Multi-Renewable Energy Agent Based Control for Economic Dispatch and Frequency Regulation of Autonomous Renewable Grid

This paper addresses frequency regulation and the economic dispatch problem of an Autonomous Renewable Grid (ARG) primarily composed of Multi Renewable Energy Agents (MRAs), interfaced through DC/AC inverters. A large number of MRAs that have an inherent fluctuating nature and frequent disturbances in inter-connected systems require fast and robust control to stabilize the frequency and to maintain cost-effective operation of an ARG. To address the above control challenges, Distributed Averaging Integrator (DAI) based control schemes were proposed in various research works. The main flaws of such schemes were slow convergence, sluggish response, poor transient performance and a difficult selection of an appropriate damping-factor. The proposed approach introduces a Distributed Sliding Mode Control (DSMC) based solution for fast convergence and improved transient response. The DSMC control is based on a distributed sliding surface, designed using a combination of local information and information from neighboring MRAs. The control is implemented locally at each MRA and achieves the asymptotic global consensus. Finally, the convergence of the proposed control scheme is proved mathematically, and performance is validated using the MRA system which has been implemented using MATLAB/Simulink. The results of the proposed control technique are compared with conventional DAI control, which shows that the proposed scheme outperforms the conventional scheme in terms of fast convergence, considering renewable resources as distributed generation.


