THE CONTROL OF VOLTAGE SOURCE CONVERTERS This invention relates to a method of controlling a voltage source converter and to such a voltage source converter. In high voltage direct current (HVDC) power transmission networks alternating current (AC) _p ower _is typically converted to direct current (DC) power for transmission via overhead lines and/or under-sea cables. This conversion removes the need to compensate for the AC capacitive load effects imposed by the power transmission medium, i.e. the _t ransmission _line or cable, and reduces the cost per kilometre of the lines and/or cables, and thus becomes cost-effective when power needs to be transmitted over a long distance. _{The conversion} between DC power and AC power is utilized in power transmission networks where it is necessary to interconnect the DC and AC electrical networks. In any such power transmission network, converters are required at each interface between AC _{and DC power to effect the} ^{required conversion; AC to DC or DC to AC.} A particular type of converter is a voltage source converter which is operable to generate an AC voltage waveform at one or more AC terminals thereof in order to provide the _a forementioned power transfer functionality between the AC and DC electrical networks. According to a first aspect of the invention there is provided a method of controlling a voltage source converter including at least one ^converter limb ^{corresponding} to a _{respective phase of the} ^converter, _{the or} ^{each converter limb extending between first} _{and s} econd DC terminals and including first and second limb portions separated by an AC _t erminal, each of which limb portion includes achain-link converter operable to provide a stepped variable voltage source, the method comprising the steps of:

(a) obtaining a respective AC current demand phase waveform for the or each converter limb which the corresponding converter limb is required to track, and a DC current demand which the or ^{each converter} limb ^is also required to ^track; and

(b) carrying out mathematical optimization to determine an optimal limb portion _current for each limb portion that the limb portion must contribute to track the corresponding required AC current demand phase waveform and the required DC current demand while minimising current conduction losses within each limb portion and additionally managing the energy stored by each chain-link converter. Carrying out mathematical optimization to determine optimal limb portion currents which track the corresponding required AC current demand phase waveform and the required DC current while minimising current conduction losses within each limb portion allows both the AC and DC power demands of a particular voltage source converter installation to be met, e.g. according to the voltage source converter owner's operational requirements, in a manner that reduces operational losses and so improves the efficiency and cost- effectiveness of the said particular voltage source converter installation. In the meantime, carrying out mathematical optimization to determine optimal limb portion currents which track the corresponding required AC current demand phase waveform and the required DC current while additionally managing the energy stored by each chain-link converter, avoids the need for separate control loops to deal with such stored energy management. The avoidance of such separate control loops is highly desirable because they otherwise adversely impact on the optimal limb portion currents determined to minimise current conduction losses, thereby degrading the associated efficiency improvements. In addition, separate control loops cause individual chain-link converters to compete with one another from a stored energy management perspective and thereby prevent the chain-link converters from achieving, e.g. near-zero energy deviation from a desired target stored energy. Preferably managing the energy stored by each chain-link converter includes balancing the energy stored by each chain-link converter. Balancing the energy stored by each chain-link converter is advantageous because it helps to ensure that the energy stored by components within each chain-link converter, e.g. respective chain-link modules having an energy storage device in the form of a capacitor, is similarly evenly balanced, i.e. the capacitors have roughly the same amount of charge as one another during operation of the associated voltage source converter. Such energy balancing of, e.g. chain-link modules, is highly beneficial as it helps to maintain correct functioning of the voltage source converter, thus maximising its lifetime, robustness, performance and stability. In a preferred method of the invention, further comprising within step (a) obtaining a target stored energy that each chain-link converter should aim to have stored therein under steady-state operating conditions, managing the energy stored by each chain-link converter includes minimising the deviation in energy stored by each chain-link converter from the target stored energy it should have stored. Minimising the deviation in energy stored by each chain-link converter from a target stored energy level is beneficial because it helps to lead to the balance of energy stored by each chain-link converter and the components therein, along with the associated benefits mentioned above. Moreover, having the energy stored by each chain-link converter conform to a desired target means that each limb portion within a given converter limb is operating in an optimal ^{manner which} helps to ensure that ^neither the overall performance nor endurance of the voltage source converter is degraded over time. Optionally step (b) of carrying out mathematical optimization to determine an optimal limb portion current for each limb portion includes applying a first weighting to the extent to which current conduction losses are minimised and a second different weighting to the degree of stored energy management carried out. Step (b) of carrying out mathematical optimization to determine an optimal limb portion current for each limb portion may include applying a second different weighting to the degree of stored energy balancing carried out and a third further different weighting to the extent to which stored energy deviation is minimised. The foregoing steps allow the method of the invention to tailor its functionality in order to accommodate different operating conditions, such as power ramping, steady-state power supply or a fault condition, while continuing to track the or each required AC current demand phase waveform and the required DC current demand, as well as minimise _current conduction losses within each limb portion and additionally _{manage the} energy stored by each chain-link converter. Preferably step (b) of carrying out mathematical optimization to determine an optimal limb portion current for each limb portion includes establishing a quadratic optimization problem of the general form

