Optimal and automated decentralised converter control design in more electrical aircraft power electronics embedded grids

In modern power systems, the proliferation of power electronics converters, and distributed generation raises important issues concerning inter-connected switching units in terms of performance, stability and robustness. Such phenomenon are more prominent in micro-grids, such as modern local low voltage distribution systems, more electric aircraft power systems etc., where many power converters are connected to the same non-stiff low power ac grid, and strongly interact with each other. Locally designed converter control systems on the same electrical bus exhibit interactive behaviour. If not taken properly into account, external disturbances to the system at given operating conditions may result in the degradation of performance, failure to meet operating conditions and, in some cases, instability. This paper presents a new approach to synthesise converter controllers in more electric aircraft ac distribution grids (which is, however, applicable to all power electronics embedded systems), keeping in consideration dynamic interactions among subsystems. An optimal control design approach based on H 2 optimisation is therefore proposed. Phase-locked loops have also been considered, and their tuning has been included in the general tuning procedure of the whole system. Simulation, and experimental results show good improvements in terms of dynamic performance, and interaction mitigation.

size of these filters [9,10].However, they contribute to approximately 25-40% of the total weight of the system [11], therefore reduction in the size of these filters must occur, at the expenses of interactions between converters that can no longer be considered negligible [12,13].Several studies analysing stability and dynamic interactions of multiple-converter systems have been published.One common approach is based on impedances matching between two sub-systems [14][15][16].This approach, however, is limited to examining a single converter at a time to verify its interaction with the rest of the power network.Other works propose to develop a state space model of the closed-loop system and analyse the poles location to evaluate the power network stability [10,17,18].Both methods are limited to stability analysis.If these approaches are to be used for tuning each of the converters controllers, an iterative procedure must be adopted where the closed-loop stability is first checked, controllers are then modified according to some rules, the stability then being checked again, and so on.Control structures based on PI controllers are very commonly employed to control power converters, especially for grid-tied converters.These standard methods are very well understood and relatively easy to design and implement [19,20].However, their design is based on local converter dynamics and the only way interactions with other converters are taken into account is through stability analysis as previously described.This paper therefore proposes a control design method for a multiple converter power system to tune each individual inverters' controllers keeping in consideration cross-converter interaction.Compared to previous works, the proposed approach is not limited to investigating the stability of the network, but aims to automatically design controllers for each of the individual inverters that compose the power system.Starting from the global state space model of the system, local converter controllers are synthesised by solving an optimal H 2 structured control problem [21].The power of this approach lies in the ability of handling complex control problems, whilst always returning a stable controller with great robustness to system disturbances and parametric uncertainty [22].Phase locked loops (PLLs) are an important component in grid connected ac systems to implement power flow control and synchronisation with the grid and between converters [23].It is of great importance to design PLLs correctly as it has a great effect on the overall stability and performance of the embedded system [24,25].Attempts to optimise the PLL to ensure system stability have been performed in the literature in light of these issues [7,27].In this paper, PLL tuning has been embedded in the general tuning procedure of the whole system.In this way its impact on the global stability of the system is considered and PLLs are automatically tuned together with converters controls, ensuring guaranteed global stability and improved dynamic performance over traditional methods.For clarity, the proposed approach is applied in this work to an ac sub-system within a power network as shown in Figure 1.However, the mathematical formulation of the presented method is easily extendible to complex power system with multiple converters.

SYSTEM MODEL
The system investigated in this study is shown in Figure 2. The model consists of a voltage source inverter (VSI) which is fed from a dc source and presents an LC output filter on the ac FIGURE 2 The notional test system in this study side.A three-phase active front end (AFE) is directly connected to VSI output via its input LR filter.The output of the AFE has a dc-link capacitor across positive and negative terminals, which is in turn connected to a constant power load (CPL).A PLL has also been implemented into the AFE's control architecture.
When developing the control of ac systems, a very common approach is to first convert from the ac abc-frame to the rotating dq-frame, which in turn converts the three-phase system into two coupled dc systems.To observe the equivalent circuits by which the following state-space expressions are derived, please refer to [28].

VSI dq model
The VSI state equations can be described in dq-reference frame as [28] In this model, I id and I iq represent the dq-axis currents, respectively, across the output filter inductors L, with R being the inductors intrinsic resistance.V cd and V cq represent the dq voltages across the VSI output filter capacitors C .I ad and I aq define the currents through the AFE input filter inductors L a (shown later), m d and m q represents the modulation indexes in the dq-frame and V dc represents the dc supply voltage to the VSI.

