real,

[10] stands in frequency domain, the current gain matrix H is passive: +H(-jco)T >0 (19) which is a sufficient condition to ensure all elements of the nodal current gain matrix are BIBO stable

[10] , One necessary and sufficient condition of (18) is positiveness of all eigen values of H(Ja>) + H(-ja))T as

[15] : min (eig[H(jM) + H(-jw)T]} >0 (20) Therefore, if (19) or (20) is satisfied, the output of I<(s) = Ir(s)Zex (s), which is a linear combination of all the voltage responses in all the blocks in Fig. 7 (a), is bounded if every transfer (impedance) function defined by: zL(s) = vriJ(s) / irtj(s) (21) is BIBO stable, as irij(s) is already bounded when (19) or (20) stands. A modified impedance matrix can be defined as Z'x(s) = where: Zi i (s) — ., -yi+Z”=ij^iyij f / \ 1 — 1 zij(s) — ~ “ . z x . j yi} yex.tjW yex,ij(.s) (i * D (22) Within (22), further defining the diagonal elements z^s) as the “nodal impedance” of Terminal / . serves to represent the apparent impedance of Terminal / when other terminals are short-circuited. Whereas, the off-diagonal impedances z'j(s) are the transfer (mutual) impedance between terminals. Applying the principle of superposition with the diagram shown in Fig. 7(a), the voltage response at Terminal j is a sum of voltage responses triggered by perturbation current at every terminal as: vrii = vri (23) In this way, the internal stability of Z'ex(s) = {z^s)}, i.e. any z- (s) is BIBO stable, becomes a sufficient condition (along with positive realness of H) to ensure BIBO and internal stability of resultant terminal voltage Zex(s)Ip(s) = K-(s) . When all the transfer impedance z-y(s) of an HVDC network are passive, the internal stability of Z'x(s) = {z-^s)} is boiled down to BIBO stability of every nodal impedance, which guarantees bounded output of I£(s) with input perturbation of Ip(sy When all terminal voltages are stable, the response of shunt current injecting to / th node bounded perturbation ipi is: kts) = ipi(s) + vrJ (s)yij(s) (24) When the admittance of every transfer branch yij(s')\Uj and every nodal voltage response vrj is BIBO stable, E”=1 vrj yij becomes bounded when vw is bounded for every node. As a result, the output current at / th terminal converter, ih is guaranteed bounded with any bounded input perturbation at / th terminal, ipi. Thus, the converter current / (s) in response to any combination of current perturbation / p(s) must be bounded. The flow of the derivation above can be summarized by Fig. 7(b), where the proposed sufficient condition for BIBO stability of terminal voltages and shunt currents in response to perturbations of nodal current is: (B-l) Positive realness of the nodal current gain matrix HQ'toy (B-ll) BIBO stability of every transfer admittance and impedance l / y^Csy^j (B-lll) BIBO stability of nodal impedances z'^s) Note, when the approximation operator in (9) is replaced by an equator as an acceptable assumption, H(s) becomes the Relative Gain Array (RGA) of Zez(s) or rex(s) [6]