where,

to is the time at which a particular period of control of a particular voltage source converter starts; and

t1 is the time at which the particular period of control of a particular voltage source converter ends. The current objective function to be minimized may take the form

where,

I is an optimal limb portion currents vector composed of individual limb portion currents that each corresponding limb portion must contribute; and

B is an average chain-link converters stored energy vector composed of individual average energy amounts that each chain-link converter is actually storing. Optionally the current objective function to be minimized is defined by a linear combination of current conduction losses, stored energy deviations between the chain-link converters, and stored energy deviations from a target stored energy. In a preferred embodiment of the invention the current conduction losses are given by

where,

I is an optimal limb portion currents vector composed of individual limb portion c ^urrents that ^each corresponding limb portion must contribute. The stored energy deviations between the chain-link converters may be _{given by}

where,

El is the average energy stored in an i-th chain-link converter; and

E~ is the average energy stored in a j-th chain-link converter. ^{Optionally the stored energy deviations from a} _{target stored energy are given by}

where,

EL is the average energy stored in an i-th chain _{-link converter; and}

Eoj is the target stored energy an i-th chain-link converter _{should have stored under} steady-state operating conditions. The various foregoing features desirably permit the utilization of mathematical _optimization in the control of a voltage source converter, and thereby provide for the associated advantages, in a manner that is readily tailored to the specific configuration of a given voltage source converter. In another preferred embodiment of the invention the current objective function is minimised subject to a first equality constraint expressed as a linear _{equation of the form}

and firstly incorporating power demands based on the respective AC _{current demand} phase waveform for the or each converter limb and the DC current demand, as well as secondly incorporating stored energy compensation factors. Such a step desirably restrains the possible set of solutions that _{minimises the current} objective function in a manner that desirably incorporates management of the energy stored by each chain-link converter. In a further embodiment of the invention the current objective function is _{minimised subject} to an additional second equality constraint expressed as a linear equation of the form

and incorporating a consideration of changes in the average energy stored by each chain- link converter. It is advantageous to take into account the effect an instantaneous level of _{optimal limb} portion current has on the average energy the corresponding chain-link _{converter stores} since the current objective function modifies such instantaneous currents to manage the time-averaged energy stored by a particular chain-link converter. In a preferred method of controlling a voltage source converter including a plurality of converter limbs, the current objective function is minimised subject to an additional third equality constraint expressed as a linear equation of the form

and incorporating a requirement that the AC current demand phase waveform for each converter limb sums to zero at the corresponding AC terminal. Such a step helps to eliminate the inclusion of AC components in the DC current demand routed between the first and second DC terminals, and so avoid the need to filter this current before, e.g. passing it to a DC network connected in use to the first and second DC terminals. Any kind of filter in, e.g. a HVDC installation, has major implications with regards to the footprint of a resulting converter station, and so avoiding such filters is very beneficial. Preferably the first, second, and third equality constraints are concatenated into a compact linear system of the form

where,

A is defined as

and b is defined as:

Carrying out such concatenation leads to a single, computationally efficient equality constraint and so reduces the processing overhead associated with the method of the invention. In a still further preferred embodiment of the invention the state vector is given by

where, I is an optimal limb portion currents vector composed of individual _{limb portion} currents that each corresponding limb portion must contribute; and

B ^{is an average chain-link converters stored energy} _{vector composed of individual} average energy amounts that each chain-link converter is actually storing. Defining the state vector, i.e x(k), in this manner beneficially unites the two unknowns, i.e. I(k) E(k) in a single equation that can then be readily constrained as required. A ^{ccording to a} second aspect ^of the invention there is provided a voltage source converter comprising at least one converter limb corresponding to _{a respective phase of the} converter, the or each converter limb extending between first and second DC terminals a ^{nd including} first and second limb portions separated by an AC terminal, each of which l ^{imb portion includes achain-link converter operable to} _{provide a stepped variable voltage} source, the voltage source converter further comprising a controller programmed to:

( ^{a) obtain} a respective AC current demand phase waveform for the or each converter limb which the corresponding converter limb is required _{to track, and a DC} current demand which the or each converter limb is also required to track; and

(b) carry out mathematical optimization to determine an optimal limb portion current for each limb portion that the limb portion _{must contribute to track the} corresponding required AC current demand phase waveform and the required _{DC current} demand while minimising current conduction losses within each limb portion and a ^{dditionally managing the energy} _{stored by each chain-link converter.} The voltage source converter of the invention shares the benefits associated with _the corresponding method steps of the invention. There now follows a brief description of preferred embodiments of the invention, by way of ^{non-limiting example, with} reference being made to the following figures in which: Figure 1 shows a flow diagram which illustrates principle steps in _{a method} according to a first embodiment of the invention of controlling a voltage source _converter;

Figure 2 shows a schematic representation of an example voltage source converter being controlled by the first method of the invention; and

Figure 3 illustrates how the voltage source converter shown in Figure 2 is controlled to manage the energy stored by respective chain-link converters within the voltage source converter. Principle steps in a method according to a first embodiment of the invention of controlling a voltage source converter are illustrated in a flow diagram 100 shown in Figure 1. The first method of the invention is applicable to any voltage source converter topology, _{i.e. a converter} including in each limb portion thereof achain-link converter _{operable to} provide a stepped variable voltage source, irrespective of the particular converter structure. By way of example, however, it is described in connection with athree-phase voltage source converter 10 which has three converter limbs 12A, 12B, 12C, each of which corresponds to one of the three phases A, B, C. In other embodiments of the invention the voltage source converter structure being controlled may have fewer than or more than three phases and hence a different commensurate number of corresponding converter limbs. In the example three-phase voltage source converter 10 shown, each converter limb 12A, 12B, 12C extends between first and second DC terminals 14, 16 that are connected in use to a DC network 30, and each converter limb 12A, 12B, 12C includes a first limb portion 12A+, 12B+, 12C+ and a second limb portion 12A-, 12B-, 12C-. Each pair of first and second limb portions 12A+, 12A-, 12B+, 12B-, 12C+, 12C- in each converter limb 12A, 12B, 12C is separated by a corresponding AC terminal 18A, 18B, 18C which is connected in use to a respective phase A, B, C of an AC network 40. Each limb portion 12A+, 12A-, 12B+, 12B-, 12C+, 12C- includes achain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C-which is operable to provide a corresponding stepped variable voltage source VA+ (only one such variable voltage source being shown in Figure ^2). Each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- includes a plurality of series connected chain-link modules (not shown). Each chain-link module includes a number of switching elements which are connected in parallel with an energy storage device in the form of a capacitor. Each switching element includes a semiconductor device in the form of, e.g. an Insulated Gate Bipolar Transistor (IGBT), which is connected in parallel with an anti-parallel diode. It is, however, possible to use other semiconductor devices. An example first chain-link module is one in which first and second pairs of switching elements and a capacitor are connected in a known full bridge arrangement to define a 4- quadrant bipolar module. Switching of the switching elements selectively directs current through the capacitor or causes current to bypass the capacitor such that the first module can provide zero, positive or negative voltage and can conduct current in two directions. An example second chain-link module is one in which only a first pair of switching elements is connected in parallel with a capacitor in a known half-bridge arrangement to define a 2- quadrant unipolar module. In a similar manner to the first chain-link module, switching of the switching elements again selectively directs current through the capacitor or causes current to bypass the capacitor such that the second chain-link module can provide zero or positive voltage and can conduct current in two directions. In either foregoing manner it is possible to build up a combined voltage across each chain- link converter 20A+, 20A-, 208+, 20B-, 20C+, 20C- by combining the individual voltage available from each chain-link module. Accordingly, each of the chain-link modules works together to permit the associated chain- link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- to provide a stepped variable voltage source. This permits the generation of a voltage waveform across each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- using a step-wise approximation. Operation of each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- in this manner can be used to generate an AC voltage waveform at the corresponding AC terminal 18A, 18B, 18C. In addition to the foregoing, the voltage source converter 10 includes a controller 22 that is arranged in operative communication with each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C-, and further is programmed to carry out the first method of the invention. More particularly the controller carries out a first step (a) of:

- obtaining a respective AC current demand phase waveform IA, IB, Ic for each converter limb 12A, 12B, 12C which each converter limb 12A, 12B, 12C is required to track;

- obtaining a DC current demand loc which the converter limbs 12A, 12B, 12C are also required to track; and

- obtaining a target stored energy value EoA+, ^Eoa_, EoB+, EoB ^_, Eoc+, ^Eoc_ that each corresponding chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- should aim to have stored therein under steady-state operating conditions. The various AC current demand phase waveforms la, IB, ^Ic, the DC ^{current demand} loc _{and the target stored energy values EoA+,} ^{EoA_, EoB+, EoB_, Eoc+, Eoc_ may be obtained directly} _from ahigher-level controller within the particular voltage source converter 10 or from some other external entity. Alternatively the particular voltage source converter may obtain them directly by carrying out its own calculations, e.g. ^{using Active} and ^{Reactive power} control loops. The various AC current demand phase waveforms la, IB, Ic and the ^DC current ^{demand loc} are expressed as a target current demand vector IABc-DcCk) as follows:

where,

Wlth,

l _A(k), le(k), Ic(k) being the respective AC current demand phase waveforms IA, IB, Ic for each converter limb 12A, 12B, 12C at time instant k, and

lock) being the DC current demand loc at that same instant of time. _{The respective} target stored energy values EoA+, EoA ^_, EoB+, EoB ^_, Eoc+, ^Eoc_ for each corresponding chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C-are expressed as _{a target} stored energy vector Eo as follows: Each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- stores energy via the capacitor included in each of the plurality of chain-link modules which make up the respective chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- and, as mentioned _{above, the target stored} ^energy _Eoa+, ^EoA_, _EoB+, ^{EoB_, Eoc+, Eoc_ is the target energy that} _{each chain-link} converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C-should ideally have stored _{when operating} under steady-state conditions. _{As such} each particular target stored energy _{EoA+, EoA-, Eog-~-, EoB-,} Eoc+, ^Eoc_ is preferably obtained by way of where,

C is the capacitance of the capacitor in each chain-link module;

N _cmax ~s the total number of capacitors in each chain-link converter; and

Vt is a predefined target voltage of each individual capacitor in the respective chain- link modules when operating under steady-state conditions. The target stored energy E r each

chain-link converter 20A+, ^{20A-, 20B+, 20B-,} 20C+, 20C-may ^differ from one another or, as is the case in the example embodiment described herein, may be the same as one another, i.e. each equal to the same target stored energy value Eo, such that the target stored energy vector Eo is given by: The controller 22 also implements a second step (as indicated by a process box 102 in the flow diagram 100), i.e. step (b), of the first method of the invention, by carrying out mathematical optimization to determine an optimal limb portion _{current la+,} IA_, _IB+, IB_, _Ic+, ^Ic_ for each limb portion 12A+, 12A-, 12B+, 12B-, 12C+, 12C- that the limb portion 12A+, 12A-, 12B+, 12B-, 12C+, 12C- must contribute to track the corresponding required AC current demand phase waveform ^IA, IB, Ic and the required DC current demand Inc while minimising current conduction losses within each limb portion 12A+, 12A-, 12B+, 12B-, ^{12C+, 12C- and additionally} _{managing the energy stored EA} ^+, _{EA-, EB} ^+, _EB-, ^{E~+ E~-} _by each chain-link converter 20A+, 20A-, 206+, 20B-, 20C+, 20C-. More particularly, additionally managing the energy stored EA+, EA-, EB+, EB-, E~+ E~- by ^{each chain-link converter 20A+, 20A-, 20B+, 20B-,} _{20C+, 20C- includes both balancing} the energy stored by each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C-, i.e. causing each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- to store substantially the same amount of energy, and minimising the deviation in energy stored ^EA+, EA-, EB+, EB-, E~+ E~- by each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, ^{20C- from} the target stored energy Eoa+, EoA ^_, Eog+, EoB ^_, Eoc+, ^Eoc_ it should have stored ⁱ .e., ⁱⁿ the embodiment described, the identical target stored energy value Eo. _{In addition to the} foregoing, as will be described in more detail below, step ^{(b) of carrying} _out mathematical optimization to determine the optimal limb ^{portion current IA+, IA_, IB+, IB_,} _{Ic+, Ic_ for each limb portion 12A+, 12A-, 12B+,} ^{12B-, 12C+, 12C- still further includes} _{applying a first weighting a to the} ^{extent to which current conduction losses are minimised,} _{a second different} weighting ~i to the degree of stored energy balancing ^{carried out, and a} _third further different weighting y to the extent to ^{which stored energy deviation is} _{minimised. In other embodiments of} ^{the invention, two or more of the weightings a, ~i, y} _{may be} identical _to one another. _{With particular reference to the type} ^{of mathematical optimization carried out, by way of} _{example (with other} types of mathematical optimization being possible), in the first ^method _{of the} invention a quadratic optimization problem is ^{established of the general form}