I. INTRODUCTION
The Autonomous Renewable Grid (ARG) is primarily composed of inverter-interfaced Multi-Renewable Agents (MRAs)/Distributed Generation [1], [2]. The increased number of MRAs is challenging the successful operation of the ARG due to their low inertia [3]. Due to the intermittent nature of MRA's as well as their widely distributed geographical nature, small capacity and the large number of them they demand a fast and robust distributed control system for frequency regulation and economically optimized power production. The Distributed Averaging Integrator (DAI) based control schemes proposed in contemporary research work exhibit a slow response especially in providing Economic Dispatch (ED). This paper proposes Distributed Cooperative Sliding Mode Control (DCSMC) to comply with the control demands of an ARG.
Unlike large synchronous generators with constant 50/60 Hz frequency, MRAs either produce power in the form of DC (PV plants, fuel cells, and storage batteries) or variable speed AC (wind and micro-turbines). In both cases a DC to AC inverter is required to connect to the wider AC network [4]. Moreover, the large synchronous generators possess inherent synchronization capabilities, while the MRAs rely on their associated control to achieve synchronization [5], [6].
The conventional approach to address the aforementioned control challenges is to divide it into four main categories namely; inner-control, primary, secondary and tertiary control [7], [8]. The inner-control is the local control of the inverter that provides the current, voltage, output impedance regulation, and synchronization with the grid. Inner-control provides high frequency switching pulses for the inverter to produces the sinusoidal voltage of desired phase/frequency and amplitude. The primary control uses a proportional droop-control technique to provide active and reactive power sharing among the MRAs but forces the frequency and voltage to deviate from their nominal values [9], [10]. This naturally requires an additional layer of control; the secondary control, to minimize the deviations produced by the primary control. The ED is provided by tertiary control that forces the MRAs to follow the optimum power injection profiles for cost effective operation of the system. However, recent research has merged the tertiary control into the secondary control, providing ED and frequency regulation at the same level [11][12][13]. Figure 1 represents the inverter with associated hierarchical control architecture and inter-MRA connection through power lines and communication links. The susceptance of the power line connecting the ℎ and ℎ MRAs, is labelled as ( = ). The secondary and tertiary controls require the power feedback from local and neighboring MRAs to generate the control input for frequency stabilization and ED. The information from neighboring MRAs is obtained through inter-MRA communication links. Based on the control input from secondary control, the primary control generates the reference values of frequency and voltage for inner control. The inner control works on a faster scale to provide synchronization with the grid system and tracks the reference values from the primary control by producing appropriate gate pulses for the inverter.
The secondary (and tertiary) control of MRAs could be centralized [14], [15], decentralized [16], [17] as well as distributed [11][12][13], [18], [19]. The centralized control provides frequency regulation and ED, with the help of communication links with individual MRAs. The communication links are used to obtain the information about frequency and local power generation and transmits the control input to the MRAs. Similarly, in distributed control, MRA units develop a co-operative control with the help of peer-to-peer communication links between the MRAs, while the decentralized control is a local control without any communication link. However, the distributed control which provides the flexibility of a plug-and-play feature as the number of MRAs increases has gained significant attention recently, especially in case of ARG [12], [13]. In centralized control a single communication link failure results in isolation/disconnection of an MRA and may compromise the frequency regulation of the system. However, in distributed control, multiple communication paths can exist in the network and failure of communication link/links would not affect the overall performance as long as the MRAs form a connected graph with active communication links.
A variety of distributed secondary control solutions were proposed in recent research and extensively discussed, among them is the DAI (or Distributed Averaging Proportional Integrator (DAPI)). The DAI based control schemes were proposed for bulk-hybrid power systems [20], [21] as well as for microgrids [10], [12], [22]. The author in [23] introduced the distributed averaging filter technique where the agents achieved a common consensus throughout the network with the help of communication links. The work formed the basis for different averaging-based techniques. The author in [21], proposed the distributed proportional integral for bulk power system, while the author in [22] presented the same technique for medium voltage, MG, with slight modifications to provide proportional power sharing. The optimum resource allocation criterion was developed in [12] and implemented using the DAI control technique for heterogeneous power systems. Detailed analysis of DAI with stability analysis in the presence of communication delays, link failures and dynamic network topologies was presented in [13]. The author in [24] discussed the performance limitation and problematic tuning of DAI.
The DAI control is based on the integration of local stateerrors and weighted-summation of inter-MRA state-errors. The inter-MRA state-errors are formed with the help of communication links. The prime short-coming of DAI control is the slow response, especially, in minimizing the inter-MRA state-errors.
The MRA based ARG experiences frequent changes in real power consumption, perturbations and MRA intermittence causing imbalance of real power, resulting in frequency deviation [25], [26]. Under such conditions, consensus development for ED becomes an even more challenging task, thus demanding a fast and robust cooperative control for low inertia ARG.
This research work focuses on the frequency stabilization and optimum resource allocation aspect of secondary control and proposes a novel DCSMC scheme. It is the foremost attempt to design DSMC based secondary control.  Figure 1. Hierarchical control structure of inverter and inter-MRA power lines throughout the network with the help of communication links between the MRAs.  Our proposed work provides fast and robust convergence in frequency regulation and ED. The control maintains the real power balance to regulate the frequency and utilizes the identical incremental cost criteria for ED.  We provide rigorous mathematical proof of the proposed control scheme using Lyapunov Candidate Function.  In order to critically investigate the performance of DCSMC, a 'seven' node system is designed and evaluated in MATLAB/Simulink that validates our control performance, compared to prior works.
The paper is organized as follows; Section 2 starts with, an introduction to various Notations, Graph Theory and a mathematical model of inverter-based ARG. The section also contains control objectives and a brief introduction of a DAI control technique. Section 3 presents the design of a distributed sliding surface for DCSMC. The stability and convergence proof are discussed in Section 4. Section 5 presents the performance analysis and comparison with DAI control using a MATLAB/Simulink based model. Conclusions and future work are presented in Section 6.

II. Mathematical Model of ARG
This section discusses the primary control dynamics and control objectives for secondary control. The section also presents a brief introduction of DAI control. The section starts with nomenclature, an introduction to the various notations used in the paper and graph theory.

A. Nomenclature
The scalar parameters are related to individual MRAs, while the vector/matrix form represents the whole network.

Scalar Value
Power production of MRA

∈ ℝ ≥0
Power set point Control input (provided by secondary control) Power flow to neighboring MRAs Voltage amplitude ∈ ℝ ≥0 Susceptance between ℎ and ℎ MRAs Power absorption of loadunits Tuning parameters of DAI
∈ ℝ ×1 denotes a column matrix of ones and ∈ ℝ × is an identity matrix. The = ( ) ∈ ℝ × , denotes a diagonal matrix, with as the diagonal elements. For a matrix ∈ ℝ × , > 0 indicates that is a positive definite matrix, while < 0 indicates that is a negative definite matrix.

