System and method for distributed control of an electrical network

Disclosed are systems and methods relating to managing and sharing resources within a spatially-distributed electrical power network in a fully distributed fashion. The electrical power network includes source nodes each having a power source and a local controller. The electrical power network includes a physical layer where the source nodes are connected to a power distribution network including one or more loads. The electrical power network also includes a communication layer for communicating power information between source nodes and neighbor source nodes of the electrical power network.

FIELD OF THE INVENTION

The present disclosure relates generally to systems and methods for electric utilities that can work in aggregation or islanded from the legacy power grid, and more specifically to distributed and real-time control and coordination of such networks.

BACKGROUND

A microgrid is a small-scale electrical power network that can operate separately from the traditional/legacy main electrical network (U.S. Power Grid). Traditional electrical power networks typically include centralized architectures and do not readily support the connection of distributed energy assets due to power system and communication network constraints. Traditional systems rely on supervisory control or central control and coordination of generation and distribution of energy between source and consumption nodes of an energy grid.

SUMMARY

Included are systems and methods related to a distributed droop-free controller for electrical power network. One embodiment of a system, among others, includes an electrical power network comprising a plurality of source nodes coupled to a respective transmission line of a distribution network, wherein each source node of the plurality of source nodes comprises a power source and a local controller, and a communication network configured to facilitate an exchange of information between a respective source node and a respective predefined subset of the plurality of source nodes, the respective subset of the plurality of source nodes comprising one or more neighbor source nodes of the respective source node.

Another embodiment of a system, among others, includes an electrical power network comprising a physical layer comprising a plurality of source nodes coupled to a transmission network via a plurality of respective transmission lines, individual ones of the source nodes comprising a power source and a local controller, and the electrical power network lacking a centralized controller, and a communication layer comprising a communication network of the plurality of source nodes grouped into a plurality of source node subsets for communicating power information, individual ones of the plurality of source node subsets comprising a respective source node and one or more predefined neighbor source nodes of the respective source node, the respective source node being connected to the one or more predefined neighbor source nodes via a respective communication link.

Another embodiment of a method, among others, includes a method for managing load sharing in an electrical power network, the method comprising receiving neighbor power information from a neighbor source node of the electrical power network, wherein the source node comprises a power source and a local controller, and the source node is designated to communicate with the neighbor source node via a predefined communication network, determining local power information of the respective source node, and updating at least one of a voltage magnitude, or a phase angle of the power source of the source node, based, at least, in part on the local power information of the source node and the neighbor power information.

DETAILED DESCRIPTION

The present disclosure relates to systems and methods for managing and sharing resources within an electrical power network (e.g., a microgrid, main power network, etc.). According to various embodiments of the present disclosure, a cooperative distributed control paradigm replaces the traditional centralized secondary control and the primary-level droop mechanism of an inverter of traditional electrical power networks. Specifically, the electrical power network of the present disclosure includes local controllers at each node of the electrical power network and does not rely on droop mechanisms of the inverters. According to various embodiments, a sparse communication network is spanned across the electrical power network to facilitate limited data exchange among inverter controllers.

Traditional electrical power networks are designed as a centralized architecture and do not readily support the connection of distributed energy assets due to power system and communication network constraints. This in turn prohibits the interconnection of additional distributed generation (e.g., renewable) and other energy resources effectively due to (1) the lack of a way to control different dissimilar assets cost effectively, (2) the lack of a way to unify systems and network asset nodes in order to manage these resources, (3) the lack of secure protocols for distributed in-field systems, (4) existing industry protocols that are inherently insecure for transport over public or vulnerable networks, (5) the lack of a system for integrating legacy protocols into a secure network, (6) the limited ability to update and deploy customized functionality to nodes over the air, and (7) the lack of system flexibility to support innovation/applications in control, analytics, and monitoring of an electrical feeder network.

Traditional systems rely on supervisory control or central control and coordination in generation and distribution of energy between source and consumption nodes of an energy grid. As the emerging energy systems offer wide integration of distributed generation and consumption of energy, there is a need for distributed control alternatives to liberate the decision making. Some distributed control systems that have been proposed subscribe to hierarchical control architecture. This hierarchical model has its own deficiencies, such as scalability and flexibility. Such architecture for the central controller or any of the communication links poses a single point of failure. Also, such models require a complex communication that make it difficult to add and subtract energy nodes in a plug-n-play fashion.

Microgrids are small-scale energy grids that are either independent energy grids on their own (islanded from a bigger energy grid (e.g., a legacy grid)) or in aggregate with the large legacy power grid. Microgrids provide some key advantages over conventional energy grids for example, improved efficiency, reliability, and expandability. Direct Current (DC) energy resources, for example, photovoltaic arrays, storage elements, and fuel cells, are conventionally connected to an alternating current (AC) microgrid distribution network via voltage-source inverters.

A three-tier hierarchical control structure is conventionally adopted for the microgrid operation. The primary control, conventionally realized through a droop mechanism, operates on a fast timescale and regulates output voltage of the inverters and handles proportional load sharing among the inverters. The primary control shares the total load demand among sources in proportion to their power ratings and is practiced to avoid overstressing and aging of the sources. In traditional microgrids, the secondary control compensates for the voltage and frequency deviations caused by the primary control by updating inverter voltage/frequency set points. Ultimately, a tertiary control carries out the scheduled power exchange within the microgrid, or between the microgrid and the main grid, over a longer timescale.

