Abstract:
Node controllers in power distribution networks in accordance with embodiments of the invention enable dynamic frequency control. One embodiment includes a node controller comprising a network interface a processor; and a memory containing a frequency control application; and a plurality of node operating parameters describing the operating parameters of a node, where the node is selected from a group consisting of at least one generator node in a power distribution network wherein the processor is configured by the frequency control application to calculate a plurality of updated node operating parameters using a distributed process to determine the updated node operating parameter using the node operating parameters, where the distributed process controls network frequency in the power distribution network; and adjust the node operating parameters.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    The present invention claims priority to U.S. Provisional Patent Application Ser. No. 62/022,861 entitled “Load-Side Frequency Control in Power Systems” to Zhao et al., filed Jul. 10, 2014. The disclosure of U.S. Provisional Patent Application Ser. No. 62/022,861 is herein incorporated by reference in its entirety. 
     
    
     STATEMENT OF FEDERALLY SPONSORED RESEARCH 
       [0002]    This invention was made with government support under DE-AR0000226 awarded by the U.S. Department of Energy and CNS0911041 awarded by the National Science Foundation. The government has certain rights in the invention. 
     
    
     FIELD OF THE INVENTION 
       [0003]    The present invention generally relates to optimal frequency control and more specifically relates to load side processes for solving for optimal frequency control. 
       BACKGROUND 
       [0004]    An incredible amount of infrastructure is relied upon to transport electricity from power stations, where the majority of electricity is currently generated, to individual homes. Power stations can generate electricity in a number of ways including using fossil fuels or using renewable sources of energy such as solar, wind, and hydroelectric sources. Once electricity is generated it travels along transmission lines to substations. Substations typically do not generate electricity, but can change the voltage level of the electricity as well as provide protection to other grid infrastructure during faults and outages. From here, the electricity travels over distribution lines to bring electricity to individual homes. The infrastructure used to transport electricity through the power grid can be viewed as a graph comprised of nodes and lines. The power stations, substations, and any end user can be considered nodes within the graph. Transmission and distribution lines connecting these nodes can be represented by lines. 
         [0005]    Distributed power generation, electricity generation at the point where it is consumed, is on the rise with the increased use of residential solar panels and is fundamentally changing the path electricity takes to many users&#39; homes. The term “smart grid” describes a new approach to power distribution which leverages advanced technology to track and manage the distribution of electricity. A smart grid applies upgrades to existing power grid infrastructure including the addition of more renewable energy sources, advanced smart meters that digitally record power usage in real time, and bidirectional energy flow that enables the generation and storage of energy in additional locations along the electrical grid. 
       SUMMARY OF THE INVENTION 
       [0006]    Node controllers in power distribution networks in accordance with embodiments of the invention enable dynamic frequency control. One embodiment includes a node controller comprising a network interface a processor; and a memory containing a frequency control application; and a plurality of node operating parameters describing the operating parameters of a node, where the node is selected from a group consisting of at least one generator node in a power distribution network wherein the processor is configured by the frequency control application to calculate a plurality of updated node operating parameters using a distributed process to determine the updated node operating parameter using the node operating parameters, where the distributed process controls network frequency in the power distribution network; and adjust the node operating parameters. 
         [0007]    In a further embodiment, the node operating parameters include a node frequency. 
         [0008]    In another embodiment, the node operating parameters include generator node parameters. 
         [0009]    In a still further embodiment, the node operating parameters include a bounded control variable. 
         [0010]    In further additional embodiments, to calculate a plurality of updated node operating parameters using a distributed process processor using the following expression: 
         [0000]    
       
         
           
             
               