C. Graph Theory
Let be a static, undirected and connected graph with = {1, … }, representing the set of nodes and ∈ × , representing the set of edges. The adjacency matrix of is represented by a symmetric matrix; ∈ ℝ × with elements = = 1, if the MRAs are connected directly by an edge and

D. Power Network
We Power measurement is an integral part of conventional hierarchical control as shown in Figure 1, while instantaneous phase can be extracted from associated inner-control of inverter-interfaced MRA units. Assumption 3: For each MRA, we have two different types of neighboring MRAs; , and , , and we relax the condition of identical power and communication neighbors, instead we assume that for ℎ , MRA ∩ , = { , ⋂ , } ∉ , ∈ We consider the real power loads and inverter-interfaced MRA, with primary control emulating the behavior of a synchronous generator [8], [10], [11]. The dynamics of ℎ MRA ( ∈ ) is given by, ̇= , ̇+ ( − ) + + = . (1) In the case of inverter interfaced MRA, the virtual inertia = , where ∈ ℝ >0 represents the time constant of low pass filter for power measurement [13]. Without loss of generality, the power set point is set to zero. The power produced by the MRA ( ) is given by, Where, . is the power flow to neighboring MRAs, while , is the power flow to local-load.
While the dynamics of ℎ load-unit is given by, Where, , is the real power consumption of the load-unit. Now, the dynamics of MRAs can be represented compactly in matrix form as, ̇= , While the load-units are represented as, Where, Now, from the dynamics of the ARG in (1) and (4), the system would operate at nominal frequency only when, While, for the complete network the frequency regulation would be achieved when, Equation (8) holds because of Assumption 1 which considers a lossless ARG model where, the sum of power delivered to the neighboring nodes is equal to the sum of power consumed by the load nodes.

E. Control Objectives
The control objectives for the secondary control are, to regulate the frequency to its nominal value ( ) and to provide ED. It is clear from (8) that frequency regulation depends upon the balance of real power. The criterion for optimum power production is based on identical incremental cost [12]. So, the control objectives can be written as, Frequency regulation of the system is represented in (9), while (10) represents the criteria for ED. The system would operate at optimum cost when the incremental cost ( ) is identical for a complete network. The system reaches equilibrium when it satisfies the control objectives in (9) and (10) simultaneously for all MRAs.

F. Distributed Averaging Integral based Secondary Control
DAI based secondary control uses peer-to-peer communication between the nearby neighboring MRAs to create the consensus in the network. The control is based on integration of error in (9) and (10). The control law DAI is given by, Where, and represents the weight on error in (9) and (10), respectively. The control objective in (10) is based on consensus to identical incremental cost, for the complete network. However, control law in (11) uses neighboring communication set ( , ) to form the error.
The control law in (11) is implemented locally at each MRA and creates consensus throughout the network. The control law in (11) provides active frequency regulation but possesses slow convergence to identical cost. In the presence of an increased number of small-scale MRA units with fluctuating outputs and frequent disturbances in the system, the DAI based secondary control would become ineffective.

