Nonlinear Hierarchical Easy-to-Implement Control for DC MicroGrids

: It was considered in this work the connection of a photovoltaics (PV) solar plant to the 1 main grid through a Direct Current (DC) MicroGrid and a hybrid storage system, composed of 2 a battery and a supercapacitor, in order to satisfy constraints of grid connection (the so-called 3 Grid-Codes). The objective, and main contribution of this paper, is to stabilize the DC MicroGrid 4 voltage in spite of large variations in production and consumption, using a nonlinear hierarchical 5 easy-to-implement control strategy. Here it is presented the MicroGrid’s control design based 6 on detailed models of the photovoltaic energy sources and the storage systems. Such DC grids 7 may present an unstable behavior created by the PV’s intermittent output power, by switching 8 ripples from the power converters and their power electronics, and oscillatory currents produced 9 by some types of loads. Therefore the system is subject to both fast and slow variations, and its 10 stabilization is based on different technologies of storage, such as battery and supercapacitor, 11 and control algorithms designed thanks to the use of time-scale separation between different 12 components of the storage systems. The obtained nonlinear results are stronger than current linear 13 controllers, allowing to keep operating margins around the voltage reference. At the same time in 14 this work, insights from power systems practice have been used, aiming to obtain a very simple 15 and easy-to-implement control scheme. Detailed simulation results are provided to illustrate the 16 behavior and effectiveness of the proposed stabilization technique. 17

the overall interconnected system is done; Section V provides simulation results about 88 the connected system behavior; finally, Section VI presents the conclusions.DC loads and grid-tied converters [22].The energy storage elements play an important 93 role for the entire power management of the DC MicroGrid as mentioned.They ensure 94 a secured network, provide high quality power and maintain the common DC grid 95 voltage stable [23].Bidirectional converters are used to charge or discharge the energy 96 storage elements such as the battery and the supercapacitor.The load (an electric vehicle 97 charge station for example) is modelled as a variable resistance, which in the present 98 case are more delicate than constant power loads.

99
In the following, the different components of the DC MicroGrid are described.A complete solar panel model with series and parallel connection is proposed 103 [24], [25].The PV is represented as a current source.Due to the variations of the tem-104 perature and the solar radiance, the current and the voltage generated by the PV panel 105 vary.

106
The PV DC/DC boost converter operates following references given by a standard 107 MPPT algorithm (not discussed in this paper) under varying levels of irradiation and 108 temperature [25], [26] and [24].

113
The mathematical model for the DC/DC boost converter that interconnects the PV 114 to the DC grid is obtained based on power electronics averaging technique (see [27], [28], 115 [29], [30]), and is given by (1).116 1.2.Storage system 117 The hybrid storage system (battery-supercapacitor) adopts the advantages of both 118 technologies, high power density from the supercapacitor and high energy density 119 from the battery.The proposed hybrid storage consists of a lead acid battery and a 120 supercapacitor, as in [31], [32], [3] and [33].The Supercapacitor is modelled by the so-called three-branch model, extracted from 126 the transmission line modelling.It is connected to the DC grid by a DC/DC bidirectional 127 boost converter as in [35] and [36].The bidirectional boost converter controlling the battery is given by (2), based on 130 the voltages on the capacitors C 3 and C 4 and the current in the inductance L 2 (see [37], 131 [29,32] and [30]).
The bidirectional boost converter controlling the supercapacitor is modelled based 133 on the voltages on the capacitors and the current in the inductance C 5 , C 6 , L 3 , shown by 134 (3), (see [29], [32] and [30]).

Interconnected Model 136
The mathematical model for the DC grid in Figure 1 is obtained based on the power 137 electronics averaging technique [27]: The measured output vector is composed by    paper, the Incremental Conductance method is applied (see [25] , [38]and [39]).We rely on the storage system (battery + supercapacitor) to act as a buffer to absorb the current (power) variations and stabilize the DC bus.Based on the physical characteristics of the storage components, and in particular on the current limitations from the battery, we indicate: 1-to consider a current reference i * ST based on power balance equations for the whole system; 2-to use a time-scale separation to decompose it in two components, a fast one (i * L3 ) and a slow one (i * L2 ), as shown in Figure 3.The slow component will be used as the reference for the battery's power supply, while the fast component will be the reference power for the supercapacitor.This time scale separation is obtained by the introduction of a first order low pass filter that decomposes the current reference in two references for the battery and supercapacitor [40] i The two components are chosen to keep sufficient time-scale separation between 162 the lower and higher level control objectives (see [41]), and to ensure that the battery 163 does not need to react faster than its specifications allow.Therefore, we target to extend 164 the lifespan of the battery [10].The power flow of the different elements of DC MicroGrid (see [22]) will depend on the grid topology.Most works consider a radial DC MicroGrid for its simplicity while yet capturing most challenges existing in DC MicroGrids.Radial systems are easier for installation and operation and meet the objective of designing a plug and play system which allows a larger penetration of renewable energy into the network.The proposed control strategy is also applicable for ring or bus [42] topologies (which are simpler than radial), and may also be implemented in meshed ones.For the latter it would be necessary to prevent congestion problems, that is not addressed in the present paper, and is usually dealt with higher-level controllers, closer to optimization and communication techniques.Bus topology is currently used in residential buildings, where low voltage DC bus is preferred to match the voltage level of many appliances and to avoid extra DC-DC conversion stages.Also in such systems, loads and AC grid interface can be located close to each other in order to reduce the distribution losses [43].A radial grid is shown in Figure 4 while a ring DC MicroGrid is shown in Figure 5.The sum of the output power of all generating units (photovoltaic system in the present case), all loads and the storage system is defined as follows: with P ST = P BAT + P SC , where P PV is the PV power, P ST is the storage power, P BAT is 166 the battery power, P SC is the supercapacitor power, P Load is the load power and P DC is 167 the DC grid power.