Droop mechanism, or its equivalents, is a decentralized approach to realize the primary control. Droop mechanism emulates virtual inertia for AC systems and mimics the role of governors in traditional synchronous generators. The droop mechanism suffers from load-dependent frequency/voltage deviation, poor performance in handling nonlinear loads, and poor reactive power sharing in presence of unequal bus voltages. Unequal bus voltages are indispensable in practical systems to perform the scheduled reactive power flow. Droop techniques cause voltage and frequency deviations, thus a supervisory secondary control is inevitable to update the set points of the local primary controls. For example, GPS-coordinated time referencing handles frequency synchronization across the microgrids. Such architecture requires two-way high bandwidth communication links between the central controller and each inverter. This protocol adversely affects the system reliability as failure of any communication link hinders the functionality of the central controller and eventually hinders the functionality of the entire microgrid. The central controller itself is also a reliability risk since it exposes a single point-of-failure. Scalability is another issue for that it adds to the complexity of the communication network and it requires updating the settings of the central controller to add or remove even one node.

Distributed control techniques are, thus, suitable solutions to the control and coordination of spatially dispersed inverter-based (also known as distributed generation) electrical power networks. Existing networked control architectures, comprising a master node (e.g., the primary node) and slave nodes, may discharge duties of a central controller while being resilient to faults or unknown system parameters. Distributed synchronization processes necessitate that each agent (e.g., the inverter) exchange information with the other agents in the microgrid according to some restricted communication protocol. These controllers can use a sparse communication network and have less computational complexity at each controller. Networked control of parallel inverters embeds the functionality of the secondary control in all inverters and, thus, requires a fully connected communication network. The master node (e.g., the primary node) in the networked master-slave methods is still a single point-of-failure.

The majority of such approaches are still based on the droop mechanism and inherit similar shortcomings, such as requiring system information (e.g., number of inverters, inverter parameters, and total load demand), requiring frequency measurement, and also handling active power sharing and frequency regulation (or, only reactive power sharing/voltage control). Recent improvements relate to distribution networks with negligible line impedances that can lack satisfactory performance in practical multi-terminal distribution systems with complex, intricate and inefficient transmission networks. In such traditional systems, a single source is also assigned as a leader that relays the rated frequency and voltage set points to other sources through a communication network. Moreover, such solutions focus on the islanded mode of operation and their extension to grid-connected mode would require some redesign or new modified control methods or both.

What is needed is a fully distributed control method for energy systems which is not affected by one or more communication link failures and has a plug and play functionality to add and remove inverters and loads immaterial of their rating. The systems and methods of the present disclosure relate to such a distributed electrical power network. Specifically, the present disclosure provides a comprehensive distribute cooperative solution that satisfies both the secondary and the primary control objectives for an electrical power network without relying on the droop mechanism. Each source node comprising a source, inverter, and controller is considered as an agent of a multi-agent system (e.g., a microgrid) and exchanges data with one or more neighbor nodes. The source nodes process the information received from the neighbors as well as the information of the local source node to update the local voltage set points and synchronize the normalized power and frequencies.

In various embodiments, cooperation among agents on a communication graph provides two voltage correction terms to be added to the rated voltage for adjustment of the local voltage set points of individual agents (e.g., source nodes). In some embodiments, cooperation among voltage, reactive power, and active power regulators effectively carry out global voltage regulation, frequency synchronization, and proportional load sharing in networks where the transmission/distribution line impedances are not negligible. In some embodiments, the rated values, embedded in a local controller, can be manipulated to achieve any desired load sharing. In some embodiments, the voltage regulator seeks to adjust the average voltage across the electrical power network, rather than the individual inverter bus, at the rated voltage value, to ensure global voltage regulation, thus eliminating the need to run a power flow analysis. In some embodiments, the control method does not employ any droop mechanism and does not require any frequency measurement. In some embodiments, source nodes do not require prior knowledge of system parameters or the number of agents. Thus, it enables seamless scalability, modularity, robustness (independent of loads), and plug-and-play capability for agents or loads. In some embodiments only a sparse communication graph is sufficient for the limited message passing among agents. This is in direct contrast with the centralized control approaches that require high-bandwidth bidirectional communication networks, or existing networked control techniques that require fully-connected communication graphs.

A power distribution network is a network that provides the physical connection between sources and loads within an electrical power network. Such a physical system may also be equipped with a physical communications or cyber-communications network to exploit different control opportunities. Interaction of the sources (e.g., inverter-augmented DC sources) in the cyber domain can offer cooperative decision making, which features scalability and improves reliability. Various embodiments of the present disclosure relate to a cyber-physical system with a communication network that facilitates data exchange among sources for control and monitoring purposes.

The feature or features of one embodiment may be applied to other embodiments, even though not described or illustrated, unless expressly prohibited by this disclosure or the nature of the embodiments. Details associated with the embodiments described above and others are described below.

Turning now toFIG. 1A, shown is a drawing of an example an electrical power network100according to various embodiments of the present disclosure. Specifically,FIG. 1Aillustrates both the physical layer103and communication layer/network106of the electrical power network. The distributed electrical power network100comprises a plurality of source nodes112(e.g.,112a,112b,112N,112i) associated with a respective power source206(FIG. 2) that are connected via the physical layer103and the communication layer106. The physical layer103comprises the plurality of source nodes112connected to a corresponding transmission bus107used to transmit power to various loads703(FIG. 7) within the distribution network109. The communication layer106, as described in greater detail inFIG. 1B, corresponds to the structure of communication between one or more source nodes112. The communication between the one or more source nodes112can be wired and/or wireless.

In various embodiments, each source node112broadcasts and/or transmits information203(seeFIG. 2) (e.g., voltage measurements, power measurements, etc.) to a subset of the plurality of source nodes112as defined via the communication network106. The subset of the plurality of source nodes112are referred to herein as neighbor nodes. The neighbor nodes112for each source node112are predefined. As opposed to the centralized/supervisory control, the electrical power network100forms a sparse electrical power network100such that not all source nodes112need to communicate either directly or indirectly with each other. It should be noted that, while the neighbor nodes112shown inFIGS. 1A and 1Bare adjacent to the respective source nodes112, the neighbor nodes112are not required to be adjacent to one another. In some embodiments, the neighbor nodes112are randomly selected. In other embodiments, the neighbor nodes112are specifically selected according to a specific criteria (e.g., physical vicinity, power generation cost, structural similarity, etc.). As can be appreciated, the neighbor nodes112can be selected in various ways so long as the distributed electrical power network100corresponds to a sparse graph with 1) at least a spanning tree, 2) a balanced Laplacian matrix, and 3) a minimum communication redundancy.