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         [0000]    where p c  is a frequency control parameter, ω is a frequency, c is a cost or disutility function,  p   j  and  p   j  are bounds on the frequency control parameter, j is the node, and           is the at least one generator node. 
         [0011]    In still another embodiment, a node controller comprising: a network interface; a processor; and a memory containing: a frequency control application; and a plurality of node operating parameters describing the operating parameters of a node, where the node is selected from a group consisting of at least one load node in a power distribution network; wherein the processor is configured by the frequency control application to: calculate a plurality of updated node operating parameters using a distributed process to determine the updated node operating parameter using the node operating parameters, where the distributed process controls network frequency in the power distribution network; and adjust the node operating parameters. 
         [0012]    In a yet further embodiment, the node operating parameters include a node frequency. 
         [0013]    In a still yet further embodiment, the node operating parameters include lode node parameters. 
         [0014]    In another embodiment, the node operating parameters include a bounded control variable. 
         [0015]    In yet another embodiment includes to calculate a plurality of updated node operating parameters using a distributed process is evaluated by the processor using the following expression: 
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         [0000]    where p is a frequency control parameter, ω is a frequency, c is a cost or disutility function,  p   j  and  p   j  are bounds on the frequency control parameter, j is the node, and           is the at least one load node. 
         [0016]    In a further embodiment again includes adjusting the node operating parameters further comprises constraining the node operating parameters within thermal limits. 
         [0017]    In another embodiment again includes a power distribution network, comprising one or more centralized computing systems; a communications network; a plurality of generator node controllers, where each generator node controller in the plurality of generator node controllers contains: a generator network interface; a generator node processor; and a generator memory containing: a frequency control application; and a plurality of generator node operating parameters describing the operating parameters of a generator node in a power distribution network; where the generator node processor is configured by the frequency control application to: calculate a plurality of updated generator node operating parameters using a distributed process to determine the updated generator node operating parameter using the generator node operating parameters, where the distributed process controls network frequency in the power distribution network; and adjust the generator node operating parameters; and a plurality of load node controllers, where each load node controller in the plurality of generator node controllers contains: a load network interface; a load node processor; and a load memory containing: the frequency control application; and a plurality of load node operating parameters describing the operating parameters of a load node in the power distribution network; where the load node processor is configured by the frequency control application to: calculate a plurality of updated load node operating parameters using the distributed process to determine the updated load node operating parameter using the load node operating parameters, where the distributed process controls network frequency in the power distribution network; and adjust the load node operating parameters. 
         [0018]    In a still further embodiment again, the generator node operating parameters include a node frequency. 
         [0019]    In still yet another embodiment, the load node operating parameters include a node frequency. 
         [0020]    In a still further embodiment again includes the generator node operating parameters include a bounded control variable. 
         [0021]    In still another embodiment again, the load node operating parameters include a bounded control variable. 
         [0022]    In a further additional embodiment, to calculate a plurality of updated generator node operating parameters using a distributed process is evaluated by the processor using the following expression: 
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         [0000]    where p c  is a frequency control parameter, ω is a frequency, c is a cost or disutility function,  p   j  and  p   j  are bounds on the frequency control parameter, j is the node, and           is the at least one generator node. 
         [0023]    In another additional embodiment, to calculate a plurality of updated load node operating parameters using the distributed process is evaluated by the processor using the following expression: 
         [0000]    
       
         
           
             
               