[16] . Also note for a passive (or every branch is minimal-phase BIBO stable) HVDC transfer network, positive realness of / / ( / &>) indicates the domination of nodal impedance in determining stability against perturbations of nodal current. Such domination implies that stability can be determined with tests on disconnected converters. Further note, the passivity of grid admittance matrix Yex (s) or impedance Zex(s) is not a prerequisite for the passivity of current gain matrix H(s) or dominance of nodal impedance. BIBO Stability Against Perturbation of Terminal Voltage Given perturbations of terminal voltage, Vp, and referring to (5), the sufficient condition to ensure BIBO stability of response shunt current Ir is the internal stability of the transfer function of grid admittance Yex(s), i.e. everyone of its element, is BIBO stable. By definition of (1)(2)(3)(4), the additive inverse of off-diagonal elements of grid admittance matrix, -y^s)^, are the transfer (mutual) admittances between Terminal i and / ; when the transfer branches are passive (physically or more broadly in control wise), these transfer admittances and the corresponding additive inverses are BIBO stable. Therefore, the BIBO stability of Yex(s) for a passive transfer network is boiled down to BIBO stability of its diagonal elements yeXiii. Substituting (3)(4) into (2)(22) yields: yex,u^ = -y^ + = (25) which is defined as the “nodal admittance” herein. Once the nodal admittance yeXiu(s) is BIBO stable, Yex(s) becomes internal stable for all elements. As a sufficient condition of BIBO stability for Yex(s), / ,-(s) = Fp(s)yex(s) becomes bounded. Applying similar analysis represented by Figs 6 and symmetrical analysis by Fig. 7 above and then referring to (24), H(s) now can be re-interpreted as the matrix of “nodal voltage gain” in stability assessment. With inputs of nodal voltage perturbations, the positive realness of H(Jm) ensures the passivity of the matrix, whose elements correspond to the voltage of all the blocks in Fig. 7(a). Again, when the transfer admittances are BIBO stable, the positive realness of H(s) indicates the dominance of nodal admittance in this sufficient condition of stability assessment with the derivations below. The response of / th terminal voltage is: ^($) = J--Vpj(s) yex,u\.s) vpi(s)yeXiii(s) ( ) k(s) = Vri(s)[yex,ii(s) where vPJ(s) represents the perturbation of nodal voltages at / th terminal. When H(s) is passive, vri(s) again becomes bounded. And then, [{(s') is BIBO stable when the nodal impedance yeXiii(s) and all transfer admittances y^Cs^^-are BIBO stable. To summarise with reference to Fig. 7(c), the proposed sufficient condition for BIBO stability of terminal voltages and shunt currents in response to perturbation of nodal voltage is: (B-l) Positive realness of the i.e now the voltage gain matrix for voltage perturbation. (C-ll) BIBO stability of every transfer admittance yi7(s)|^7. (C-lll) BIBO stability of nodal admittance yex,ii(s) = l / z',(s). Summary of Sufficient Condition for Small Signal Stability Combining the stability conditions for perturbation of nodal current and of nodal voltage above, the sufficient condition proposed to ensure BIBO stability of all terminal voltages and shunt currents in response to perturbations of both nodal current and terminal voltage may be aggregated as: (B-l) Positive realness of the current (voltage) gain matrix (B-ll) BIBO stability of every transfer admittance y^(s)|and impedance z'y(s) = i (D-lll) BIBO stability of nodal impedance Za'(s) and admittance yex>u(sy Note that the above conditions show that the BIBO stability of an MVMT-HVDC grid can be ensured in a decentralized manner as long as (B-l) is satisfied. Generalisation Bipolar HVDC Grid The stability condition can be generalized to an HVDC grid with n bipolar nodes, i.e. 2n DC terminals, with (converters) or without (switching stations) actively controlled elements, in shunt or transfer branches, etc.. In such scenario, the network transfer admittance (in s domain) can be expressed as [8]: -M2,i -Mt,2 ^j=l,j*2 ^2, / —^2,(n-l) ~Mi,n ~^2,n ^net,2nx2n(^) *—* J — 1., J 7- ll 1 1 j,j (27) ~Mn,l Y" M . where: ^ij- + ^ijN ^ijN ^ijN %ij+ + ZljN (28) where Zij+, Z^-, and ZijN are the transfer impedances between / th and jth bipolar nodes at positive pole, negative pole and neutral return, respectively. Substituting (27)(28) into (2) to replace Ynet, the derivations which follow shall still stand, but the transfer admittance y^ are generalized as the off-diagonal elements in (27)(28), respectively. From (28), it can be determined when the transfer network is passive, i.e. every transfer impedance, Zij+, Zu , or ZijN , has a positive real component over all frequencies, the generalized transfer impedances y^s) and its inverse z0(s) remain BIBO stable. Referring to (28), the nodal admittance at / th bipolar terminal at positive pole yu+(s) and negative pole y^(s) may be generalized as: fy / t+OO = y / +(s) + z„ tz~+^+z.. Z..N = 777 I V-- tsi = V- + 7" ________zti++ztjN___________i__ 1. / 11 V J yI v J / —1, / ^ 1 7 <7 j-7 7 -1-7 7 (o'l y 1] + ^-ljN+^lJ-^ljN ZH-\S) where yi+(s) and y^(s) are the impedances of converter of / th bipolar node at the positive pole and negative pole, respectively; zii+'(s) and Zii-'(s) are the generalized nodal impedance of / th bipolar node at the positive pole and negative pole, respectively. Generalised Criteria for Transfer Network with Low Impedance Comparing with (18), construct generalized current gain matrix H'(s) = {h'ijfs)} as: ( h- ii(s) zex,iiCs)yex,iiCs) (30) ij(^)lt^j Crijzex,ij(.s)yex,&ijhij(s) where is a positive co-efficient for every element of lT(jw) across all frequencies, such that: 0 <O^j <6; min “ to -»co Ij* i(| h' ij (j a)) | +1 h' ji |) (31) when every elementHr(Jto) is finite overall frequencies, defined by (31) is a finite positive real constant; hence must exist as a frequency-independent constant, which guarantees diagonal dominance of H'Qco) + H'(-ja))T overall frequencies