Turning now toFIG. 1B, shown is a drawing of an example of the communication layer106of the distributed electrical power network100according to various embodiments of the present disclosure. In some embodiments, the electrical power network100as a multi-agent cyber-physical system can be expressed with a graphical representation with active agents (sources) modeled as nodes112of the graph and communication links115mapped to edges connecting node112.

In some embodiments, the communication links115are not reciprocal and a directed graph (digraph) is formed. Each node112and edge inherits the dynamic model of the corresponding node112and communication channel115, respectively. Communication links115may exchange information203(SeeFIG. 2) with different gains referred to as the communication weights. In one non-limiting example, if Node j broadcasts and/or transmits data xjto Node i through a link with designated a weight of aij>0, then, the information203received at Node i is aijxj. Generally, aij>0 if Node i receives information203from Node j and aij=0, otherwise. Such a graph is usually represented by an associated adjacency matrix AG=[aij]∈N×Nthat carries the communication weights, where N is the number of dispatchable sources206.

In some embodiments, the communication weights are time varying and can include some channel delay. In other embodiments, the communication weights are time-invariant and a scalar adjacency matrix is assumed. N, denotes the set of all neighbors of Node i. The in-degree and out-degree matrices Din=diag {diin} and Dout=diag{diout} are diagonal matrices with diin=Σj∈Niaijand diout=Σi∈Njaij, respectively. The Laplacian matrix is defined as LDin−AG, whose eigenvalues determine the global dynamics of the entire electrical power network100. The Laplacian matrix is balanced if the in-degree and out-degree matrices are equal; particularly, an undirected (bidirectional) data network satisfies this requirement. A direct path from Node i to Node j is a sequence of edges that connects the two nodes112. A digraph is said to have a spanning tree if it contains a root node, from which, there exists at least a direct path to every other node112. A graph is called to carry the minimum redundancy if it contains enough redundant links that, in the case of any single link failure, the electrical power network100remains connected and presents a balanced Laplacian matrix. Thus, the electrical power network100as described in this embodiment is resilient to failure due to a broken communication link115.

Various embodiments of the present disclosure relate to a control method. The control method requires a communication network106with the adjacency matrix AG=[aij]∈N×Nthat has a spanning tree, may be undirected or directional, yet with a balanced Laplacian matrix, and that carries the minimum redundancy. Communication weights, aij, are design parameters. Each source node112exchanges a vector of information203, Ψi=[ēi,pinorm,qinorm], with its neighbor source nodes112on the communication network106, where ēiis the estimation of the averaged voltage magnitude across the microgrid, processed at Node i. pinormpi/piratedand qinormqi/qiratedare the normalized active and reactive powers supplied by Node i. piand qiare the measured active and reactive powers supplied by Node i112, respectively, and piratedand qiratedare the rated active and reactive powers of the same source206. The control method attempts to share the load among sources in proportion to their rated powers.

Moving on toFIG. 2, shown is an example of a detailed schematic of a source node112of the electrical power network100, according to various embodiments of the present disclosure. The source node112comprises a controller209, a source206, and an inverter212in communication with one or more neighboring nodes112as defined by the communication network106. In various embodiments, objectives of the controller209of a respective node112may comprise global voltage regulation, frequency synchronization, active power sharing, reactive power sharing, and/or other features. Fine adjustment of the voltage magnitude and frequency may satisfy all or at least some objectives. Particularly, active and reactive power flow can be managed by tuning the frequency and voltage magnitude, respectively.FIG. 2shows an example schematic of the control policy for a particular node112. For the purpose of example, the node112ofFIG. 2will be referred to herein as Node i.

The controller209is configured to processes local information and information203received from one or more neighbor nodes112to update the voltage magnitude and frequency (or, equivalently, phase angle) set points via the electrical power network100of the particular source node112. The controller209can comprise a voltage regulator215, a reactive power regulator218, an active power regulator221, and/or any other appropriate device. The voltage regulator215is configured to regulate the average voltage of the electric power network100. A voltage estimator224of the controller209finds the average voltage across the electrical power network100, which is then compared to the rated voltage to produce the first voltage correction term. For example, if the set voltage of the electrical power network100is predefined to be 120 V, the voltage regulator215is configured to regulate the average voltage of the particular node112to be 120 V. As such, each controller209for each node112in the electrical power network100regulates the voltage to be substantially equivalent to the predefined setting.

The reactive power regulator218is configured to manage the reactive power generation between different sources206to ensure that all the nodes112are generating substantially the same amount of reactive power. The reactive power regulator218at each node112compares its normalized reactive power with those of its neighbor nodes112. The difference is fed to a subsequent PI controller227that generates the second voltage correction term. In some embodiments, the controller209adds the voltage correction terms to the electrical power network rated voltage (provided by the tertiary control403(seeFIG. 4)) to generate the local voltage magnitude set point. The regulators of the controller209collectively adjust the average voltage of the electrical power network100at the rated voltage. The regulators allow different set points for different bus voltages and, thus, account for the line impedance effects. Moreover, the reactive power regulator218adjusts the voltage to achieve proportional reactive load sharing.

The active power regulator221regulates the active power and the frequency to ensure that the frequency is the same for all nodes112in the electrical power network100. The active power regulator221compares the local normalized active power of each node112with the local normalized active power of the neighbor nodes112and uses the difference to update the frequency and, accordingly, the phase angle of the inverter212of the node112. Thus, the controller209can accurately handle the global voltage regulation and proportional load sharing. The features of the electrical power network help in plug and play capability of more nodes112to the electrical power network100or plug and play of the entire network as islands to an existing grid or a legacy grid406(seeFIG. 4) and resiliency to failures of some communication links115.