168
In general, the DC MicroGrid is connected to the main AC Grid then P DC = P GRID , 169 and the balance power is supplied to or obtained from the AC Grid (P GRID being the 174 The energy storage elements can switch between charge and discharge mode in order to maintain the DC grid power balance.V DC (t) must be kept around its nominal value V * DC and inside voltage limits following the power balance given by equation ( 4).It is possible to rewrite equation (4) as: In the following, i * ST as in ( 5) is designed for i L2 and i L3 , in order to steer V DC to its 175 reference V * DC using backsteeping techniques [44].Let us define the output tracking error Based on the error dynamics, we suggest the following current 177 reference: with: α10 = K α 10 e V DC where α 10 represents an integral error term, and K 10 , K 10 and K α 10 are positive tuning 179 gain parameters. 180 Due to time-scale consideration between the voltage and current dynamics, in the sequel we consider V DC as a zero dynamics of the system, so i ST can be considered already equal to i * ST , such that inside an operation region (D = {e V DC ∈ R | e V DC ≤ V DC Max } for a desired constant V DC Max ) and we can write: As a consequence, the closed-loop dynamics of V DC results to be linear and expo-181 nentially stable, with desired dynamics given by the stable poles assigned by the tuning Taking the i * L2 reference value for i L2 , it is possible to compute the control law by feedback linearization.First, we define the output tracking error: and consider the control law [11]: where i * L2 is obtained through a derivative filter, α 6 represents an integral term and K 6 , K 6 and K α 6 are the positive tuning gain parameters.By technological reasons, the following condition is always fulfilled: Consequently, the closed loop system is shown to be linear and exponentially stable [11]: Consequently, the closed loop system results linear and exponentially stable [11]: We remark that i * L3 is obtained as i * L2 in (11), and we need to consider a condition similar to the one in with (12) (always fulfilled by technological reasons): The control inputs u 2 and u 3 feedback linearize the dynamics of i L2 and i L3 .Because 199 of the linearity of the stable closed loop system, there exist matrices P 6,9 = P T 6,9 > 0 200 such that P 6,9 A 6,9 + A T 6,9 P 6,9 = −I 2 , where I 2 is an identity matrix in R 2 .Therefore, we 201 introduce the candidate Lyapunov function: 202 which time derivative is:

204
Based on (1), the control target for the PV array is to stabilize the voltage capacitor V C1 around the time varying (piecewise constant) reference value V * C1 , provided by the MPPT algorithm.Similarly to [10], we state the assumption that the current can be set to have much faster dynamics than the capacitances' voltages.This allows the current and voltage regulation to be in a cascaded control scheme, with an inner current loop and an outer voltage loop.Consequently, a reference i * L1 for i L1 will be designed to steer V C1 to its reference V * C1 .To this purpose, we define the tracking errors: Then, similarly to [11], we consider the tracking reference as: and the control law: where α 1 and α 3 represent the integral terms assuring zero error in steady state between the dynamics of the states and their references: and where K 1 , K 1 , K α 1 , K 3 , K 3 and K α 3 are positive tuning gain parameters.As before, the control input u 1 feedback linearizes the dynamics of V C1 and i L1 , and the closed loop results to be linear and exponentially stable [11]: As a consequence, there exist matrices Then, we introduce the Lyapunov function for the dynamics of V C1 and i L1 and its derivative as: 207

208
It is possible to split the states into controlled variables V C1 , i L1 , i L2 , i L3 and V DC and 209 the zero dynamics (uncontrolled variables) V C2 , V C3 , V C4 , V C5 and V C6 .With respect to 210 the equilibrium points given by the references, the calculation of the missing equilibrium 211 points V e C2 , V e C3 , V e C4 , V e C5 and V e C6 can be easily done by steady-state considerations for 212 the closed loop system.
213 Lemma 4.1.Under the assumption that for each time the conditions: and u 3 given by ( 20), ( 11) and ( 14) such that the system (1), ( 2), ( 3), ( 4), has in closed loop an 216 equilibrium point x e and any initial condition exponentially converges to it.