[17] , After replacing the auxiliary transfer matrix H with H' in the methodology presented above (see “BIBO Stability against perturbation of nodal current’’), the condition of (19)(20) can be now replaced by: (32) And min (eig[H'(ja)) + H'(-»T]} >0 (33) When the diagonal dominance of + is ensured by (31), the positive definiteness of + H'(-ja>)T in (32)(33) can be ensured by the positive realness of every diagonal element

[17] as: Relh'aCj^ + = Retfh'uCja^O (34) As the gains of any constant do not change Condition (B-ll) or (D-lII), the BIBO stability of z'j(s) via (21), a generalized stability criterion is now updated as Condition (E-l), (B-ll), (D-lll), where Condition (E-l) refers to “positive realness of every current gain (or equivalently voltage gain), h'u(ja)) = Zex.uO^yex.uCj^’ at / th terminal”. Note, according to (30), (19)(20) is a special case of (31)(32)(33) when = 1. Thus, (E-I) is a generalized condition of (B-l). Also note, when every physical branch of a bipolar HVDC grid, Zij+, Zi}_, ZijN are passive or infinite (open circuit), the resultant off-diagonal elements in (27)(28) must be finite. When the grid admittance matrix Yex(Jo)) is not singular in frequency domain, in (31) must exist for 7th row and therefore (E-l) still applies to a bipolar HVDC grid to ensure dominance of nodal impedance. Test Schemes of the Stability Criteria Having established stability criteria above, there follows a practical assessment in the commercial context of MVMt-HVDC design. By following the procedure of calculating impedance and admittance matrix in frequency domain (1-4)(9)(27-29) and then (18), (E-l) can be naturally examined by aggregating frequency scans over disconnected converters and transfer branches [8]. (B-ll) can be tested by time domain simulations, or frequency scans with Nyquist Criterion when the admittance (or impedance) of every transfer branch are proved to be minimum phase by time-domain test. Test of (D-lll) Nodal Impedance / Admittance Stability It is impossible to make the nodal impedance positive real when a converter is under digital control across all operating scenario, especially under the DC Constant Power Load (CPL) or voltage dependent control [2], Among them, CPL mode will make an HVDC converter unstable on its own by introducing negative resistance starting from 0 Hz [2], This will lead to unstable zeroes (of nodal impedance / admittance), whose number is difficult to obtain from frequency domain response; hence Nyquist Criteria is inadequate to assess the nodal admittance consisting of unstable converter admittance. Considering this, time-domain test is proposed for terminals both under linear control (voltage droop) and non-linear control (constant power or equivalent). As shown in Fig. 4(a), the nodal admittance is composed of converter impedance in parallel with all the transfer impedances connected to the converter. (1) Terminal under linear control with respect of voltage (voltage droop) When the small-signal impedance of the converter is linear voltage independent terminal, the stability of zexil(s) can be assessed by applying time domain test with step current source applied in parallel with the overall zexii(s), which is shown in Fig. 8(a) and then tested with step voltage source applied in series foryex^(s). Note that in an alternative approach, the stability can be assessed with Nyquist Criteria provided the converter admittance y((s) and impedance zt(s) have been assessed stable with stable transfer impedance / admittance to confirm non-existence of right-half-plane zeroes for the nodal impedance / admittance. (2) Terminal under non-linear control with respect of voltage (constant power or eguivalent) Given the converter admittance can be a function of operating point when under non-linear control, i.e. constant power or equivalent, the expected operating equilibrium of voltage must be traversed in a time domain test. However, since a converter under a given constant power control cannot traverse all eligible equilibrium voltages across the nodal admittance on its own, auxiliary sources must be constructed reach the equilibriums to test without changing the nodal admittance to test. To fulfill this, a time domain test scheme is proposed as Fig. 8(b) shows. As shown in Fig. 8(b), with all the transfer admittance to Terminal i aggregated as £yy (i * j,j = 1,2,---,n), a voltage controlled DC source, V, is connected in series with the aggregated transfer branch; and then a shunt DC source, I, under current control is placed in parallel with the / th converter. When both controlled sources are under open-loop control, the impedance of the voltage source is zero over all frequencies and the current source infinite. Thus, the scheme shown in Fig. 8(b) does not change the apparent admittance measured across the converter terminal yeX)u, as shown. When the converter is under constant power control, equivalent, the admittance of the / th converter is a function of the equilibrium as: y; = f(yio.Pio) (35) where vi0 and pi0 are the equilibrium voltage and power at the / th terminal to test, respectively. Thus, time domain test has to be carried out to traverse all eligible operating point over a pre-defined domain, e.g. defined by the CX-index Q. Assigning an active power order of pi0 to the circuit, the equilibrium voltage vi0 can be solved from test circuit in Fig. 8(b) as: Vio = I A ± A2 + (36) where: (37) when the converter is controlled to push active power pi0, to the DC grid, the values of V and I can be tuned to reach an expected operating voltage vi0 to satisfy the inequities of (36)(37). In this way, the expected domain of (v^Pio) can be traversed with step changes of (¼ / ) inputs. Representing Nodal Impedance / Admittance fora Bipolar HVDC Terminal in Time Domain Test To simulate the nodal impedance / admittance for a bipolar HVDC terminal, the diagonal elements of (27), i.e. the aggregation of the generalized transfer admittances, need to be represented by equivalent admittance composed of physical branches. Comparing (28) with the “star-delta” transformation