The source206of the source node112can comprise any type of power source such as, for example, a PV panel, a battery, a wind turbine, a fuel cell, a diesel generator, and/or any other AC or DC power source. The inverter212is configured to convert DC to AC according to various embodiments of the present disclosure.

The controller209at Node i receives information203from the neighbor nodes112, Ψjs. This information may comprise a frequency, a voltage, a current, a power, and/or any other appropriate information associated with the particular neighbor node112. The controller209processes the information203of the neighbor nodes112as well as its own local information, Ψi, to update its voltage set point. ei* and wi* are the set points of the (line to neutral) voltage magnitude (root mean square value) and frequency, respectively. Accordingly, a space vector pulse width modulation (SVPWM) module generates the actual voltage set point, vi*
vi*(t)=ei*(t)√{square root over (2)} sin(∫0twi*(τ)dτ)  (1)
and assigns appropriate switching signals to drive the inverter module. It should be noted that the controller is assumed activated at t=0. As seen inFIG. 2, each inverter212is followed by an LCL filter230to attenuate undesired (switching and line-frequency) harmonics. The set point in Equation (1) may be the reference voltage for the output terminal of the filtering module or, equivalently, the microgrid bus that corresponds to Source i206.

The voltage regulator215and the reactive power regulator218adjust the set point of the voltage magnitude by producing two voltage correction terms, δei1and δei2, respectively, as
ei*(t)=erated+δei1(t)+δei2(t)  (2)
where eratedis the rated voltage magnitude of the microgrid. Regardless of the operating mode of the electrical power network100, (i.e., islanded or grid-connected modes), the rated voltage can be safely assumed equal for all active nodes112(e.g., dispatchable sources). The voltage regulator215at Node i includes a voltage estimator224that finds the global averaged voltage magnitude, i.e., the averaged voltage across the electrical power network100. This estimation is compared with the rated voltage, erated, and the difference is fed to a proportional integral (PI) controller, Gi,233to generate the first voltage correction term, δei1, and thus handle global voltage regulation. Accordingly, the voltage regulator215collectively adjusts the average voltage of the electrical power network100on the rated value. Individual bus voltages may slightly deviate from the rated value. This deviation is essential in practice to navigate reactive power across the electrical power network100. Therefore, the reactive power regulator218at Node i adjusts an additional (i.e., the second) voltage correction term, δei2, to control the supplied reactive power.

The reactive power regulator218calculates the neighborhood reactive loading mismatch, mqi,

mqi=∑j∈Ni⁢baij⁡(qjnorm-qinorm)(3)
which measures how far is the normalized reactive power of the Node i from the average of its neighbor sources206. The coupling gain b is a design parameter. The mismatch in Equation (3) is then fed to another PI controller, Hi,227to adjust the second voltage correction term, δei2, and, accordingly, mitigate the mismatch. All the mismatch terms, in the steady state, converge to zero and, thus, all normalized reactive powers synchronize, satisfying the proportional reactive power sharing among sources206.

The active power regulator221controls frequency and active power of the source206. The active power regulator221calculates the neighborhood active loading mismatch to assign the frequency correction term, δwi,

δ⁢⁢ωi=∑j∈Ni⁢caij⁡(pjnorm-pinorm)(4)
where the coupling gain c may be a design parameter or have a value of “1.” As seen inFIG. 2, this correction term is added to the rated frequency, wrated,
ωi*(t)=ωrated+δωi(t)  (5)
and, thus, Equation (1) can be written as
vi*(t)=ei*(t)√{square root over (2)} sin(ωratedt+∫0tδωidτ)  (6)
Equation (6) helps to define the phase angle set point for Node i,

According to Equations (6) and (7), the active power regulator221keeps the frequency at the rated value and fine tunes the phase angle set point, δi*, to reroute the active power across the electrical power network100and mitigate the neighborhood active loading mismatch. All phase angles, δi*, will converge to their steady-state values and, thus, all frequency correction terms, δwi, decay to zero. Therefore, the frequency of the electrical power network100synchronizes to the rated frequency, wrated, without any frequency measurement loop, while the controller209stabilizes the phase angles, δi. Indeed, transient variations in the inverter frequency adjust the phase angle of the inverter212and control the active power flow. The frequency will not deviate from the rated value in the steady state and normalized active powers will synchronize, which provides the proportional active load sharing.

In various embodiments, the controller209of the present disclosure is a general solution that can handle load sharing for variety of distribution networks109, such as, for example, predominantly inductive networks, inductive-resistive networks, primarily resistive networks and/or any other appropriate type of network. The nature of the line impedances defines the role of the active power regulators221and reactive power regulators218(seeFIG. 2) for load sharing. For example, a predominantly inductive network naturally decouples the load sharing process and the reactive power regulator218must handle the reactive load sharing by adjusting voltage magnitude while the active power regulator221handles the active load sharing through adjusting the frequency (or, equivalently, the phase angle). However, for other types of distribution networks, active and reactive power flows are entangled to both voltage and phase angle adjustment. For such cases, the load sharing is a collaborative task where the active power regulator221and reactive power regulator218would work together to generate the desired set points.

Turning now toFIG. 3, shown is a drawing of an example of a node112where the source206is non-dispatchable (e.g., renewable energy sources with stochastic power output) according to various embodiments of the present disclosure. In such embodiments, the controller209may be augmented with the methodology shown inFIG. 3. Supplied power by each stochastic source206is measured via a data measurement module303and reported to an auxiliary control unit306. The auxiliary control unit306can run optimization scenarios (e.g., Maximum Power Point Tracking (MPPT)) to decide the desired operating points. The controller209can compare the desired generation with the actual supplied power and update the rated powers, piratedand qirated, to address any mismatch. The control method ofFIG. 2uses the tuned rated powers to adjust the voltage and frequency set points. With the modification inFIG. 3, the stochastic sources206can be pushed to exploit individual potentials (e.g., to produce maximum power) while the controller209inFIG. 2can proportionally share the remaining load demand among dispatchable sources206.