where,

10 starts; and

t _{1 is the time at} which the particular period of control of the voltage source ^converter 10 ends. _The current objective function to be minimized is ^{then defined as taking the form}

where,

I _{is an optimal limb portion currents vector} ^{composed of the individual limb portion} _{currents hat each}

^{corresponding limb portion 12A+, 12A-, 12B+, 12B-,} 12C+, 12C- must contribute; and

E _{is an} average chain-link converters stored energy vector ^{composed of individual} _{average energy amounts EA} EA EB EB E E that each chain-link ^{converter 20A+,}

2 _0A-, 20B+, 20B-, 20C+, 20C- is actually storing. _More particularly I takes the form of a column vector, i.e.

where,

is the optimal limb portion current flowing through limb portion 12A+ _{at time} instant k, with the same nomenclature applying to the _rest of the optimal _{limb portion} currents, i ^_ The sign convention for the limb portion currents IA+, la_, IB+, IB_, _Ic+, Ic_ is shown in Figure 2. In this regard the limb portion currents IA+, IA ^_, IB+, IB ^_, Ic+, ^Ic_ represent controllable variables ^{in an overall control strategy, which} _{means that they can be freely determined, i.e. optimal} limb portion currents IA+, IA_, IB+, IB_, Ic+, Ic_ determined, in order to fulfil _{the power demands} and other current conduction and stored energy management constraints required to be fulfilled by the method of control. Meanwhile the average chain-link converters stored energy vector _{E, at time instant k, is} fully defined as:

T ^{hereafter, the current objective function} _{be minimized is further defined by a}

linear combination of current conduction losses, stored energy _{deviations between the} chain-link converters, and stored energy deviations from a target stored energy. More particularly the current objective function is, in the embodiment shown (although ^{other definitions are also} _{possible), defined by}

where

(i) the current conduction losses are multiplied by the _{first weighting a and are} given by

with 1 being the optimal limb portion currents vector described hereinabove; (ii) the stored energy deviations between the chain-link converters are multiplied by the second weighting ~i and are given by

with,

Et being the average energy in an i-th chain-link converter _(where i = A+, A-, B+, B-, C+, C-); and

E~ being the average energy in a j-th chain-link converter (where j = A+, A-, B+, B-, C+, C-); and (iii) the stored energy deviations from a target stored energy are multiplied by the third weighting y and are given by

Wlth,

E~ again being the average energy stored in an i-th chain-link converter; and Eon being the target stored energy an i-th chain-link converter should have stored under steady-state operating conditions. Following the aforementioned steps the current objective function, i.e.

is minimised subject to:

(i) a first equality constraint expressed as a linear equation of the form b

(ii) an additional second equality constraint expressed as a linear equation of the form (iii) an additions! third equality constraint expressed as a linear equation of the form

I _{n each} of the _foregoing instances the state vector, i.e. x, is given by

where, as set out above,

I is the optimal limb portion currents vector composed of individual limb portion currents that each corresponding limb portion must contribute; and

E is the average chain-link converters stored energy vector composed of individual average energy amounts that each chain-link converter is actually storing. Meanwhile the first, second, and third equality constraints are concatenated into a compact linear system of the form

with,

A being defined as

and b being defined as:

Meanwhile, the first equality constraint Al• x = b1 firstly incorporates power demands based on the respective AC current demand phase waveforms IA, IB, Ic, for each converter limb 12A, 12B, 12C and the DC current demand loc. More particularly,

with the matrix Al incorporating the power demands by way of matrix M6 that is defined as

and which is based on the following system of linear equations that include the AC current ^d emand phase ^{waveform IA,} IB, Ic, for each converter limb 12A, 12B, 12C and the DC current demand Ipc:

w ^ith the variables aA, aA, aB, aB, a~, a~ representing, respectively, the operating state of the corresponding chain-link converter 20A+, 20A-, 20B+, 208-, 20C+, 20C- in each limb portion 12A+, 12A-, 12B+, 12B-, 12C+, 12C-, i.e. the binary variables a indicating ^{whether the respective limb portion 12A+,} _{12A-, 12B+, 12B-, 12C+, 12C- is modulating} normally a requested reference voltage ^~ r is blocked (a