AFE dq model
The AFE state equations can be described in dq-reference frame as [28] ̇I  where R a is the intrinsic resistance of the filter inductances L a , C a is the dc link capacitance on the AFE output.p d and p q depict the dq-axis AFE modulation indexes.P l depicts the power desired at the load of the AFE at steady state, and V dc a is the dc voltage across the dc-link capacitance.The superscript p identify quantities represented in AFE dq-reference frame.This aspect will be further explained in Section 2.3.The non-linear system (2) has two equilibrium points: only one is in the operative range of the converter and it is the one considered in this paper hereafter [28].
VSI dq voltages are controlled such that the d-axis controls the peak voltage of the 3-phase grid, whilst the q-axis is controlled to 0 to maintain unity power factor.Similarly, to ensure unity power factor on the AFE side, the q-axis current of the AFE is controlled to zero.As standard on MEA, the frequency of the ac grid is set to 400Hz [1].Generally the ac grid is at 115 Vrms, but due to limitations of the experimental setup (described in Section 6) this has been reduced to 100 Vrms.This is deemed as acceptable for proof-of-concept purposes.

PLL model
A synchronous reference frame phase locked loop (SRF-PLL) has been used in this work to synchronise the AFE reference frame to that of the VSI reference frame.Figure 3 shows the generalised block diagram for the SRF-PLL [12].Its dynamic equations can be derived as where  p is the PLL estimated angle and T  p is the transformation from  to dq.The superscript p identifies quantities in the PLL dq-reference frame.All quantities without superscript are represented in the grid reference frame.The PLL control equations can be evaluated as where K p and K i represent the proportional and integral gains of the PLL, respectively.It is convenient to rewrite PLL equations in terms of angle error between the estimated angle and the true Figure 4 shows the relation among these quantities.Since a rotation by  e +  0 can be split in two rotations by  0 and subsequently by  e , substituting Equation (5) into Equation ( 4) results in the following state-space expressions for the PLL: The exogenous term  0 represents the nominal grid frequency and can be dropped during control design.Equation ( 6) can be linearised around the equilibrium steady-state point where superscript * identifies quantities at system equilibrium point.Due to the presence of the PLL, VSI and AFE equations are expressed in two different reference frames.The cross-coupling terms must be therefore transformed in the correct reference frame before merging the two models.AFE currents found in the VSI state Equation (1) have to be rotated by − e to be expressed in the VSI reference frame, resulting in  Similarly, VSI voltages found in the AFE state Equation (2) must be rotated by  e to be expressed in the AFE reference frame, resulting in Substituting Equations ( 8) and (9) in Equations ( 1) and ( 2), respectively, it is possible to merge the two sub-systems together into one global model.

H 2 SYNTHESIS AND CONTROLLER DESIGN
The main contribution of this work is to present a strategy to design inverter controllers of a power network with full consideration of interactions between sub-systems.On authors' knowledge, proposed works in the literature describe methodologies to tune converters controllers independently or to check the global system stability after the tuning has been performed.Here, we are presenting a strategy to automatically tune all the systems controllers (including PLL) at the same time ensuring the resulting global system will be stable.Hereafter the methodology will be described for the model presented in previous section.However its mathematical formulation permits to extend the concept to all converters presents in the power network.
The proposed approach is based on the optimal linear quadratic regulator (LQR).Given a linear time invariant system, LQR theory defines the optimal control action as (10) where x and u are system state and control action, respectively, whilst Q  is the diagonal positive semi-definite state weighting matrix, and R  is the positive definite diagonal input weighting matrix.It can be shown the solution to problem (10) is in the full state feedback form where K is the control gain matrix.LQR offers different advantages such as good dynamic response, robustness against parameters variation and tuning simplicity, especially when applied to multi-variate systems [29].However, feedback matrix K is, in general, full.This implies all system control actions are a function of all system states.In the case study analysed in this work, it causes VSI control action to be a function of both VSI and AFE states requiring additional communication between converters.This also applies for AFE control action and, in general, for any converter connected to the power system.Communication links between converters are not desired as they increase the system cost and complexity and reduces reliability.A solution is to impose a diagonal block structure to the matrix K in order to impose each converter control action as a function of the con-verter's own states.To do this, it is necessary to reformulate the LQR problem as an H 2 optimal control problem.Let us consider the following system: where x represents the system states, w are the plant disturbances, u is the plant control action, z is the performance output and finally y are the measured outputs.The matrix A is the state matrix while B 1 is the disturbance input matrix.B 2 represents the control matrix.In this study, the state measurement matrix C 2 is set to be an identity matrix since all states are directly measurable.Let us assume to adopt a full state feedback control approach as described in Equation (11).In addition, let us consider the following assumptions: The H 2 optimal control problem can then be formulated as follows: where ‖P‖ 2 is the H 2 norm of system (12).It can be shown that problems (10) and ( 14) are equivalent and return the same optimal control action [30].
In the following, a brief explanation about how to numerically solve problem (14) is provided.For more details please refer to [30].The H 2 norm in Equation ( 14) can be computed as where Σ is the solution of the Lyapunov equation The gradient of ‖P‖ 2 with respect to K can be computed as where Ψ is the solution to the Lyapunov equation Problem ( 14) can then be formulated as a non-linear programming problem and solved using a gradient descent algorithm [31].The advantage of H 2 formulation compared to LQR is that the former can handle constraints on matrix K .It is therefore possible to impose a block diagonal structure to matrix K ensuring each converter control action is only a function of states of the converter itself.Problem (14) can then be reformulated as where S is the desired structural constraint.Please note problems ( 10) and ( 19) are no longer exactly the same.However, in most cases, the advantages of LQR are maintained also in the structured H 2 formulation.Another drawback is that problem ( 14) is non-convex, and therefore, this work uses a multiple starting point approach to avoid local minima.