Moving on toFIG. 4, shown is an electrical power network100including a tertiary control unit403and connected to another electricity grid406according to various embodiments of the present disclosure. In the embodiment of islanded mode of the electrical power network100, the system operational autonomy requires preset (fixed) values for the rated voltage magnitude and frequency, eratedand wrated, in all controllers209. The voltage and frequency settings typically follow the standard ratings of the nearby other electricity grid. To further extend operational range of the controller209to the grid-connected mode, adjustable voltage magnitude and frequency ratings are considered. A tertiary control unit403fine-tunes such ratings when connecting to the other electricity grid406and/or other electrical power network100.

In this embodiment, there is a single tertiary control unit403for the entire electrical power network100. The tertiary control unit403can use the same communication network106as the local controllers209, to propagate updated voltage and frequency ratings to all the controllers209across the electrical power network100. The tertiary control unit403runs cost/efficiency optimization to determine the desired active and reactive powers to be exchanged between the electrical power network100and the main electricity grid406, pd* and qd*, respectively. The optimization scenarios can also account for auxiliary services such as, for example, frequency regulation, reactive power support, and/or other type of auxiliary service. The power flow between the electrical power network100and the other electricity grid406can be bidirectional and, thus, the desired powers pd* and qd* can be either positive or negative. In some embodiments, the power flow can be determined via a power measurement unit404.

The tertiary control unit403compares the actual powers supplied to the other electricity grid406, pd* and qd*, with the desired values and, accordingly, updates voltage and frequency ratings sent to the controllers209of the electrical power network100. In various embodiments, the steady-state rated voltage assignment, erated, may have slight deviation from the standard value, however, the steady-state value of the rated frequency, wrated, will always converge to the standard value (e.g., 60 Hz in the North America). In some embodiments, the transient variations in the rated frequency adjust the phase angles across the electrical power network100and manage the active power flow.

For the voltage regulation in the embodiments ofFIG. 4, each node112may have a voltage estimator224(seeFIG. 2) that develops the estimation of the averaged voltage magnitude across the electrical power network100, (e.g, ēi, for Node i), and exchanges this estimation with its neighbor nodes112. A voltage estimation policy, according to various embodiments of the present disclosure is demonstrated inFIG. 5.

Accordingly, the voltage estimator224at Node i updates its own output, ēi, by processing the neighbor information203including the neighbor node estimates, ēis(j∈Ni) ējs (j∈Ni), and the local voltage measurement, ei,

This updating policy is known as the dynamic consensus protocol. As seen in Equation (8), the local measurement, (e.g., ei) is directly fed into the estimation protocol. Thus, in case of any voltage variation at Node i112i, the local estimate, ēi, immediately responds. The change in ēipropagates through the communication network106and affects all other estimations. If e=[e1, e2, . . . , eN]Tand ē=[ē1, ē2, . . . , ēN]Tare the measured voltage and the estimated average voltage vectors, respectively. E and Ē are the Laplace transforms of e and ē, respectively. Accordingly, global dynamic response of the estimation policy may be formulated as
Ē=s(sIN+L)−1E=HestE(9)
where IN∈N×N, L, and Hestare the identity, Laplacian, and the estimator transfer-function matrices, respectively. If the communication graph106has a spanning tree with a balanced Laplacian matrix, L, then, all elements of s converge to a consensus value, which is the true average voltage, i.e., the average of all elements in e. Equivalently,
ēss=Mess=eSS1(10)
where M∈N×Nis the averaging matrix, whose elements are all 1/N. xssexpresses the steady-state value of the vector x∈N×1.xis a scalar that represents the average of all elements in the vector x.1∈N×1is a column vector whose elements are all one.
System-Level Modeling

System-level modeling relates to the dynamic/static response of the entire electrical power network100with the controller209in effect. As shown inFIGS. 1A and 1B, the electrical power network100encompasses interactive cyber and physical subsystems (e.g., the physical layer103(FIG. 1A) and the communication layer106(FIGS. 1A and 1B)). The communication graph topology defines the interaction among controllers209, the functionality of the controllers209determines output characteristics of the sources206, and the transmission/distribution network109rules the physical interaction among sources206and loads703(seeFIG. 7). Thus, a system-level study involves the mathematical modeling of each of the subsystems and establishment of mathematical coupling between the interactive subsystems.

Dispatchable sources206(FIG. 2), transmission lines107(FIG. 1A), and loads703form the physical layer103of the electrical power network100according to various embodiments of the present disclosure. Referring back toFIG. 1A, the physical layer103is shown where, in some embodiments, source nodes112can be considered as controllable voltage source inverters. The controller209(FIG. 2) of the present disclosure determines the voltage set points (both magnitude, ei*, and phase, δi*) for each source inverter212by processing the supplied active and reactive powers. The controller209acts on the physical layer103, which is a multi-input/multi-output plant with the voltage set points as the inputs and the supplied active and reactive powers as the outputs. The output variables (i.e., the supplied powers) are expressed in terms of the input variables (i.e., the voltage set points).