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         [0000]    where p is a frequency control parameter, ω is a frequency, c is a cost or disutility function,  p   j  and  p   j  are bounds on the frequency control parameter, j is the node, and           is the at least one load node. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0024]      FIG. 1  is a diagram conceptually illustrating a power distribution network in accordance with an embodiment of the invention. 
           [0025]      FIG. 2  is a diagram conceptually illustrating nodes utilizing node controllers connected to a communications network in accordance with an embodiment of the invention. 
           [0026]      FIG. 3  is a block diagram illustrating a node controller for optimum frequency control in accordance with an embodiment of the invention. 
           [0027]      FIG. 4  is a flow chart illustrating a process to solve for optimal frequency control in a power distribution network in accordance with an embodiment of the invention. 
           [0028]      FIG. 5  is a flow chart illustrating a process to adjust generator and load parameters in accordance with an embodiment of the invention. 
           [0029]      FIG. 6  is a diagram illustrating an IEEE New England test system for simulations in accordance with an embodiment of the invention. 
           [0030]      FIG. 7  is a diagram illustrating generator frequencies in a simulated optimal frequency control processes in accordance with an embodiment of the invention. 
           [0031]      FIG. 8  is a block diagram illustrating a node controller for optimal load control in accordance with an embodiment of the invention. 
           [0032]      FIG. 9  is a flow chart illustrating a process to solve for optimal load control in a power distribution network in accordance with an embodiment of the invention. 
           [0033]      FIG. 10  is a flow chart illustrating a process to adjust an aggregate controllable load in accordance with an embodiment of the invention. 
           [0034]      FIG. 11  is a diagram illustrating an IEEE 39 bus system for simulations in accordance with an embodiment of the invention. 
           [0035]      FIGS. 12A-12B  are diagrams illustrating a cost or disutility function and an aggregate controllable load respectively in accordance with an embodiment of the invention. 
           [0036]      FIGS. 13A-13C  are diagrams illustrating the evolution of node frequencies in simulated area 1 in accordance with an embodiment of the invention. 
           [0037]      FIGS. 14A-14C  are diagrams illustrating the evolution in node frequencies in simulated area 2 in accordance with an embodiment of the invention. 
           [0038]      FIGS. 15A-15B  are diagrams illustrating LMPs and inter area line flows respectively without thermal limits in optimal load control simulations in accordance with an embodiment of the invention. 
           [0039]      FIGS. 16A-16B  are diagrams illustrating LMPs and inter area line flows respectively with thermal limits in optimal load control simulations in accordance with an embodiment of the invention. 
           [0040]      FIGS. 17A-17E  are diagrams illustrating node frequency evolution for optimal load control simulations in accordance with an embodiment of the invention. 
           [0041]      FIGS. 18A-18E  are diagrams illustrating location marginal prices evolution for optimal load control simulations in accordance with an embodiment of the invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0042]    Turning now to the drawings, systems and methods for dynamic load control in power networks in accordance with embodiments of the invention are illustrated. Power frequency control maintains frequency in a power distribution network as power supply and demand change. Frequency control traditionally occurs on the generator side of the power distribution network, but it can also occur on the load side of the network. A power distribution network can be partitioned into generator side and load side portions based on supply or demand roles. 
         [0043]    An optimization problem can be utilized to increase the performance of the network with respect to particular goals. The optimal frequency control (OFC) problem typically seeks to minimize total generation cost, user disutility, and/or other operational constraints, and will be the focus of SECTION 1 below. In several embodiments, the OFC problem can maintain power balance across an entire power distribution network by controlling frequency on the generation side and/or the load side. 
         [0044]    The OFC problem can be solved when equilibrium points are found. In various embodiments, an approach to solving for these equilibrium points is through the use of a primal-dual algorithm approach. An extension of this approach is to use partial primal-dual algorithms. Primal-dual algorithms can lend nicely to distributed solutions and these distributed solutions can then be utilized to develop control processes for components of the power distribution networks. Specifically, distributed solutions frequently do not require explicit communication between portions of the network and/or knowledge of overall system parameters. In many embodiments, generator side and load side frequency control occur simultaneously to solve for OFC. Generating power in a power distribution network can be a complicated process. In various embodiments, models used for solving the OFC problem can include power generation with the addition of turbines and governors. 
         [0045]    An alternative optimization problem, the optimal load control (OLC) problem will be the focus of SECTION 2 further below. The OLC problem seeks to minimize aggregate disutility. In many embodiments, the OLC problem can balance power across an entire network by controlling aggregate load on the load side. Similar to SECTION 1, a primal-dual algorithm can be used, which lends itself to a distributed solution. In various embodiments it can be useful to add additional constraints to an increasingly complicated model, for example (but not limited to) to constrain the power line flow within thermal limits. 
       Frequency in Power Networks 
       [0046]    A power distribution network in accordance with an embodiment of the invention is shown in  FIG. 1 . Electricity is generated in power generator  102 . Power transmission lines  104  can transmit electricity between the power generator and power substation  106 . Power substation  106  additionally can connect to large storage battery  108 , which temporarily stores electricity, as well as power distribution lines  110 . The power distribution lines  110  can transmit electricity from the power substation to homes  112 . The homes can include solar panels  114 , a house battery  116 , and/or an electric car  118 . In a number of embodiments, processes are utilized that achieve OFC and/or OLC in power distribution networks. 
         [0047]    The power generator  102  can represent a power source including those using fossil fuels, nuclear, solar, wind, and/or hydroelectric power. In various embodiments, multiple power generators  102  may be present. Substation  106  changes the voltage of the electricity for more efficient power distribution. Solar panels  114  are distributed power generation sources, and can generate power to supply the home as well as generate additional power for the power grid. House battery  116  can store excess electricity from the solar panels to power the home when solar energy is unavailable, or store electricity from the power grid to use at a later time. Substations  106 , large storage batteries  108 , homes  112 , solar panels  114 , house batteries  116 , and electric cars  118  can all be considered to be nodes within the power distribution network and the distribution lines  110  can be considered to be lines within the power distribution network. Additional items that draw electricity may also be located within house  112  including (but not limited to) washing machines, dryers, refrigerators, hair dryers, computers, lamps, and/or televisions. In some embodiments these additional items can also be nodes. In combination, nodes and lines form a power distribution network. 