[19] , every generalized transfer admittances, i.e. the top right element and bottom right elements in (28), can be simulated as Fig. 9 shows. And then, nodal admittance can be simulated in time domain by aggregating all the generalized transfer admittances and converter admittance to obtain the diagonal elements of (2) via (27) and (28). Case Studies of Achievability: Dominance of Nodal Impedance vs Passivity of Impedance Figure 10 shows in schematic form the layout of a 3-terminal bipolar DC grid. Terminals (bipolar VSC converters) 1 and 2 are assigned as voltage (droop) controlled, whereas Terminal 3 is assigned as the power controlled terminal. Setting the control of both pole ends of each station are under identical control, the block diagram of the illustrative control strategies are shown in Fig. 11. As shown, for both types of pole ends, the main circuit are modelled as a controllable current source I in parallel with an aggregated passive impedance Z(s). For a voltage controlled converter (pole end), cascaded closed loop control is adopted such that the output of inner voltage control loop connects to the input of inner current loop. Similar to voltage controlled converter, the inner current loop is cascaded with the output of the outer power control. By setting the voltage regulator as a proportional control, i.e. droop control, the rest of the control responses are modeled as Fig. 12 shows. As shown in Fig. 12 (a), the current loops of both voltage control and power controlled pole ends are modelled as a per unitized 2nd order process cascading an aggregated unit delay, where Kpj and Kn represent the proportional gain and integral gain of the current regulator, respectively. The power control is modeled as an ideal power control cascaded with a 1st order delay to represent its slow dynamics as an outer loop, which is shown in Fig. 11(b). The damping enhancement control in Fig. 11(b) is optional and used to counteract the negative resistance introduced by possible CPLs. In Fig. 12(c), it is illustrated as an extra voltage proportional gain Kv, i.e. a virtual conductance, cascaded with a washout regulator to avoid impacting the DC power flow