Optimal H 2 problem implementation
To apply the control design technique just described, it is required to merge all sub-systems equations presented in Section 2 into a single linear state space form as in Equation ( 12).VSI and AFE equations are in open-loop form and therefore it is straightforward to put them in this form.PLL equations ( 7), however, are in a non-canonical closed-loop form.Some additional manipulation is therefore necessary.
A change of variable is firstly executed as Substituting Equation (20) into Equation (7a) Substituting Equation (22) into Equation ( 21), and Equation (20) into Equation (7b), the new PLL state equations are obtained as where Equation ( 23) can be now rewritten as 1 and  2 are dummy inputs defined to obtain the final form (24).
It is now possible to obtain the global open loop system model in Equation ( 12) form merging Equations ( 1), ( 2), ( 8), ( 9), ( 20) and (24a), and linearising them about the equilibrium point.The resulting matrices are and detailed matrices are provided in appendix.The system states are defined as where Please note that system states have been extended with  states as integral states in order to remove steady-state error [32].System inputs are defined as A full state feedback in the form of ( 11) can now be synthesised solving problem (19) as explained before.The structure adopted for the feedback matrix K is where K vsi ∈ ℝ 2x6 and K a fe ∈ ℝ 2x5 are VSI and AFE controllers gains, respectively.K pll ∈ ℝ 2x2 are PLL gains as defined in Equation (24b).
The model used to tune inverter controllers is a global model of all the considered power networks.Solving the structured H 2 problem for this system permits to synthesise all inverters controller gains (including PLL) at the same time, keeping automatically in consideration dynamic interaction between sub-systems.The stability of the power network is guaranteed, and therefore there is no need for further stability analysis after the tuning.Imposing structure (33) permits to avoid any communication between sub-systems.Even if the design as demonstrated is for a simple power network sub-system, and its mathematical formulation is easily extendible to any number of converters.