         [0048]    Nodes within the power distribution network may have different roles. Nodes that are creating power (such as power generator  102  or solar panels  112 ) can be generator nodes on the generator side of the network, and nodes which consume power (such as electric cars  118  or additional items connected to the network within a house) can be load nodes on the load side of the network. In many embodiments, OFC and/or OLC can be attained through generator side frequency control, load side frequency control, and/or a combination of generator and load side frequency control. 
         [0049]    In many embodiments, node controllers are located at nodes throughout the network to control the operating parameters of different nodes to achieve OFC and/or OLC. Connected nodes can be nodes within the power distribution network that are connected by distribution and/or transmission lines and can be controlled by a node controller. The type of control utilized can depend on the specifics of the network and may include distributed, centralized, and/or hybrid power control. Although many different systems are described above with reference to  FIG. 1 , any of a variety of power distribution networks including node controllers may be utilized to perform power distribution as appropriate to the requirements of specific applications in accordance with embodiments of the invention. Nodes utilizing node controllers connected to a communication network in accordance with various embodiments of the invention are discussed further below. 
       Node Controller Architectures 
       [0050]    Nodes utilizing node controllers connected to a communication network in accordance with an embodiment of the invention are shown in  FIG. 2 . Nodes  202  can connect to communication network  204  using a wired and/or wireless connection  206 . In some embodiments the power distribution network can act in place of the communication network. The communication network may also be connected to one or more centralized computer systems  208  to monitor calculations made by or to send instructions to multiple node controllers to, for example, control power distribution in the network at a global level. Additionally, in many embodiments a database management system  210  can be connected to the network to track node data which, for example, may be used to historically track power usage and/or frequencies at various locations over time. Although various system configurations are described above with reference to  FIG. 2 , any number of systems can be utilized to achieve control of nodes within a power distribution network as appropriate to the requirements of specific applications in accordance with embodiments of the invention. Node controllers for OFC in accordance with various embodiments of the invention are discussed further below. 
       Section 1 
     Node Controllers for Optimal Frequency Control 
       [0051]    A node controller in accordance with an embodiment of the invention is shown in  FIG. 3 . In various embodiments, node controller  300  can perform calculations in a distributed manner at a node in a power distribution network. The node controller includes at least one processor  302 , an I/O interface  304 , and memory  306 . The at least one processor  302 , when configured by software stored in memory, can perform calculations on and make changes to data passing through the I/O interface as well as to data stored in memory. In many embodiments, the memory  306  includes software including frequency control application  308  as well as node frequency  310 , node operating parameters  312 , generator node parameters  314 , and load node parameters  316 . A node can measure node frequency through the I/O interface and/or receive this value from a centralized computer. The frequency control application  308  will be discussed in greater detail below and can enable the node to perform calculations to solve for optimal frequency control in a distributed manner. These distributed calculations can be performed such that only node operating parameters  312  and node frequency  310  need to be known by the node, and no communication with other nodes or knowledge of system parameters are required. 
         [0052]    Node operating parameters will be discussed in detail below but may include (but are not limited to) a cost function, a utility function, and/or bounds of control variables. Distributed calculations will generate generator node parameters  314  in the case of a generator node and load node parameters  316  in the case of a load node. In various embodiments, the same node controller can be used for both generator and load nodes. It should be readily apparent node controller  300  can be adapted to generator node or load node specific applications, or a hybrid controller that can switch between specific node types as requirements of specific applications require. Although a number of different node controller implementations are discussed above with reference to  FIG. 3 , any of a variety of computing systems can be utilized to control a node within a power distribution system as appropriate to the requirements of specific applications in accordance with various embodiments of the invention. Graph representations of a power distribution network in accordance with several embodiments of the invention are discussed further below. 
       Power Network Graph Representations 
       [0053]    In various embodiments, the following graph representation is utilized to represent at least a portion of a power distribution network. Let           denote the set of real numbers. For a set          , let           denote its cardinality. A variable without a subscript usually denotes a vector with appropriate components, e.g., ω=(ω j , j∈         )∈         . For a, b∈         , a≦b, the expression [•] a   b  denotes max {min({•, b}, a}. For a matrix A, A T  can denote its transpose. For a square matrix A, the expression A         0 indicates it is positive definite and A         0 indicates the matrix is negative definite. For a signal ω(t) of time t, let ω denote its time derivative dω/dt. 
         [0054]    The classical structure preserving model can be combined with generator speed governor and turbine models. The power network is modeled as a graph (         , ε) where          ={1, . . . , |                   } is the set of nodes (buses) and ε  ⊂           ×          is the set of lines connecting those nodes (buses). The line connecting nodes (buses) i, j∈          can be denoted by (i, j), and in several embodiments it can be assumed that (N, ε) is directed, with an arbitrary orientation, so that if (i, j)∈ε then (j, i)∉ε. “i:i→j” and “k:j→k” can respectively be used to denote the set of nodes (buses) i that are predecessors of node (bus) j and the set of nodes (buses) k that are successors of node (bus) j. In many embodiments, it can be assumed without loss of generality that (N, ε) is connected, and the following assumptions can be utilized: lines (i, j)∈ε are lossless and characterized by their reactances x ij , voltage magnitudes |V j | of nodes (buses) j∈          are constants, and reactive power injections on nodes (buses and reactive power flows on lines are not considered. 
         [0055]    A subset          ⊂          of the nodes (buses) are fictitious nodes (buses) representing the internal of generators. Hence, the set           can be called the set of generators and the set          =         \          can be the set of load nodes (buses). All the nodes (buses) in           can be called load nodes (buses) without distinguishing between generator nodes (buses) (nodes connected directly to generators) and actual load nodes (buses), since they can be treated in the same way mathematically. The nodes (buses) can be labeled so that          ={1, . . . , |         |} and          ={|         |+1, . . . , |         |}. 
         [0056]    The voltage phase angle of node (bus) j∈         , with respect to the rotating framework of nominal frequency ω 0 =120π rad/s, can be denoted by θ j . The frequency deviation of node (bus) j from the nominal frequency ω 0  can be denoted by ω j . Hence 
         [0000]      {dot over (θ)} j =ω j    j∈             .   (1)
 