217
Proof.The proof is based on a Lyapunov function W, which is a composition of the 218 previous Lyapunov functions.We consider the closed loop system with respect to the 219 control inputs u 1 , u 2 and u 3 already defined for controlling the dynamics of V C1 , i L1 , 220 i L2 , i L3 and V DC .They were proven to be stable by the Lyapunov functions W 1.3 , W 6.9 221 and W 10 .These Lyapunov functions, provided for each step, will be used for the entire 222 system by a composite Lyapunov function.The Lyapunov function V is then defined as: where W 13 , W 69 and W 10 have already been introduced and W 23456 refers to dynamics V C2 , V C3 , V C4 , V C5 and V C6 .We introduce the errors e V C2 , e V C3 , e V C4 , e V C5 and e V C6 between the remaining voltage dynamics and their equilibrium points, and consider a candidate Lyapunov function W 23456 as: 224 Its time derivative is: Finally, we consider the composition of the Lyapunov functions for the whole +([α 9 e i L3 ] T P 9 [α 9 e i L3 ]) + ([α 10 e V DC ] T P 10 [α 10 e V DC ]) > 0 which time derivative is: Then, exponential convergence of all states of the interconnected system towards 228 their equilibrium points is ensured.

230
In this section we present simulations using detailed switching models on Sim-231 scape Electrical from Matlab Simulink.These simulations allow to isolate the control 232 and dynamic response phenomena, while still representative for the remaining power 233 electronic aspects. 234 Two simulations are presented.The first one is done during 300s of simulation time, irradiance.It can be seen that the output power follows the irradiance variations.Figure

275
These results confirm that the proposed scheme is effective in achieving the goals 276 defined for this application.verter ( [45][46][47]).The model is formulated under the assumption that all the converter 284 elements are ideal.The PI controllers are designed similarly to the ones presented in 285 [48,49], [50], [51], [52] 286 In general the DC MicroGrid is connected to the main AC Grid and then P DC = 287 P GRID , and balance power is supplied to or obtained from the AC Grid.In our work we 288 will study the more difficult case when the DC MicroGrid is in islanded mode, and we 289 can not rely on the AC network to balance the DC grid.In this case, any unbalance on 290 power flow will create a voltage excursion on the DC grid.

291
From equation ( 6) we define the dynamic equation of voltage of DC MicroGrid : The energy storage elements can switch between charge and discharge mode in order to 293 maintain the DC grid power balance.V DC (t) must be kept around its nominal value and 294 inside voltage limits following the power balance To present a performance comparison 295 of the proposed controller, Figure 13 shows the results for the same simulation as above   nonlinear control has a much smaller overshoot (10%) than the linear PI.In the same 320 way, settling time is much faster for the nonlinear control.

89 1 .
DC Microgrid Model90 A general DC MicroGrid as in Figure 1 is composed of Distributed Generation (DG) 91 such as photovoltaic arrays, energy storage elements like supercapacitors and batteries, 92

109 1 . 1 . 2 .
DC/DC Boost Converter 110 The DC/DC converter used to control the PV array is an unidirectional boost 111 converter.It is represented by three state variables: the voltages on the capacitors C 1 and 112 C 2 and the current in the inductance L 1 .

121 1 . 2 . 1 .
Battery model 122 The chosen model for the study is based on the resistive Thevenin model for a Lead 123 Acid Battery, considered as a voltage generator [34].1241.2.2.Supercapacitor 125

L3 and 145 V 2 .Figure 2 )
Figure 2).First, we define the high level controller to give reference values for lower local 153 155

2. 1 .
High Level Controller for PV array source 156 A Maximum Power Point Tracking (MPPT) algorithm is used to extract the maxi-157 mum power from a photovoltaic array of solar cells.To accomplish this task, a V * C1 time 158 varying (piecewise constant) desired voltage reference is provided by the MPPT.In this 159

Figure 3 .
Figure 3.The adopted time-scale depending control strategy.

Figure 4 .
Figure 4.The Power Flow scheme.

235Figure 6
Figure 6  in the left shows the output power for the photovoltaic system with varying

Figure 7 Figure 8 Figure 9 Figure 10
Figure7shows the signal decomposition from equation (5), with fast and slow

Figure 8 .
Figure 8.Control inputs u 2 and Output power of the battery.