H 2 controller tuning procedure
Apart from the constraint imposed in the feedback gain matrix for the H 2 optimisation, the optimisation problems defined in Equation ( 10) and ( 19) can adopt the same cost functions.For this reason, the tuning procedure can be explained referring to the LQR control problem (10).There is no direct relation between the Q  and R  weights and the time and frequency domain performance, however, each of the assigned weights does designate how we wish the system to behave, since the larger the weight in each respective matrix, the more the states and inputs are penalised, respectively, as per Equation (10).As an example, the larger the R  weights, the more the inputs are penalised to change during transients, reducing the controller bandwidth.Conversely, decreasing the weight of R  results in less penalisation on the inputs, allowing the inputs to move more flexibly which, in turn, results in an overall increase in the controller bandwidth.With regards to the Q  matrix, larger values of the Q  weights increases the penalisation of system states, resulting in the controller trying to stabilise the system with as little change to the states as possible, resulting in an increase in overall bandwidth.Conversely, with a decrease in Q  weights, the less penalisation of the states and thus implies less concern about the changes of the states, allowing them to move further from their equilibriums to stabilise the system, and thus exhibits a reduction in controller bandwidth.In general, it is often necessary to perform a trial and error approach in the weight selection in order to get the desired performance.However, some simplifications can be made in order to develop a simple tuning procedure to attain desired performance from the grid.First of all, when a system is extended with integral states, it is convenient in the Q  matrix to weight only these terms to ensure good reference tracking and disturbance rejection.Another simplification is that for the VSI, there is no need to have alternate weights for each the d and q axes; as is shown in [12] the equivalent circuits are almost identical.Since there is a trade-off between weighting either the Q  or R  matrix, and the system performance, the starting point to any LQR-based optimisation is to initially set all the concerned weights in each matrix to 1, the most simplest form these weights can be.Incorporating this, and the above simplifications on Q  , the initial weighting matrices are When designing LQR-based controllers, typically the R  matrix is first tuned by adjusting a constant , such that R  = I .This constant is adjusted across all the input weights until satisfactory performance of the system is attained.This is effective when optimising a single sub-system, but in this case there are many sub-systems whose input characteristics are different from one another and therefore should not be weighted using the same universal constant .Therefore, for the initial tuning step it is helpful to split  into weighting constants for each subsystem.
where m is the number of inputs attributed to sub-scripted sub-system.As a general rule when it comes to tuning these weighting matrices, increasing the corresponding weights of R  reduces the control effort from the respective input during transients and therefore decreases the corresponding converter bandwidth.For example, if the weighting constant  a fe corresponding to the inputs p d and p q is increased, the AFE bandwidth decreases accordingly.For the system described in Table 1 the R  matrix selected which brought the system to satisfactory performance was selected as Even with the Q  matrix left at its default setting in Equation (34), one should find this ought to bring the system close to the desired performance.Therefore, the Q  matrix can then be used to adjust and fine tune the controller to as close to the specification required.For an increase in a given weight in Q  the less the respective state can vary during transients, effectively increasing the bandwidth of said state.Simply, if one finds the dynamic response of the dc-link voltage V dc a induces too large a undershoot for a given transient, and too slow to recover back to equilibrium using the R  matrix in Equation (36), one can simply increase the weight to effectively speed up the response of V dc a .This example is an instance that actually occurred when designing the controller for this work.Consideration should also be made as to which states should be prioritised for increasing the bandwidth.For instance, increasing the bandwidths of all the states can lead to large values in K , which for some application can greatly reduce the robustness of the control.Bearing this in mind, the selected Q  matrix where the dynamics of V dc a and I P aq were prioritised for improved performance is the following:

IMPLEMENTATION OF THE CONTROLLER
In Figure 5, the block diagram of the H 2 decentralised controller is presented.As it can be seen, the general structure of the controller is not trivial, and shares similarities to that of a more complicated PI controller.In order to generate the decentralised H 2 controller, the structured H 2 problem (14) has been solved using HIFOO Matlab toolbox [35].As the output global controller from the H 2 optimisation is constrained to the structure imposed by (33), several independent but globally optimal controllers are generated for each of the sub-systems.The resultant output is a decentralised controller where each diagonal element in Equation (33) are the independent state-feedback controller gains for each of the sub-systems.Each of these controllers can be further separated down into their constituent proportional gains (K vsi P & K a fe P ) which provide a desired actuation based on the value of the states for each respective converter directly; and integral gains (K vsi I & K a fe I ) which act on the integrals of the error of the references at that instant moment in time.Therefore, each of the decentralised controllers presented in (33) can be further broken down into their proportional and integral controllers such that and where K pll follows the structure as presented in Equation (24b).With the separation of the control across proportional and integral actions, the control law for each converter as shown in (11) can also be further split down in to their respective parts such that where Since in this study the VSI is assumed the generator of the grid, the desired angle of the grid () is set in software, and fed into each of the dq transformations.The AFE on the other hand has the angle computed by the SRF-PLL.To compute the angle, the three-phase voltages are measured, and fed into a dq The resultant modulation indexes calculated from Equations ( 40) and ( 41) is atypical from standard modulation schemes in that these variables vary between −1 and 1. Therefore in each of the PWMs, which dictate the demanded switching action, the carrier wave fed into the comparator action also varies by the same amount, as shown in Figure 6.This does not affect the performance of the converter as opposed to conventional modulation schemes.