         [0000]    The system dynamics are described by the swing equations 
         [0000]        M   j {dot over (ω)} j   =−D   j ω j   +p   j   −F   j (θ) j∈             (2)
 
         [0000]    where M j &gt;0 for j∈          are moments of inertia of generators and M j =0 for j∈         , and D j &gt;0 for all j∈          are (for j∈         ) the damping coefficients of generators or (for j∈         ) the coefficients of linear frequency dependent loads, e.g., induction motors. The variable p j  denotes the real power injection to node (bus) j, which is the mechanic power injection to generator if j∈         , and is the negative of real power load if j∈         . The net real power flow out of node (bus) j is 
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         [0000]    are the maximum real power flows on lines (j, k)∈ε. 
         [0057]    In various embodiments, a system of governor and turbine can be associated with a generator j∈         , and their dynamics can be described by 
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         [0000]    where a j  is the valve position of the turbine, p j   c  is the control command to the generator, and p j , as introduced above, is the mechanic power injection to the generator. The time constants τ g,j  and τ b,j  characterize respectively the time-delay in governor action and the approximated fluid dynamics in the turbine. Traditionally, there is a frequency feedback term 
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         [0000]    added to the right-handside of (4), known as the frequency droop control. Here this term is merged into p j   c  to allow for a general form of frequency feedback control. 
         [0058]    Equations (1)-(5) specify a dynamical system with state variables (θ, ω, a         , p         ) where 
         [0000]      θ:={θ 1 , . . . ,         }, ω:={ω 1 , . . . ,         }
 
         [0000]        a           :={a 1   , . . . ,             }, p             :={p   1 , . . . ,         } 
         [0000]    and input variables (         ,          ) where 
         [0000]                :={ p   1   c   , . . . ,p   |             |   c },          :={         , . . . ,         }. 
         [0000]    (         ,          ) are feedback control to be designed based on local measurements of frequency deviations, i.e., (         (ω),          (ω)). Parameters  p ≦  p   j  specify the bounds of the control variables, i.e.,  p   j ≦p j   c (ω)≦  p   j  for j∈         , and  p   j ≦p j (ω)≦  p   j  for j∈         . Note that if  p   j =  p   j  then they specify a constant, uncontrollable input on node (bus) j. This can create a closed-loop dynamical system, and equilibrium points for this system will be described below. 
       Equilibrium Points 
       [0059]    Definition 1. An equilibrium point of the system (1)-(5) with control (         (ω),          (ω)), (referred to as a closed-loop equilibrium for short), is (θ*, ω,          ,          ,          ,          ), where θ* is a vector function of time and (ω*,          ,          ,          ,          ) are vectors of real numbers, such that 
         [0000]                =         (ω*),          =         (ω*)  (6)
 
         [0000]        dθ*   j   /dt=ω*   j    j∈             (7)
 
         [0000]      ω* i =ω* j   =ω* i,j∈             (8)
 
         [0000]      − D   j ω* j   +p*   j   −F   j (θ*)=0  j∈             (9)
 
         [0000]        p*   j   =a*   j   =p   j   c,   * j∈             .   (10)
 