In addition the first equality constraint Al• x = b1 secondly incorporates stored energy compensation factors by way of matrix ME and vector b1. Matrix ME is defined by

where,

c ^onstants _{KpAc~ Kiac~ Kpvc~} ^{KiD~ are energy correction} gains with each of _KpAc ^{and KiA~ being (3,6) matrices and each of Kpo~} _{and Kio~ being (1,6) matrices; and} Tti is a predefined integration time. ^T he foregoing matrix ME is based on the following proportional-plus-integral feedback loops (although other control loops may be used) that relate stored energy deviations and ^{corresponding energy} correction currents to one another in the following manner:

where,

I _ABcE (k) establishes the AC correction currents needed to balance the energy stored in each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- and minimise the deviation in stored energy from the target stored energy value; and

IDcE(k) establishes the DC correction current needed to achieve the same aforementioned stored energy management result. I _{ABcECk) and} ^ID~E _(k) ^{are derived by considering an energy balancing} _current ^vector j ^ABC-DCgCk) which maps energy deviation dE(k) into correction currents, is defined as:

and follows from an understanding that a total current demand vector IABc-Deck) is obtained as a combination of the target current demand vector IABc-Deck) (as defined hereinabove) and the aforementioned energy balancing current vector IABc-ocECk), i.e:

Meanwhile,

dE(k) is an energy deviation vector that is obtained as the difference between the target stored energy vector Eo and the average chain-links stored energy vector E(k), i.e.

and Int D E(k 1) are accumulated energy correction values

that are used to achieve a smooth convergence of the stored energy EA EA EB EB E of each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+,

20C- to its corresponding target stored energy Eoa,+, _Eoa-, Eog+, _{Eog-, Eoc+, Eoc-.} In the meantime, vector b1 is defined by

where,

Eo is the target stored energy vector as defined hereinabove. ^The second equality constraint A2• x = b2 incorporates a consideration of changes in the average energy stored by each chain-link converter. More particularly

where,

f and re linear vector functions _{that take as}

arguments the voltage V~aps of each capacitor in the various chain-link converters 20A+, 20A-, 20B+, 20B-, 20C+, 20C- at time instant k; and

I ^{dentity(6) is a square matrix of dimension 6 x} _{6, composed of 1's in the main left-} to-right diagonal and 0's everywhere else. ^{The third equality constraint A3• x = b3 incorporates a} _{requirement that the AC current} ^{demand phase waveform for each converter limb} _{sums to zero at the corresponding AC} terminal, i.e. the third equality constraint incorporates the following _requirement:

which can be written in the aforementioned matrix form, i.e. as A

with and

In use the controller 22 determines, using the above-described mathematical optimization, an optimal limb portion current IA+, IA ^_, Ig+, ^Ig_, Ic+, ^Ic_ for each limb portion 12A+, 12A-, 12B+, 12B-, 12C+, 1 ~C- that the limb portion 12A+, 12A-, 12B+, 12B-, 12C+, 12C- must contribute so as to: - minimise current conduction losses within each _limb portion 12A+, _{12A-, 12B+,} 12B-, 12C+, 12C-; - additionally balance the energy stored EA+, EA-, EB+, EB-, E~+ E~- by each chain- l ^{ink converter 20A+, 20A-, 20B+, 20B-, 20C+,} _{20C-, i.e. cause each chain-link} converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- to store substantially the same amount of energy Eo; and - minimise the deviation in energy stored EA ^+, EA-, EB ^+, EB-, ^E~+ E~- by each chain- link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- from the target stored energy EoA+, EoA ^_, EoB+, EoB ^_, Eoc+, ^Eoc_ it should have stored i.e., in the embodiment described, the identical target stored energy value Eo. As a consequence of the latter two outcomes the energy stored EA+, EA-, EB+, EB-, E~+ E~- by each chain-link converter 20A+, 20A-, 20B+, 20B-, 20C+, 20C- converges on a desired target stored energy value Eo, e.g. zero joules (J), as shown in Figure 3. Meanwhile the controller 22 achieves the foregoing while continuing _to track _{the required} AC current demand phase waveforms la, IB, Ic and the required DC current demand loc.