SIMULATIVE ANALYSIS
In order to characterise and ascertain the performance of the proposed decentralised controller, simulative tests will be carried out within the MATLAB 2020a Simulink environment.To best determine the performance of the control, the proposed controller will be compared against a standard PI controller, whose control structure is that presented in Figure 7, and an LQR controller, which has the exact same control structure as that presented in Figure 5.For the LQR augmented system, however, unlike the proposed H 2 controller, each converter has an LQR controller locally optimised to its own dynamics, since LQR controllers cannot be structured into decentralised form, and a centralised control architecture is undesired in this study.Since the PI is the controller with the slowest stable bandwidth, the decentralised H 2 and LQR controllers are tuned to achieve similar performance as that with the PI controller to maintain fairness in the analysis between each of these controllers.The parameters used for the simulation, and the experimental setup for which these controllers will be tested upon are presented in Table 1.In designing the PI controller, the procedure from [12] was used, where the PI controller bandwidths which achieved the best overall dynamic performance, as well as stable performance on the experimental setup described in Section 6, and are presented in Table 2. Any higher bandwidth resulted in steady state oscillatory behaviour or instability.By general con-  vention, internal current loops are designed to be 10x the bandwidth of the outer voltage loop.It was however observed that to get the best stable performance from the AFE PI controller, the inner current control loop had to have a bandwidth of approximately 20x that of the outer voltage loop.The damping factor  is set to 0.7 for all PI controllers.The tuning of the LQR control is similar to that of the proposed H 2 controller, however performed on the local level of each converter instead of the global system.The cost function defined in Equation ( 10) is used to synthesise each of the controllers.
To synthesise the LQR controller for the VSI, B vsi from the appendix was used, as well as the VSI state-matrix A vsi lqr defined in (44) Likewise, for the AFE LQR controller, the AFE input and state-matrices B a fe and A a fe , respectively, as defined in the appendix are used for the synthesis.The Q lqr and R lqr weighting matrices used to achieve similar dynamic performance to the H 2 and PI controllers are given in Table 3; selected to attain as close as dynamic performance attained for both the PI and H 2 controllers.
In the proposed controller, as per Equation ( 33), the PLL is automatically tuned with consideration to the global dynamics of the system.However, for the LQR controller, due to that the output controller cannot be constrained, the PLL cannot be augmented in the AFE LQR controller synthesis.Therefore, for both the LQR and PI controllers in this study, and to ensure the same dynamics for each test, the PLL in both cases is designed with a bandwidth of 100 Hz, and a damping of 0.707, based on the PLL transfer functions as defined in [37].Therefore, the PLL controller gains for the LQR and PI controller tests are Two different simulative tests will be investigated.The first being a step load test from when the system is initially at noload steady state; and the second being a parametric robustness test to analyse the performance of each controller when faced with non-nominal parametric quantities.

Simulative tests -1kW step load
In this test, the dynamic performance of each controller shall be compared, when each system is brought to no-load steady state, and at an instant a 1 kW step load is applied at the CPL of the AFE converter.To start, the average simulation of the traditional PI controller is presented in Figure 8.
For each of the simulative tests to be investigated, the load step is initiated at 0.3 s and the dynamic performance of each controller will be analysed.For the PI controller in Figure 8, the interactive effects between converters are very evident.A locally sourced disturbance on the AFE perturbed across all the system states.In fact, for this particular system configuration, this particular disturbance perturbed m d so much that it inflicted the VSI to enter saturation, due to large deviations incurred caused on V cd and V cq .Due to the nature of the CPL, large decreases in voltage results in large increase in current.The large 3A overshoot incurred in I ad in turn brought down the dc-link voltage 50 V below reference, which, due to the controllers not being designed with consideration to the other controllers dynamics, inflicted large perturbations on the VSI states, and thus perturbations directly on the grid currents and voltages.This in turn caused saturation of the VSI modulators for a time, which caused prolonged effect of the disturbance as the controller attempted to return the system to operate at its new steady state point.Therefore, the disturbance on the AFE influenced the grid voltages greatly, and in turn extended the non-alignment of the PLL to the grid, further prolonging the disturbance dynamic.As will be shown, optimal controls greatly mitigates these issues, however, the LQR not being optimised to the network globally starts showing similar behaviours to this PI control.
In these types of power systems, the PI controller is by far the most popular methodology of controlling these power converters, however, LQR controllers, due to ease of design and their superior optimal performance having been gaining popularity.The performance of the system under the influence of an LQR controller is shown in Figure 9. Dynamic performance across almost all states is improved in the LQR controller, due to the controllers being optimal to the local dynamics of their associated converter, whilst PI is design based on specified bandwidths.To quantify the performance improvement of the LQR controllers, distortion across V cd and V cq has been reduced to magnitudes of 4 V each, whilst for PI it was 50 V and 30 V, respectively.The dc-link also exhibits a dramatic improvement, with a undershoot magnitude of 20 V in comparison to the PIs 60 V deviation.The only state, where the LQR performs worse is in I aq , where disturbance magnitude was increased from 0.2 A in PI to 2 A in LQR.As with any design improvements, there is always a necessary draw back, and for use of this optimal controller, I aq does unfortunately suffer performance degradation, in order to attain performance benefits across the other states.Additionally, note that for the PI and LQR, the PLLs are identical, yet the LQR achieved upto 20x performance boost in terms of error mitigation, and provided no oscillatory response; all due to the vastly improved performance of the grid voltages under LQR control.
On the contrary, however, is the proposed H 2 controller.As explained, the LQR and H 2 share some similarities in terms of their generation, but the H 2 controller allows for constraints to be imposed on the output control structure, as per (33), such that independent decentralised controllers are generated, that are not only global optimal to all grid dynamics, but are also optimal concerning the closed-loop performance of each subsystem of the network since all the controllers in K are optimised simultaneously.The simulation results for the H 2 controller are shown in Figure 10, and the performance improvement attained over the LQR is clear.Firstly, the magnitudes of state deviations are significantly reduced.V cd and V cq for instance have their undershoots reduced by half to 2 V, and incur no oscillatory overshoots, albeit a more damped response.This in turn results in less significant distortion on the on the grid voltages, as will be observed later in the experimental setup.Since the grid voltages become less distorted, this also means the optimised PLL error is reduced by 60%, resulting in better tracking of the AFE to the grid dq angle.With better tracking of the grid, comes better transmission between converters, and as such, the deviation in the reactive VSI current I iq is significantly reduced to 2.3 A, compared to the LQRs 3.5 A. Comparing the H 2 and LQR controller performance across other states, one can observe for I id overshoots of 1.2 A and 2.2 A, respectively, and I aq undershoots of -2.2A and -3.2 A, respectively.Performance improvement for the H 2 controller is most notably improved on the dc-link voltage.Whilst both controllers undershoot by 20 V during the transient, the H 2 quickly brings the voltage to equilibrium with minimal overshoot of 1 V; compared to LQR whose overshoot was 8 V and exhibited more oscillatory response upon reaching equilibrium than the response of H 2 .These at first might seem minimal improvements, however, this is just that start, and the performance improvement is even clearer, when the system is subject to even greater load step.