         [0060]    Notation can be abused by using ω* to denote both the vector (ω* 1 , . . . ,          ) and the common value of its components. Its meaning should be clear from the context. 
         [0061]    In the definition above, (8) ensures constant F(θ*) at equilibrium points by (3), and (9)(10) are obtained by letting right-hand-sides of (2)(4)(5) be zero. From (8), at any equilibrium point, all the nodes (buses) are synchronized to the same frequency. The system typically has multiple equilibrium points as will be explained in detail below. An equilibrium point can also be written as (θ*, ω*,          ,          , p*) where p*:=(         ,          ), when state variables           and control variables           do not need to be distinguished. 
       Decentralized Primary Frequency Control 
       [0062]    An initial point of the dynamical system (1)-(5) corresponds to the state of the system at the time of fault-clearance after a contingency, or the time at which an unscheduled change in power injection occurs during normal operation. In either case, the system trajectory, driven by primary frequency control (         (ω),          (ω)), converges to a desired equilibrium point. The criteria for desired equilibrium points can be formalized by formulating an optimization problem called optimal frequency control (OFC), and using OFC to guide the design of control processes for various components within a power distribution network. 
       Optimal Frequency Control Problem 
       [0063]    In several embodiments, the objective in the power distribution network can be to rebalance power after a disturbance at a minimum generation cost and user disutility. This can be formalized by requiring any closed-loop equilibrium (p*, d*) to be a solution of the following OFC problem, where d j =D j ω* j  for j∈         . 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    The term c j (p j ) in objective function (11) is generation cost (if j∈         ) or user disutility for participating in load control (if j∈         ). For simplicity c j  can be called a cost function for j∈          even if it may be a user disutility function. The term 
         [0000]    
       
         
           
             
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         [0000]    implicitly penalizes frequency deviation on node (bus) j at equilibrium. The constraint (12) can ensure power balance over an entire network, and (13) can be bounds on power injections. These bounds can be determined by control capacities of generators or controllable loads, as well as uncontrollable power injections as an exogenous input. 
         [0064]    In many embodiments, the following assumptions can be utilized: 
         [0000]    Condition 1. OFC is feasible. The cost functions c j  are strictly convex and twice continuously differentiable on ( p   j ,  p   j ).
 
Remark 1. A load −p j  on node (bus) j∈          results in a user utility u j (−p j ), and hence the disutility function can be defined as c j  (p j )=−u j (−p j ). The user disutility functions or generation cost functions usually satisfy Condition 1, and in many cases are quadratic functions.
 
Condition 2. For any optimal solution (p*, d*) of OFC, the power flow equations
 
         [0000]        F   j (θ)= p*   j   −d*   j    j∈             (14)
 
         [0000]    are feasible, i.e., have at least one solution θ*∈         .
 
Remark 2. Condition 2 ensures the existence of a closed-loop equilibrium of the dynamical system (1)-(5) with the feedback control described below.
 
       Design of Decentralized Feedback Control 
       [0065]    An overview of a process for solving for optimal frequency control in a power distribution network is illustrated in  FIG. 4 . In process  400 , the frequency of each node is measured  402 . Optimal frequency control is achieved  404  for the power network by controlling generator node parameters and load node parameters using the measured node frequency. 
         [0066]    A process for a node to adjust generator node parameters and/or load node parameters is illustrated in  FIG. 5 . Referring above to  FIG. 3  and  FIG. 4 , the process can be similar to parts of process  400  and can be performed by a node controller similar to node controller  300 . Process  500  includes measuring  502  frequency of the node. If the node is a generator node  504 , generator node parameters are calculated  506  using the node frequency and node operating parameters. Generator node parameters are adjusted  508  and the process completes. If the node is not a generator node  504 , and is a load node  510 , load node parameters are calculated  512  using the node frequency and node operating parameters. Load node parameters are adjusted  514  and the process completes. If the load is neither a generator node  504  nor a load node  510 , the process completes. This process will be described in detail below. 
         [0067]    As noted above, OFC can be used to guide controller design. In various embodiments, (         (ω),          (ω)) can be designed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                   16 
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         [0000]    The function (c′ j ) −1 (•), which is the inverse function of the derivative of the cost function, is well defined if Condition 1 holds. Note that this control is completely decentralized in that for every generator and load indexed by j, the control decision is a function of frequency deviation ω j  measured at its local node (bus). Only its own cost function c j  and bounds [ p   j ,  p   j ] need to be known. No explicit communication with other generators and loads is required, nor is any knowledge of system parameters. Moreover, the following theorem shows that this design fulfills OFC objectives.
 