III. Distributed Cooperative Sliding Mode Control
DCSMC is proposed in the work for fast convergence, effectively complying with the control requirements of ARG. The design of DCSMC possesses two essential parts; distributed sliding surface design and reaching law design. Firstly, an intermediate local sliding surface is designed based on state-errors, then distributed sliding surface is designed using weighted summation of local and neighboring intermediate sliding surfaces. The control law forces the system trajectories to asymptotically converge to the distributed sliding surface.
For ease of formulation, we start with translating the dynamics in (5) into deviation variables. Let, ∆ = − and ∆ = − . So, (5) becomes, Let, = [Δ Δ ] ∈ ℝ 2×1 collectively represent the states in (12) and * = [Δ * Δω * ] ∈ ℝ 2×1 represents the optimum or the equilibrium value. The optimum state * is defined to satisfy the control objective in (9) and (10) simultaneously. From (9), Δω * = 0, the equilibrium value of angular frequency is the same for each MRA (ω * = ). Now, the equilibrium information of Δ * should be obtained using (10). Using the same technique of near-neighbor communication adopted in (11), (10) can be represented as: Here, is used to represent the neighboring communication set , . Now, rewriting (13) using the result in (7) and then representing it in matrix form. Note that, (7) is based on nominal operational frequency of the system.
Remark 1: Equation (15) does not possess a unique solution in terms of ∆ , as is the positive-definite diagonal matrix, while the Laplacian matrix ℒ ∈ ℝ × has a rank equal to − 1 [27].
Since the global equilibrium value Δ * is not available, each MRA finds a local intermediate-optimum state value ̃ * = [Δ̃ * Δω * ] ∈ ℝ 2×1 where Δω * = 0. ̃ * is calculated with the help of local and neighboring information, and locally satisfies the control objectives in (9) and (10). Now for Δ̃ * , we represent (14) using (2), Now, rewriting (16) using Assumption 3 to divide the neighboring set , into ∩ , ; the neighboring MRAs with both the communication and power links d − , ; the neighboring MRAs with power link but no communication.
Where, ́= ∑ sin(∆̃ * − ∆ ) ∈ − , . The value of power ( ) and phase deviation (∆ ) of neighboring MRAs are obtained through communication links, while , and ́ are measured locally. The value of desired phase deviation for ℎ MRA (∆̃ * ) can be calculated as, Now, the control objectives in (9) and (10) transform to; Where ∈ ℝ >0 is constant. The intermediate sliding surface is given by, The sliding surface is the combination of local and neighboring intermediate sliding surfaces [28]. Where, represents the sliding surfaces of neighboring MRAs, obtained through communication links. The sliding surface for complete system can be represented in matrix form as, The reaching law for individual MRAs is designed as, Where and are tuning parameters. In matrix form the above equation be represented as, Comparing (22) and (23), From (21), Using (12), (24) and (25), the control law for individual MRAs is given by,

Remark 4 (Chattering Effect):
The secondary control, in the proposed control strategy, is based on control law (26), containing chattering effect. However, the second order dynamics of primary control (12) smooths-out the fluctuations in the control law. Moreover, a low-pass filter is also present with each inverter to remove the harmonics produced due to high frequency switching in inner control.

Remark 5:
The proposed control is designed to provide fast and robust convergence for low inertia MG. However, following the contemporary secondary control solutions, the dynamics of inner-control, inverter, low pass filter and delays in communication links are not modelled.

IV. Stability Analysis
The stability analysis comprises the stability proof of reaching phase and sliding phase. The stability of the former is proved using Lyapunov function, while stability of sliding phase is intuitively obvious. Now, the following Lemma would be helpful in stability proof. Lemma 1: The sum of Laplacian matrix; ℒ ∈ ℝ × , corresponding to connected communication graph of MRAs and Identity matrix , is positive definite (ℒ + > 0). Proof: The proof of lemma is similar to the ones used for the sum of Laplacian matrix and pinning matrix in [27]. ∎

V. Performance Analysis
The performance of the proposed control scheme is evaluated using a seven node MATLAB/Simulink model. The simulation is performed using DCSMC and DAI based secondary controls and performances are compared critically.