5.1.1
Analysis of performance at 1.6kW step load Figures 11 and 12 present the LQR and H 2 controllers of each system, respectively, with each controller re-optimised with P l = 1.6kW .Since this load step rendered the PI controller unstable, the PI results are not presented.Comparing the results between the LQR and H 2 controls, it can be seen the the LQR starts incurring a larger oscillatory dynamics back to equilibrium than that of the H 2 .Most evidently are that of the currents and the dc-link voltage.Extrapolating some numbers for the LQRs increase in oscillations, I id overshoot is 3.5 A greater; I iq undershoot is 3 A lower; V cd overshoot is 6 V higher; V cq undershoot So, increasing the power, the deviations in performance between the two controllers become more evident, and increasing the load step, causes further deterioration of LQR performance in comparison to the H 2 .In fact, during simulative testing, LQR could cope with step loads of upto 8kW before instability, whilst the H 2 controller was capable of 10kW.This largely results from the fact that since the non-linear dynamics of the CPL in the AFE are not considered in the LQR VSI controller, system interaction becomes ever prevalent as the non-linear term grows.The un-modelled non-linear dynamic results incurs further oscillatory responses from the LQR controller, comparable in effect to the oscillations observed with the PI control dynamic response.
On the other hand, for the H 2 controller, each controller is optimal to the dynamics of every other closed loop sub-system, and thus, accounts for the non-linear CPL dynamic in all the controller designs.Therefore, sub-system interaction between sub-systems is largely mitigated, resulting in the non-oscillatory behaviour between sub-systems, and the overall performance increase when subject to greater non-linear CPL dynamics.As a result, the drawback is a more slower, damped response to state-deviations, but attains more robust and non-oscillatory behaviour across the system states as a whole.