Theorem 1. Suppose Conditions 1 and 2 hold, it can be proven that:
       1. There is a unique optimal solution of OFC and its dual.   2. For the system (1)-(5) with control (48)(48), there exists at least one equilibrium point. Let (θ*, ω*, a*         , p           c, *, p*) be any of its equilibrium point(s). Then (p*, d*; ω*) is the unique optimal solution of OFC and its dual, where d* j =D j ω* for j∈                 
 
       Optimal Frequency Control Simulations 
       [0070]    The performance of the proposed control can be illustrated through a simulation of the IEEE New England test system shown in  FIG. 6 .  FIG. 6  illustrates an IEEE New England Test system. This system has 10 generators and 39 nodes (buses), and a total load of about 60 per unit (pu) where 1 pu represents 100 MVA. Compared to the model (2)-(4), the simulation model is more detailed and realistic, with transient generator dynamics, excitation and flux decay dynamics, changes in voltage and reactive power over time, and lossy transmission lines, et cetera. The test system in  FIG. 6  is similar to the test system described below with respect to in  FIG. 11 . 
         [0071]    The primary frequency control of generator or load j is designed with cost function 
         [0000]    
       
         
           
             
               
                 
                   c 
                   j 
                 
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                     j 
                   
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         [0000]    where p j   set  is the power injection at the setpoint, an initial equilibrium point solved from static power flow problem. By choosing this cost function, the deviations of power injections from the setpoint can be attempted to be minimized, and have the control 
         [0000]    
       
         
           
             
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               j 
             
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         [0000]    from (48)(48). Only the load control p j  for j∈          is written since the generator control p j   c  for j∈          takes the same form. The following two simulated cases can be considered in which the generators and loads have different control capabilities and hence different [ p   j ,  p   j ]:
       1. All the 10 generators have [ p   j ,  p   j ]=[p j   set (1−c), p j   set (1+c)], and all the loads are uncontrollable;   2. Generators 2, 4, 6, 8, 10 (which happen to provide half of the total generation) have the same bounds as in case (1). Generators 1, 3, 5, 7, 9 are uncontrollable, and all the loads have [ p   j , {dot over (p)} j ]=[p j   set (1+c/2), p j   set (1−c/2)], if p j   set ≦0 for loads j∈         .
 
Hence simulated cases (1) and (2) have the same total control capacity across the network. Case (1) only has generator control while in case (2) the set of generators and the set of loads each has half of the total control capacity. c=10% can be selected, which implies the total control capacity is about 6 pu. For all j∈         , the feedback gain 1/R j  is selected as 25 p j   set , which is a typical value in practice meaning a frequency change of 0.04 pu (2.4 Hz) causes the change of power injection from zero all the way to the setpoint. Note that this control is the same as frequency droop control, which implies that indeed frequency droop control implicitly solves an OFC problem with quadratic cost functions used here. However, the simulated controller design can be more flexible by allowing a larger set of cost functions.
       