FIG. 1Ahelps to formulate the supplied current of each source node112. By formulating the supplied current by Node i,

Ii=Yii⁢Vi+∑j=1⁢(≠i)⁢N⁢⁢Yij⁡(Vi-Vj)(11)
where Iiand Viare the phasor representation of the supplied current and phase voltage of the Node i, respectively. Yiiand Yijand are the local load admittance at Bus i (Node i)107and the admittance of the transmission line connecting busses i and j, respectively. With no loss of generality, the distribution network is assumed reduced (e.g., by using Kron reduction) such that all non-generating busses are removed from the network. Thus, the complex power delivered by the Node i is,

Assume Vi=ei∠δiand Yij=yij∠δijwhere ei, yij, δiand θijare the magnitude of Vi, magnitude of Yij, phase of Vi, and phase of Yij, respectively. Yij=gij+jbijis the rectangular representation of the admittance Yij. One can use Equation (12) to derive the active and reactive powers delivered by Node i (piand qi, respectively),

The secondary control typically acts slower than the dynamic of the electrical power network100(e.g., microgrid), as its objectives are voltage and power regulation in the steady state. Accordingly, one can safely neglect the fast dynamic transient responses of the microgrid and use the phasor analysis in Equations (13) and (14) to model the power flow. Equations (13) and (14) express nonlinear relationships between the voltages and supplied powers. In time domain, any variable x can be represented as x=xq+{circumflex over (x)} where xqand {circumflex over (x)} are the quiescent and small-signal perturbation parts, respectively. Thus, one can write,

Referring back toFIG. 2, the communication network106(i.e., cyber domain) is where the controllers209exchange information203(e.g., measurements), process information203and, update the voltage set points. The voltage regulators215and reactive power regulators218cooperate to adjust the voltage magnitude set points, ei*. In the frequency domain,

It is assumed that for t<0 all source nodes112of the electrical power network100operate with identical voltage set points, i.e., for all 1≤i≤N, ei*=eratedand wi*=wratedand thus, vi(t)=eratedsin(wratedt). The controller209is activated at t=0 such that the quiescent value of any variable x, xq, represents its steady-state value for t<0, i.e., before activating the controller209, and the small-signal part, {circumflex over (x)}, captures the variable response to the controller activation for t>0. Therefore, δe1,q=[δe11,q, δe21,q, . . . , δeN1,q]T=0, δe2,q′q=[δe12,q, δe22,q, . . . , δeN2,q]T=0, and eratedq=erated1eratedq=erated1and, accordingly, simplify Equations (30) and (31),

G⁡(E^rated-Hest⁢E^)=Δ⁢⁢E^1(33)-bHLqrated-1⁡(qqs+Q^)=Δ⁢⁢E^2(34)
where q=[q1q, q2q, . . . , qNq]Tcarries the reactive powers supplied by individual sources for t<0. Since the rated voltage does not change before and after activating the controller209, Êrated0. The voltage set points dynamics can now be found by substituting Equations (33)-(34) into Equation (32),

As seen, Equation (35) has two terms. The first term, −GHestÊ, represents the effort of the controller209to achieve the global voltage regulation, and, the second term, −bHLQrated−1(qq/s+{circumflex over (Q)}), explains how the controller209balances reactive load sharing across the electrical power network100.

Active power regulators221(FIG. 2) adjust the active power flow by tuning the phase angles. The controller209at each node112, e.g., Node i, compares the local normalized active power with those of the predefined neighbor nodes112and, accordingly, updates the phase angle set point as in Equation (7). Controller activation at t=0 implies that ωi*(t<0)=ωratedand, thus, δiq=δiss(t<0)=0. Accordingly,

Equivalently, in the frequency domain,

Δ^i*=1s⁢(∑j∈Ni⁢⁢caij⁡(Pjnorm-Pinorm))(37)
where {circumflex over (Δ)}i* is the Laplace transform of δi*. Equation (37) in matrix format,

Δ^*=⁢-cs⁢Lprated-1⁢P=⁢-cs⁢Lprated-1⁡(pqs+P^)(38)
where {circumflex over (Δ)}*=[{circumflex over (Δ)}1*, {circumflex over (Δ)}2*, . . . , {circumflex over (Δ)}N*]Tand prated=diag {pirated} is a diagonal matrix that includes the rated active powers of the sources. pq=[p1q, 2, . . . , pNq]Tcarries the active powers supplied by individual sources206before activation of the controller209. Equation (38) represents the phase angles dynamic response to mitigate and, eventually, eliminate the active load sharing mismatch.

Referring next toFIG. 6, shown is a model diagram of an example of the entire electrical power network100according to various embodiments of the present disclosure.FIG. 6illustrates the electrical power network100separated into a quiescent model603and a small-signal model606. The entire system in the small-signal model606can be treated as a multi-input/multi-output plant, where pq/s and qq/s are the inputs and Ê, {circumflex over (P)}, and {circumflex over (Q)} are the outputs. Equations (35) and (38) show how the controller209adjusts the voltage set points by processing the load sharing mismatches. Accordingly, for the inverter212driving Node i,
{circumflex over (Δ)}i=GiΔ{circumflex over (Δ)}i*  (39)
Êi=GiEÊi*  (40)
where GiEand GiΔare the magnitude and phase transfer functions, respectively. Each inverter accommodates an output filter to eliminate the switching harmonics, whose dynamic is included in the transfer functions GiEand GiΔ. Equivalently, in the matrix format,
{circumflex over (Δ)}=GΔ{circumflex over (Δ)}*  (41)
Ê=GEÊ*(42)
where GE=diag{GiE} and GΔ=diag{GiΔ} are diagonal matrices of the inverter transfer functions. The entire system can be formulated by substituting Equations (35) and (38) in Equations (25) and (26), and also using Equations (41) and (42).

The transmission/distribution network is assumed to be predominantly inductive and, thus, The active and reactive powers are mainly controlled by adjusting the voltage phases and magnitudes since the transmission/distribution network109is predominately inductive. Accordingly, in Equations (25) and (26), kep≅0 and kδq≅0, respectively, which helps to find the reduced-order dynamic model of the entire system. Substituting Equation (41) in Equation (38) and Equation (42) in Equation (35) yields

Substituting the reduced form of Equations (25) and (26) in Equations (43) and (44) yields

Equations (43) through (48) describe dynamic response of the entire electronic power network100with the controller209in effect. Equations (45) and (46) describe that if the power (either active or reactive) was proportionally shared prior to activating the controller209, i.e., prated−1pq=n1or qrated−1qq=m1, the power flow would remain intact after activation of the controller209, i.e., {circumflex over (p)}=0 or {circumflex over (q)}=0.