[20] , Note, the models of control response shown in Figs. 11 and 12 are mimicking the terminal performances HVDC converters may achieve rather than detailed design. This case study is to demonstrate a better achievability of (E-l) dominance of nodal impedance than impedance passivity. It is envisaged that a more complex model will incorporate more specific network topologies, converter models supplied by vendors, fault current limiting arrangements, HVDC cable models, etc.. Table I - Parametric Settings Transfer impedance type Purely resistive and inductive Transfer Inductance Between Terminals 1 and 2: (positive pole, negative pole, neutral) (mH) (150, 150, 50) for balanced condition; (150,150, 50) for unbalanced condition. Transfer Inductance Between Terminals 2 and 3: (positive pole, negative pole, neutral) (mH) (120, 120, 50) for balanced condition; (120, 120, 50) for unbalanced condition. Transfer Resistance Between Terminals 1 and 2: (positive pole, negative pole, neutral) (Q) (3, 3, 5)for balanced condition; (3, 1.5, 5)for unbalanced condition. Transfer Resistance Between Terminals 2 and 3: (positive pole, negative pole, neutral) (Q) (4, 4, 6)for balanced condition; (4, 2, 6)for unbalanced condition. Voltage Regulator: Proportional (droop) gain 13.2 A / kV Bipolar Terminals 1 and 2 Current Regulator: PI regulator for all terminals Natural frequency at 50 Hz; damping coefficient at 1. Gain of damping control Bipolar Terminals 3 (A / kV) 13.2 A / kV Time constant of low-pass process of power control Tp Bipolar Terminals 3 (s) 0.1 Time constant of wash-out filter of damping control for Bipolar Terminal 3 Tv (s) 1 Terminal impedance Z for all pole ends Fully capacitive at 42 pF Control delay (sec) 1 / 10000 Based on initial parameters shown in Table I above, results of a power ramp test is presented in Fig. 13 (a) to showcase the performance with an adverse constant power load drawn from the DC grid. Initiating the DC grid in no load condition from t = 0 sec to 1 sec, all 6 pole ends of the 3 terminals settle at the nominal voltage, 525 kV. From t = 1s, Terminal 3 starts to invert an active power at a ramp rate to - 0.1 MW I sec till 1 MW at t = 11s. During this process, voltage of every pole drops. Since there is no secondary voltage control in place, the voltage regulation is relatively shallow, but also forms a very adverse condition of introducing negative conductance. Despite the voltage deviation, the system is stable after the ramp stops from t = 11 s. Since the network is balanced, the voltage profiles of the two pole ends at each terminal overlap and so do the powers at terminal 3. As a comparison of achievability, the nodal current gains corresponding the settling point of Fig. 13 (a) are plotted in Fig. 15(a), again covering frequencies from 0.1 Hz to 5000 Hz. As seen, all the current gains starts with a positive real component at 0.1 Hz and finally converges to 1 on the real axis as the dominance of inductive element in the transfer network will prevent propagation of perturbation current / voltages over high frequencies. During this process, the current (voltage) gains stays positive throughout the frequencies despite that the corresponding impedance became negative in real component. This shows a better achievability of positive realness of current gain than the impedance while the system remains stable in time-domain. The ramp test in Fig. 13(a) is repeated by introducing imbalance in the transfer network, shown in Table I. As shown in Fig. 13 (b), this time the voltage profile and power between poles are not overlapping anymore, except for the terminal under a constant power control. The impedances are again plotted in Fig. 14(b), where the impedance again trespass the imaginary axis when the frequency reaches around 300 Hz and destroy passivity of grid impedance. As a similar contrast, the nodal current gains stays positive in real components from frequencies of 0 Hz to 5000 Hz. This result again shows that in unbalanced HVDC grid, the dominance of nodal impedance, i.e. positive realness of current gain, is more achievable than passivity of impedance. Conclusions For an MVMT-HVDC grid, the invention allows stability to- be achieved without framing number of unstable zeroes to accommodate Nyquist Criteria or passivity of impedance / admittance. Once condition (E-l) is satisfied, every nodal current (voltage) gain, / i'u( / <o) = Zex.utj^yex.iitj0*)’is positive real across all frequencies, the stability of the grid is dominated by nodal impedance. In this case, stability can be ensured in a decentralized manner in complying with the following conditions: (B-ll) BIBO stability of every transfer impedance and admittance; (D-lll) BIBO stability of every nodal impedance and admittance. As criteria on disconnected components / subsystems, (B-ll) (D-lll) can be tested in timedomain for TSOs. They can also be tested with NC, state space, or passivity, whichever is more convenient or applicable, for vendors. The invention can be extended to scenarios of unbalanced bipolar HVDC grid with generalized transfer impedance and nodal impedance in (27) and (28), respectively. The verifications herein are based on a simplified model that an HVDC grid might try to mimic, however as the stability criteria is based on a “black-box” model for every component, the skilled person will realise that the criteria can be applied to more complex models with, for example, frequency-dependent cable models, transfer networks of other topologies, and more detailed HVDC converter model supplied by converter OEMs, etc. The skilled person will also realise that this type of black-boxed approach can be further applied to model reduction of HVDC grid or assessing reinforcement of an existing HVDC grid, without undue burden. The invention enables for the first time a set of standardized specifications for an MVMT-HVDC grid without requiring knowledge or disclosure of any internal design of HVDC converters. This will greatly reduce the cost of offshore HVDC grid design in commissioning a new or expand an existing HVDC grid. It will particularly enable transmission system operators to deploy offshore power grids to transmit bulk wind power to onshore transmission systems. Through this process, the supply chain will be de-risked due to this vendor agnostic design. Revenue will be generated through connecting to new wind farms after deployment of offshore grid. This methodology can be used to quantify the dynamic performance of an existing HVDC grid or power grid. 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. References [1] "Net Zero Strategy: Build Back Greener." HM Government Department for Business, Energy &Industrial Strategy. U.K. [Online]. Available: https: / / www.gov.uk / qovernment / publications / net-zero-strategy [2] D. Chen, B. Marshall, C. Foote, S. 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Claims