Figure 9 .
Figure 9.Control inputs u 3 and output power of the supercapacitor.

Figure 11 Figure 12
Figure 11 presents the DC voltage V DC response for this longer simulation with 277

Figure 12 . 311 Figure 14
Figure 12.Dynamic response of the system.

Figure 13 .
Figure 13.V DC dynamics applying classical PI linear control.

322
The efficient integration of distributed renewable power sources, especially fast 323 intermittent sources as solar, is an essential but difficult task.It depends on developing 324 flexible integration strategies to the network.325Inthis paper, a DC MicroGrid composed of PV panels and storage devices evolving 326 in different time scales is considered to favor this integration, and controllers for each 327 element are designed.A hierarchical control strategy is proposed such that it absorbs the 328 maximum available renewable power while keeping the DC grid's voltage stable with 329 respect to variations due to power production and consumption.The proposed control 330 algorithm splits the power reference into fast and slow components, to be tracked by 331 the supercapacitor and the battery respectively.As a consequence, the storage devices 332 keep the DC voltage at the desired value and guarantee the balance between power 333 generation and consumption.Moreover, in order to guarantee a suitable lifespan for the 334 battery, the control strategy respects physical constraints.335 Detailed simulation results corroborate our claims, and illustrate the good behavior 336 of each element and the overall interconnected system.Compared to other results in 337 the literature, this paper focuses on proposing easily implementable control algorithms, 338 based on physical characteristics of the grid's elements, in order to obtain desired 339 closed-loop dynamics and a formal stability proof.Future works will be dedicated to 340 implementation in test-beds and the extension of the result to more complex meshed DC 341 grids.

Figure 14 .
Figure 14.Comparison of the V DC dynamics by applying nonlinear control (in blue) and classical PI control (in red).
I PV Output current of solar cell P PV Output power of solar cell V PV Terminal voltage of PV cell I bat Output current of the battery V bat Output voltage of the battery P bat Output power of the battery I sc Output current of the super capacitor V sc Output voltage of the super capacitor P sc Output power of the super capacitor P DC Output power of the DC microgrid T Cell's reference Temperature G Solar irradiation SOC State of charge of battery R 01 ,R 02 internal resistances of DC/DC converter for the PV R 1 ,R 2 resistances of DC/DC converter for the PV L 1 inductance for the boost converter for the PV C 1 C 2 capacitance of DC/DC converter for the PV V C1 ,V C2 Voltage of capacitance C 1 and C 2 of DC/DC converter for the PV i L1 current of inductance L 1 for the boost converter for the PV R 03 ,R 04 internal resistances of DC/DC converter for the battery R 3 ,R 4 resistances of DC/DC converter for the battery L 2 inductance for the boost converter for the battery C 3 C 4 capacitance of DC/DC converter for the battery V C3 ,V C4 Voltage of capacitance C 3 and C 4 of DC/DC converter for the battery i L2 current of inductance L 1 for the boost converter for the battery R 05 ,R 06 internal resistances of DC/DC converter for the super capacitor R 5 ,R 6 resistances of DC/DC converter for the super capacitor L 3 inductance for the boost converter for the super capacitor C 5 C 6 capacitance of DC/DC converter for the super capacitor V C5 ,V C6 Voltage of capacitance C 5 and C 6 of DC/DC converter for the super capacitor i L3 current of inductance L 1 for the boost converter for the super capacitor C dc DC-link capacitance of DC micro grid f Frequency of the AC grid f c cutoff frequency for the filter K 1 ...... K 10 positive tuning gains parameters u 1 , u 2 and u 3 Duties cycle of DC/DC converter C 5 , C 6 and C DC are known positive 140 values of resistances, inductances and capacitors respectively, and P load represents the DC 141 demanded power load.To model the switches, small resistances (R 01 , R 02 , R 03 , R 04 , R 05 142 and R 06 ) are included to take into account conduction losses.V PV , V BAT , V SC and V DC are 143 the photovoltaic panel, battery, supercapacitor and DC MicroGrid voltages, respectively.144 .1.2.Supercapacitor Control Law 191We define the output tracking error e i L3 = (i L3 − i * L3 ) and its derivative ėi L3 =

Table 2 :
(2) maximum DC bus voltage error is equal to 55V for linear 314 control in red.By using the PI linear control, the voltage V DC oscillates during 1, 2s 315 before reaching steady state while the steady state response for the proposed nonlinear 316 control is reached at 10ms as designed (it could be chosen to be faster though).Performance comparison with linear control and non linear control Table(2)shows a comparison of voltage overshoot and settling time for several 318step changes in load demand and photovoltaic power.There one may remark that the

Table 3 :
List of Symbols