Parametric robustness analysis
When developing controllers on the assumption of the system having specified parametric values, it is important to evaluate the FIGURE 13 PI step load performance with parameter variation of±55% subject to 1kW step load performance when some parameters deviate from their nominal value, as is typical in industrial design due to component tolerances, or due to degradation of devices through continuous use or ageing.Parametric test shall be conducted using the same simulative model as the previous tests, using a 1kW step load.In each of the tests, all parameters will collectively altered by ±55%, and the resultant performance will be analysed.The PI controller test are shown in Figure 13.
As it is to be expected, as parameters are lowered, the variation of the states increases, and thus, performance worsens.Once the system goes beyond a −55% variation, the PI controlled system incurs instability.This is mainly attributed to V dc a as when the step load initiates, the dc-link drops to 0 V. Additionally, the overshoots of the currents I ad and I aq increase rapidly as parameters are reduced, with any further variation may potentially cause safety measures to trip on the converters.On another note, I iq will increase as the parameters increase in value, due to the impedance of the grid increases requiring greater reactive current.This too could also make the system more susceptible to tripping if not properly account for.
Figure 14 presents the results for the LQR controller following the same parametric variation as that for the PI test.In this test, it becomes markedly clear that the oscillatory behaviour, which was starting to become evident during the 1.6-kW tests, becomes even more pronounced in the presence of uncertain parametric values.In fact, in some case due to the incurred oscillations present on the LQR controller, states such as I id , I ad and in some regards V dc , at a +55% variation are worse in terms of their oscillatory dynamic performance than that of the PI controller.This again, is not only due to the interaction between un-modelled sub-system FIGURE 14 LQR step load performance with parameter variation of±55% subject to 1kW step load dynamics, and un-modelled non-linear CPL dynamic of the AFE in the VSI controller, as was discussed earlier, it is also due to the fact that each LQR controller, like the PI, has not accounted for possible uncertainty elsewhere on the grid.To explain this latter point, let's first analyse the H 2 controller performance.
The proposed H 2 controllers' parametric variation results are presented in Figure 15.When compared to the PI and LQR controllers, it is very evident how no oscillatory behaviour is present across any of the states during the transitions.In fact, the H 2 controller shows little in the way of dynamic performance change across the whole range of parameter variations.All results show the same damped, stable, non-oscillatory responses, with only greater deviations occurring in some states, as inductive and capacitive parametric values change, in turn changing the dynamic behaviour of the current and voltages, respectively.The performance is maintained due to another feature present in the H 2 optimisation procedure, which also explains the poorer performance observed in the LQR.The optimisation as defined in (14), whereby the H 2 norm of the global system is minimised, can also be defined as the minimization of the transfer function between any external disturbances of the system, and the errors (or performance output) of the system.For more information, please refer to [30].For the H 2 controller, where all the system dynamics are accounted for, this results in all the synthesised closed loop controllers globally minimizing all disturbances to the system and the resultant errors inflicted on the states.As parameters vary, so can the disturbances across sub-systems increase.But since the decentralised H 2 controllers account for these disturbances globally, detrimental effects to the states are min-FIGURE 15 H 2 step load performance with parameter variation of±55% subject to 1kW step load imised, sub-system interaction mitigated, and thus, oscillation are not present on the H 2 controlled system.The LQR, as previously mentioned, shares similar features to the H 2 controller, including the just described minimization of disturbance to system error.However, due to the inability of the LQR controller to be constrained to a structure, the global dynamics of all the sub-systems cannot be accounted for.Whilst on the local level, the transfer function from disturbance to error is minimised [30], resulting in the improved performance over the PI, this does not happen globally, and thus each LQR controller is not designed to minimise globally such disturbances.As parameters vary in value, so will the disturbances increase or decrease across the states of the other sub-systems, which are unaccounted for by the independent LQR controllers and thus results in an increase in cross-converter interactions.Therefore, oscillations across the states incur, suffering similar the PI controller which also does not incorporate dynamics of other sub-systems in its design.Therefore from the simulative tests, the proposed H 2 controller when compared to the LQR and PI controller has been shown to be more robust, and offer better dynamic performance in the presence of a non-linear step load; and better performance to parametric uncertainty, whilst having the minor drawback of more damped system response to disturbances.

EXPERIMENTAL RESULTS
Experiments are conducted on the setup shown in Figure 16.
Both the VSI and AFE are commercially available inverters.
The VSI is a 2kW, 300V converter from BMT (Best Motion Technology).The AFE is a 3kW converter from Semicron.The The University of Nottingham [33].The nominal values of the system in this study are shown in Table 1, as well as reference values for the control.
It is of greater interest to compare the performances of the LQR and H 2 controller experimentally in this study.As it was shown in Section 5.1.1,exhibiting the system system to higher powers highlights the performance differences between these optimal controls in greater detail, and thus experiments shall be analysing the dynamic performance with a step of 1.6kW.Since the PI controller was not stable at these powers, and that the performance was worse and incomparable to that of the optimal controls and finally due to space requirements, experimental results of the PI have not been been presented.The aim of the experiment is to validate that the simulative analysis accurately portrays the controllers on hardware, and to analyse any further performance differences.
In each case, the AFE dc-link capacitor was pre-charged while the VSI was at no-AFE steady state, as to procedure detailed in [34].This is to safely soft-start the AFE, and minimise inrush currents.