Controller Design

In various embodiments of the present disclosure, converter transfer function matrices, GΔand GE, rated active and reactive matrices, pratedand qrated, respectively, and p−δ and q−e transfer matrices, kδpand keq, respectively, are predefined for the electronic power network100. In various embodiments, the communication networks106for exchanging information and defining neighbor nodes112comprises a sparse graph with 1) at least a spanning tree, 2) balanced Laplacian matrix, and 3) minimum communication redundancy. Communication weights of the graph, aij, and, thus, the Laplacian matrix, L, directly determine the voltage estimator dynamic, Hest.

In some embodiments, the controller matrices G=diag{Gi} and H=diag{Hi} and the coupling gain b may be adjusted by evaluating Equation (48) to place all poles of TQin the Open Left Hand Plane (OLHP). Smaller gains help to stabilize the entire electrical power network100while larger gains provide a faster dynamic response. In some embodiments, the parameters can be predefined based in part on at least one of stability, settling time, and/or other factors. In some embodiments, the estimator dynamic is faster than the dynamics of the electrical power network100. Therefore, evaluating Equation (48), Hest≅M can be assumed. In some embodiments, the switching frequency of the inverter212can be suitable to provide a prompt response to the voltage command, i.e., GE≅IN.

Referring back toFIG. 2, the voltage regulator215and the reactive power regulator218, adjust the voltage magnitude, ei*, by generating two voltage correction terms, δei1and δei2, respectively. Since the voltage regulator215is tasked to maintain average voltage across the electrical power network100at the rated value, the speed of the voltage regulator215can ensure voltage stability/regulation. In some embodiments, the voltage control loops (e.g., voltage estimator224, PI controller233, voltage measurement filters, etc.) can be designed for a bandwidth higher to the reactive power control loops (e.g., PI controller227, reactive power measurement filters, etc.). In some embodiments, the voltage measurement filters can remove the switching harmonics and filter out much lower frequency terms of the line-frequency harmonics and other contents caused by load nonlinearity or unbalance. In such embodiments, the power measurement process and the overall active/reactive load sharing control loops are slowed down. In some embodiments, the reactive power PI controller, Hi,227can be slower than the voltage PI controller, Gi,233.

Equations (45) and (47) provide dynamic response of the active load sharing mechanism. Given the fast response of the inverter212, GΔ≅INcan be assumed, which simplifies Equation (47). In some embodiments, the coupling gain c can be sweeped and the stability and dynamic response through Equation (47) can be assessed to find an appropriate choice for c.

Steady-State Performance Analysis

A performance analysis of the electrical power network100investigates load sharing and voltage regulation quality in the steady state. Voltage regulation and reactive load sharing is first to study. In the steady state, the voltage estimators215converge to the true average voltage of the electrical power network100. Equivalently, ēss=Mess=ess1. Thus, based on the various embodiments ofFIG. 2

Given the balanced Laplacian matrix,1TL=0, which simplifies Equation (52),

Since all entries of the matrix U are positive, Equation (53) yields erated=ess, which implies that the controllers successfully regulates the averaged voltage magnitude of the microgrid,ess, at the rated value, erated. Moreover, by substituting erated−ess=0 in Equation (51),
Lqrated−1qss=0.  (54)

If L is the Laplacian matrix associated with a graph that contains a spanning tree, the only nonzero solution to Lx=0 is x=n1, where n is any real number. Thus, Equation (54) implies qss=nqrated1, which assures that the controller209shares the total reactive load among the sources in proportion to their ratings.

Frequency regulation and active load sharing is the next to study. The controller209guarantees the convergence of the voltage magnitude vector, e, and phase angle vector, δ to steady-state values. Thus, Equations (6) and (7) suggest that all sources206would synchronize to the rated frequency, ωrated. Moreover, based on Equation (7), stabilizing the phase angles across the electronic power network100implies that all the frequency correction terms in Equation (4) should decay to zero. Equivalently,
cLprated−1pss=0  (55)
which offers, pss=mprated1, where m is a positive real number. Thus, the controller209successfully handles the proportional active load sharing.

Experiments

FIG. 7illustrates a schematic drawing of an example of an electrical power network100according to various embodiments of the present disclosure. The electrical power network100comprises four inverter-driven sources206placed in a radial connection to supply two loads703(e.g.703a,703b), Z1and Z4. In one non-limiting example, assume that the inverters212(FIG. 2) of the sources206have similar topologies but different ratings, i.e., the ratings of the inverters212of sources206aand206bare twice those for the inverters212of sources206cand206d. Each inverter212is augmented with an LCL filter230(FIG. 2) to eliminate switching and line-frequency harmonics. For the given experiment, an RL-circuit model is used for each transmission line107(FIG. 1A).

In various embodiments, the controller209of the present disclosure is a general solution that can handle load sharing for variety of distribution networks109, such as, for example, predominantly inductive networks, inductive-resistive networks, primarily resistive networks and/or any other appropriate type of network. For the given experiment, an inductive-resistive distribution network109is adopted to investigate collaborative interaction of the active and reactive power regulators in load sharing.

Structure of the communication network106is highlighted inFIG. 7. While the communication network ofFIG. 7comprises a ring structure, the communication network can be designed in other structures so long as the communication network106a sparse network that carries the required minimum redundancy where no single communication link failure would hinder the connectivity of the communication network106. Communication links115are bidirectional to feature a balanced Laplacian matrix.

Performance Assessment

FIGS. 8A-8Hillustrate examples of graphical representations of performance evaluations of the controller209of the electrical power network100during experimentation according to various embodiments of the present disclosure. In the experiment, the inverters212are initially driven with fixed voltage command, i.e., e*=120 V and w*=120π rad/s. It should be noted that no voltage feedback control had been initially in action to compensate the voltage drop across the LCL filters230and, thus, the resulting bus voltages ofFIG. 8Amay be less than the desired set point, i.e., e*=120 V.FIGS. 8E and 8Fillustrate that the total load is not shared among sources206in proportion to their rated power.