A. Simulation Model
The

B. Effect of Damping Coefficient
The per unit values of different parameters of the network (including the damping factor = {1.6 1.22 1.38 1.42 1.30} ) are obtained from [29]. Since the performance of DAI is largely affected by the value of damping factor ( ) used in dynamic (12), different values of damping factor (multiple of D; 10 , 20 , 40 , 100 , 200 ) are used and the performance of both the control schemes are compared. The results of DAI obtained for different damping values are in accordance with classical control theory. Large values of result in smaller frequency deviation (or over-shoot) and slower convergence to steady state, while the smaller values of exhibit the oscillatory behavior. However, the DCMSC is unaffected by the choice of and possess constant convergence time. Figure 3 represents the total power demand ( ,6 + ,7 + ,4 ) of the network with abrupt changes at '1 second' and '3 seconds'. At a simulation time of '1 second', the load ,7 increases from 2.482 to 4.86 , while the value of ,6 increases from 1 to 2 at simulation time of '3 seconds' causing fluctuations in frequency and power. The simulation starts with steady state values, the DAI control (11) and DCSMC (26) are used to achieve the control objectives in (9) and (10). Both the control schemes are analyzed at different values of damping factor. Figure 4 shows the frequency ( 2 = 2 2 ) of a single MRA ( 2 ) at different values of damping factor ( , 10 , 20 , 40 ), for DAI control. Figure 4 represents large deviation, slow convergence and oscillations in frequency at a damping value of . However, increasing the damping value significantly improves the performance of DAI. Increase in damping value decreases the magnitude of frequency deviation but reduces the convergence speed and the system takes a long time to reach the steady state. Figure  5 represents the response of DAI at significantly large damping values (100 , 200 ), it shows a small deviation in frequency due to a disturbance at '1 second' and negligible deviation at '3 seconds'. However, large damping values result in proportionally slow convergence to steady state.

Identical Incremental Cost
The DAI is particularly slow in achieving the second control objective (10) that is, convergence to identical incremental cost. The convergence to identical incremental cost is analyzed at damping values of 100 and 20 . Figure  10 presents the slow response of DAI at a damping value of (100 ). The control fails to reach the identical incremental cost after both the disturbances. The performance of DAI improves at a damping value of 20 , as represented in Figure 11. However, the control still consumes more than '2 seconds' to reach the identical incremental cost for both the disturbances. The response of DCSMC, for 100 and 20 is represented in Figure 12 and Figure 13, respectively. Figure  12 and Figure 13 are almost identical and possess the same convergence time, which is less than '0.3 seconds'. Hence, the performance of DCSMC is unaffected by the choice of damping value.
The performance of DCSMC and DAI is summarized in Table 1. In Table 1, the DCSMC has comparatively greater   , Where * represents the identical incremental cost value of the network, such that; 1 1 * = 2 2 * = ⋯ = * = * . The IAE clearly illustrates the effectiveness of DCSMC the values are quite low as compared to DAI.

D. Continuously Varying Load
The performance of DCSMC is also tested for continuously varying power demand. Figure 14 presents the fluctuating power demand in the network. Due to the slow response, the DAI fails to converge to identical cost in the presence of fluctuating power demand, as shown in Figure 15. While the DCSMC in Figure 16, exhibits fast response and maintains ED in the system by forcing MRAs to operate at identical incremental cost. Figure 17 illustrates the instantaneous phase deviation (∆ ) of MRAs for DCSMC at a damping value of 20 . The second order dynamics of primary control together with an increase damping value smooths out the chattering effect in control law (26). Hence, the chattering in DCSMC does not affect the power quality in the network.

F. Impact of DCSMC on ARG
The simulation results highlight the fast convergence of DCSMC in the presence of abrupt as well as continuous fluctuations in the power demand. The robustness of DCSMC to the choice of damping factor is also presented. While the DAI fails to maintain the ED in low inertia ARG with continuous fluctuations in power demand, resulting in   Figure 17. Instantaneous phase deviation with DCSMC increased production cost. The comparative analysis highlights the superiority of DCSMC over conventional DAI control.

VI. Conclusion and Future Work
In this work, we considered an ARG with MRA to meet the power demand of the network. Being different in nature, the MRA operates the AC network by emulating the behavior of a synchronous generator with the help of associated control. This implies that the stability of ARG depends upon the efficiency of the control. In AC networks, stability of the system depends upon stability of frequency, which originates from the balance of active power. This assigns a prime importance to the role of secondary control providing frequency stabilization. Another aspect of the secondary control is to provide ED for cost effective optimum operation of the system. The DAI based secondary control displays a slow response to the non-linear dynamics of the system. This work proposes the use of DSMC based, non-linear technique to provide fast convergence, robustness and improved transient response. It stabilizes the frequency quite effectively, while providing the ED for cost reduction. The possible extensions to the presented work include analysis of DCSMC for communication delays, link failures, and dynamic (power and communication) network topologies. The proposed control scheme can also be used for the other aspect of secondary control that is voltage regulation and reactive power distribution.