1. A method of stabilising a multi-terminal HVDC system, the multi-terminal HVDC system comprising a plurality of HVDC converters and a plurality of transfer branches, the method comprising:characterising the dynamic performance of the multi-terminal HVDC system;optimising the multi-terminal HVDC system to avoid instability caused by interactions between HVDC converters;configuring a transfer network wherein each of the plurality of transfer branches is bounded input bounded output stable; andconfiguring each of the plurality of HVDC converters to be bounded input bounded output stable.

2. The method of claim 1, wherein characterising the dynamic performance of the multi-terminal HVDC system comprises isolating each of the converters and determining its response to a current and / or a voltage perturbation.

3. The method of claim 2, wherein the response of each converter is determined for a plurality of current and / or voltage perturbations.

4. The method of claim 3, wherein the responses of each converter to each current and / or voltage perturbation is aggregated in a respective converter admittance matrix.

5. The method of claim 3 or claim 4, wherein a frequency response is obtained by applying current and / or voltage perturbations at a plurality of different frequencies.

6. The method of any preceding claim, wherein characterising the dynamic performance of the multi-terminal HVDC system comprises isolating each of the transfer branches and determining its response to a current and / or voltage perturbation.

7. The method of claim 6, wherein the response of each transfer branch is determined for a plurality of current and / or voltage perturbations.

8. The method of claim 7, wherein the responses of each transfer branch to each current and / or voltage perturbation is aggregated in a respective branch admittance matrix.

9. The method of claim 7 or claim 8, wherein a frequency response is obtained by applying current and / or voltage perturbations at a plurality of different frequencies.

10. The method of any preceding claim, wherein optimising the multi-terminal HVDC system to avoid instability caused by interactions between HVDC converters comprises ensuring that the real component of each complex current gain value of the converter admittance matrix is positive.

11. The method of any preceding claim, wherein optimising the multi-terminal HVDC system to avoid instability caused by interactions between HVDC converters comprises ensuring that the real component of each complex current gain value of the branch admittance matrix is positive.

12. The method of any preceding claim, wherein the method comprises determining a nodal current gain matrix to be positive and real.

13. The method of any preceding claim, wherein the method comprises determining bounded input bounded output stability of the admittance and impedance of each of the transfer branches.

14. The method of any preceding claim, wherein the method comprises determining bounded input bounded output stability of the admittance and impedance of each of the converters.

15. The method of any preceding claim, wherein configuring the transfer network comprises isolating each of the transfer branches, applying a current perturbation to each transfer branch, and ensuring that the voltage response to the current perturbation is finite.

16. The method of any preceding claim, wherein configuring the transfer network comprises isolating each of the transfer branches, applying a voltage perturbationto each transfer branch, and ensuring that the current response to the voltage perturbation is finite.

17. The method of any preceding claim, wherein the method comprises ensuring that current perturbation does not lead to instability by isolating each converter and each transfer branch connected to it, applying a controllable voltage source and a controllable current source to the converter.

18. The method of claim 17, wherein the orders of the controllable voltage source and the controllable current source are varied to achieve predetermined DC operating points.

19. The method of claim 17 or claim 18, wherein a converter control is adapted or configured to avoid an infinite current response at any operating point.

20. The method of any preceding claim, wherein the method comprises determining converter admittance as a function of operating point in a time domain test.

21. The method of claim 20, wherein the time domain test comprises testing the converter with a step current source applied in parallel and a step voltage source applied in series.

22. A method for controlling (or operating) a plurality of HVDC converters within a multi-terminal HVDC system (which may be 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 / (or pole specific) feedback signals comprise a modification to the DC voltage reference for the respective converter;further comprising stabilising the multi-terminal HVDC system in accordance with any preceding claim;wherein determining the respective (converter / (or pole specific) feedback signals comprises analysing terminal characteristics of each converter and determining an index indicative of a margin against loss of equilibrium; orwherein determining the respective converter (or 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.

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

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

25. The system of 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 claim 24 or claim 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 22.

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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