LQR control system performance
For the experimental test conducted in this study, initially both the VSI and AFE are running at no-load steady state condition.At 25 ms, a 1 kW CPL step is applied on the output of the AFE.The system response to this disturbance for the LQR controller can be observed in Figure 17.From first observations, it is clear that the experimental results reflect that of the simulative results presented in Figure 11 relatively accurately, with only some minor differences to note.Firstly, the dc-link voltage did drop slightly further than was predicted, resulting in a drop of 45 V, as opposed to 40 V achieved in simulation.Due to the nature of the CPL, this in turn resulted in a slightly higher I ad overshoot than predicted, of around 2 A. Other than these minor differences, the rest of the system states are fairly accurate to that obtained in simulation.The tests also confirmed what was observed in simulation, in that the more oscillatory

Decentralised H 2 control system performance
For comparison, the experimental performance of the structured H 2 controller under a 1.6 kW step load, using the weights from Equations ( 36) and ( 37) is shown in Figure 18.When compared with the simulative results presented in 12 it is clear that each complement on another with similar performance attained.In contrast to the LQR results, it is clear that in many regards the proposed H 2 controller outperforms that of the LQR control.Firstly, whilst the drop in dc-link voltage is less than that of the LQR controller by about 10 V, consistent with the simulations, it is clear a less oscillatory response has been achieved, reducing distortion, and in turn improving the dynamic speed o the response.This is also seen with I ad where the more damped, speed response, and the undershoot of I aq is reduced.Transferring to the VSI and its a similar story.V cd and V cq only incur minor perturbations which results in less distortion on the grid voltages.The grid currents, albeit more damped than that of the LQR provide less distortion, and is immediately clear the improvement of performance in three-phase obtained over the LQR controller.As was mentioned before, the H 2 and LQR controllers share much of the theory in how each is generated, and the response obtained in hardware is more typical to the response of an LQR controller; fast damped response and non-oscillatory.It is therefore immediately clear, and can be confirmed that the interactive behaviour between sub-systems, when utilising the proposed controller in this paper can largely be mitigated.Additionally, the THD performance of each controller has been analysed.Since each uses the same control architecture, differences were minimal.Grid voltages with H 2 had 2% THD, whilst LQR had 3%.Currents were comparable to each other, suggesting slightly better quality output signals were attained adopting the proposed H 2 approach.As was observed in simulation, the PLL performance for each controller fits perfectly to that predicted.Again, the H 2 has shown that the magnitude of the error can be reduced significantly by 35%, but in order to attain this performance, the PLL does incurs a very damped response, however this does not seem to overly effect performance of the system, since error is kept so low throughout the disturbance.

CONCLUSION
This paper has presented an approach to design a global decentralised H 2 optimal control for converters forming an embedded power network, taking into account their interactions on the common ac grid.From the global model of the system, the proposed method synthesises individual standalone controllers of each converter keeping in consideration dynamic interactions.The PLL, which often can cause stability issues, was successfully tuned as part of the whole system.There is no need to additional stability check after tuning procedure as a global stable system is guaranteed in the adopted method.The method has been successfully been tested through experiment and compared to traditional PI control, as well as another form of optimal control, LQR.The results have presented significant improvements in the decentralised controller design.For instance, significant reduction in the effect of sub-system interactions, which opens further possibility of reducing the need of bulky passive filters, often used as decoupling elements, paving way for the possibility of reduced system size and cost.It was also shown to have improved robustness to parametric uncertainty, and against highly non-linear CPL dynamics.For clarity the proposed approach has been presented using as an example a notional sub-system consisting of two converters on the same ac grid.However its formulation is easily scalable to a more complex network, and will be presented in future works.

FIGURE 1
FIGURE 1 Electrical sub-system under investigation

FIGURE 3
FIGURE 3 Block diagram of the SRF-PLL

FIGURE 4
FIGURE 4 DQ frame operational angles between estimated and true grid frequencies

I
ad = I p ad cos( e ) − I p aq sin( e ) I aq = I p ad sin( e ) + I p aq cos( e ).

FIGURE 5
FIGURE 5 Decentralised H 2 controller block diagram

FIGURE 6
FIGURE 6 Configuration of the PWM comparator for modulator range

FIGURE 7
FIGURE 7 The cascade PI control architecture for the VSI(a) and AFE(b)

FIGURE 8 FIGURE 9
FIGURE 8 PI controller step load test -No load steady state to 1 kW

FIGURE 10
FIGURE 10 H2 controller step load test -No load steady state to 1 kW

FIGURE 11
FIGURE 11 Performance of LQR controller under 1.6kW step load

FIGURE 12
FIGURE 12Performance of H 2 controller under 1.6kW step load

FIGURE 16
FIGURE 16 Experimental test rig

FIGURE 17
FIGURE 17 Experimental performance results of the LQR controller under 1.6 kW step CPL

FIGURE 18
FIGURE 18 Experimental performance results of the H 2 controller under 1.6 kW step CPL

TABLE 1
Nominal system parameters

TABLE 2
PI controller gains used for testing

TABLE 3
LQR controller weights for the individual converters