Assume that the voltage PI controllers, G,233are designed slightly faster than the reactive power PI controllers, Hi,227. In the experiment, the cut-off frequencies of the power measurement filters are as low as 3 Hz to damp all undesired low-frequency harmonics. These design considerations set the dynamic responses of the two voltage and reactive power regulators apart enough to dynamically separate the two resulting voltage correction terms, i.e., δei1and δei2. The controller209is activated at t=8 s. The voltage correction terms δei1and δei2have been added to the voltage set points to help with the global voltage regulation and reactive load sharing.FIG. 8Aillustrates an example of the controllers209boosting the bus voltages across the electronic power network100to satisfy the global voltage regulation; i.e., for t>8 s, the average voltage across the electronic power network100is successfully regulated at the desired 120 V. As seen inFIGS. 8B and 8C, the first and the second voltage correction terms respond at two different time scales; the first correction term δei1(output of the voltage regulator215) responds four times faster than the second correction term δei2(output of the reactive power regulator218).FIG. 8Bshows that the controllers209have varied the frequency set points in transients to adjust individual phase angles and provide the desired active load sharing. In the experiment, the active power regulator221is proven to only enforce transient deviations in frequency and that imposes no steady-state deviation. In the experiment, all inverter frequencies synchronize to the rated frequency of 60 Hz in the steady state.FIGS. 8E and 8Fshot e filtered power measurements and explain how the controllers209have effectively rerouted the power flow to provide proportional load sharing. Individual and total reactive and active load demands are plotted inFIGS. 8G and 8H. The loads have drawn more power once the controller209is activated since the voltages are boosted across the entire electrical power network100.

With respect to the controller performance in response to the load change, assume that the load at Bus d, Z4, has been unplugged at t=20 s and plugged back in at t=35 s. As illustrated inFIGS. 8A-8H, the controller209has successfully maintained global voltage regulation, frequency synchronization, and proportional load sharing, despite the change in load. As shown inFIGS. 8E and 8F, the inverters212of Nodes c and d respond faster to the load change than the other two inverters212as they are in closer vicinity of Z. Soft load change is performed in this study for safety purposes. In fact, the load inductor at Bus d features an air-gap control knob. Using this control opportunity, at t=20 s, the load inductance is manually increased to its maximum value to provide an ultimate current damping feature. Then, the load is physically unplugged. A reverse procedure is followed at t=35 s to plug the load, Z4, back in. This soft load change procedure, besides the damping effect of the power measurement filters, explains why the supplied powers inFIGS. 8E and 8Fand the load demands inFIGS. 8G and 8Hshow slow and gradual profile rather than sudden changes.

Communication Delay and Channel Bandwidth

Communication is indispensable to access neighbor information203and, thus, to the operation of distributed electrical power networks100. Accordingly, channel non-idealities, e.g., transmission/propagation delay and limited bandwidth, and channel deficiencies such as, for example, packet loss may compromise the overall system performance. Thus, low delay and high bandwidth communication protocols are of paramount value for distributed control structures. For example, WiFi and Ultra Wide Band (UWB) protocols offer bandwidths up to 5 GHz and 7.5 GHz, respectively, with delays less than 1 μs. It should be noted that the length of the communication link107directly affects the channel delay. Channel non-ideality effects have a negligible impact on the overall system performance if the channel delay is negligible compared to the controller dynamics. According to various embodiments of the present disclosure, the system dynamics of the electrical power network100exhibit different time constants for the voltage, active, and reactive power regulation. Therefore, the controller209can operate safely with most of the existing communication protocols.

Turning now toFIG. 9, shown is an example of a schematic drawing illustrating the plug- and play capability of the electrical power network100according to various embodiments of the present disclosure. In stage 2, assume that the inverter212(FIG. 2) of Node c112has been unplugged. Assuming this inverter212is turned off instantly, the power measurements can exponentially decay to zero because of the existing low-pass filters.

It should be noted that a source failure also implies loss of all communication links115connected to that particular node112. Accordingly, when Node c212cfails, it automatically renders the links b-c (between Nodes b and c) and c-d inoperable. However, as seen inFIG. 9, the remaining links115still form a connected graph with balanced Laplacian matrix and, thus, the control methodology should remain functional. In an experiment performed, the controllers209successfully respond to the inverter loss and share the excess power among the remaining inverters212in proportion to their power ratings. After the loss of the inverter212of Node c212c, the voltage measurement for bus c may be unavailable. Thus, the controllers209collectively regulate the new average voltage, i.e., the average voltage of the remaining three inverters212, at the rated value of 120 V. However, in some embodiments, bus c can experience voltage sag due to the loss of generation. It should be noted that although the inventor212of Node c is disconnected from bus c the bus voltage is still available. In the experiment, the inverter212of Node c is plugged back in and the controller209of Node c112cis activated. In some embodiments, the synchronization procedure can delay inverter engagement. After successful synchronization, the controller209is activated and performs successfully in the global voltage regulation and readjusting the load sharing to account for the latest plugged-in inverter.

Failure Resiliency in Cyber Domain

The original communication network106is designed to carry a minimum redundancy, such that no single communication link115failure can compromise the connectivity of the communication network106. When a communication link115is disabled, there is no impact on the voltage regulation or load sharing, and the communication network106is still connected with a balanced Laplacian matrix. In some embodiments, the receiving-end controller209updates its set of neighbor nodes112by ruling out the node112on the transmitting end of the failed communication link115. This reconfiguration ensures that the misleading zero-valued data associated to the failed link (e.g., zero active and reactive power measurements) will not be processed by the receiving-end controller209and, thus, the electrical power network100remains functional.