 
         [0074]    In several embodiments, the simulated system is initially at the setpoint with 60 Hz frequency. At time t=0.5 second, nodes (buses) 4, 15, 16 each makes 1 pu step change in their real power consumptions, causing the frequency to drop.  FIG. 7  illustrates the frequencies of all the 10 generators under case (1) only generators are controlled (red) and case (2) both generators and loads are controlled (black). The total control capacities are the same in these two cases. It can be seen in both cases that frequencies of different generators have relatively small differences during transient, and are synchronized towards the new steady-state frequency. Compared with simulated generator-only control, the combined simulated generator-and-load control improves both the transient and steady-state frequency, even though the total control capacities in both cases are the same. 
         [0075]    In contrast to the optimal frequency problem discussed above in SECTION 1, a similar optimization problem, the optimal load problem is discussed below in SECTION 2. 
       Section 2 
     Node Controllers for Optimal Load Control 
       [0076]    A node controller in accordance with an embodiment of the invention is shown in  FIG. 8 . In many embodiments, node controller  800  can perform calculations at a node in a power distribution network in a manner similar to the node controller described above with respect to  FIG. 3 . The node controller includes at least one processor  802 , an I/O interface  804 , and memory  806 . The at least one processor  802 , when configured by software stored in memory, can perform calculations on and make changes to data passing through the I/O interface as well as to data stored in memory. In many embodiments, the memory  806  includes software including load control application  808  as well as aggregate controllable load  810 , power network dynamics parameters  812 , and dynamic load control parameters  814 . The load control application  808  will be discussed in greater detail below and can enable the node to perform calculations to solve for optimal load control subject to line flow limits. These calculations can be performed such that power network dynamics parameters  812  and dynamic load control parameters  814  (both to be discussed in detail below) can be used together to calculate the aggregate controllable load. Although a variety of node controllers are described above with reference to  FIG. 8 , any of a variety of computing systems can be utilized to control a node within a power distribution system as appropriate to the requirements of specific applications in accordance with various embodiments of the invention. 
       Power Network Graph Representations for OLC 
       [0077]    In many embodiments, the following graph representation is utilized to represent at least a portion of a power distribution network. While largely similar to the graph representation used above in SECTION 1, it should be noted that SECTION 2 uses different notations.          can be the set of real numbers and           can be the set of natural numbers. Given a finite set S⊂         , |S| to denotes its cardinality. For a set of scalar numbers a i ∈         , i∈S, a S  can be the column vector of the a i  components, i.e. a S :=(a i , ∈S)∈           |S| ; the subscript S is typically dropped when the set is known from the context. Similarly, for two vectors a∈           |S|  and b∈           |S′|  the column vector x can be defined as x=(a,b)∈           |S|+|S′| . Given any matrix A, its transpose can be denoted as A T  and the ith row of A can be denoted as A i . A S  can be used to denote the sub matrix of A composed only of the rows A i  with i∈S. The diagonal matrix of a sequence {a i , i∈S}, is represented by diag(a i ) i∈S . Similarly, for a sequence of matrices {A h , h∈S}, blockdiag(A h ) h∈S  can denote the block diagonal matrix. Finally, 1 (0) can be used to denote the vector/matrix of all ones (zeros), where its dimension can be understood from the context. 
         [0078]    A power distribution network described by a directed graph G(         , ε) where          ={1, . . . , |         |} is the set of nodes (buses) and ε⊂         ×          is the set of transmission lines denoted by either e or ij such that if ij∈ε, then ji∉ε. 
         [0079]    The nodes (buses) can be partitioned          =         ∪          and           and           to indicate the set of generator and load nodes (buses) respectively. In many embodiments it can be assumed that the graph (         , ε) is connected, and additionally the following assumptions can be made which are well-justified for transmission networks: Node (bus) voltage magnitudes |V i |=1 pu for j∈         . Lines ij∈ε are lossless and characterized by their susceptances B ij =B ji &gt;0. The analysis can be extended to networks with constant R/X ratio. Reactive power flows do not affect node (bus) voltage phase angles and frequencies. 
         [0080]    In various embodiments, it can be further assumed that the node (bus) frequency ω i  and line flows P ij  are close to schedule values ω 0  and P ij   0 . In other words, P ij =P ij   0 +δP ij  and ω i =ω 0 +δω i  with δP ij  and δω i  small enough; without loss of generality, take ω 0 =0. The evolution of the transmission network is then described by 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    where d i  denotes an aggregate controllable load, {circumflex over (d)} i =D i ω i  denotes an aggregate uncontrollable but frequency-sensitive load as well as damping loss at generator i, M i  is the generator&#39;s inertia, P i   m  is the mechanical power injected by a generator i∈         , and −P i   m  is the aggregate power consumed by constant loads for i∈         . Finally, C i,e  are the elements of the incidence matrix C∈          of the graph G(         , ε) such that C i,e =−1 if e=ji∈ε, C i,e =1 if e=ij∈ε and C i,e =0 otherwise. 
         [0081]    For notational convenience the vector for can be used whenever needed of (17), i.e. 
         [0000]    
       
         
           
             
               
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         [0000]    where the matrices           and           by splitting the rows of C between generator and load nodes (buses), i.e. C=[         ] T , D=diag(         , D B =diag (B ij ) ij∈ε  and M         =diag         . 
       Operational Constraints for OLC 
       [0082]    Each control area can be indicated using k and          ={1, . . . , |         |} can indicate the set of areas. Within each area, the Automatic Generation Control (AGC) scheme seeks to restore the frequency to its nominal value as well as preserving a constant power transfer outside the area, i.e. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
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                       k 
                       ∈ 
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                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where            k ⊂          is the set of nodes (buses) of area k∈         , e k ∈         , k∈         , is a vector with elements (e k ) i =1 if i∈          and (e k ) i =0 otherwise, {circumflex over (P)} k  is the net scheduled power injection of area k. 
         [0083]    In many embodiments, it can be shown that 
         [0000]        Ĉ:=             C   (19)
 
         [0000]    with          :=[e 1  . . .           and Ĉ∈         , then constraint (18) can be compactly expressed using 
         [0000]        ĈP={circumflex over (P)}   (20)
 
         [0000]    where {circumflex over (P)}=         ∈         . It is easy to see that Ĉ k,e (e=ij) is equal to 1 if ij is an inter-area line with i∈         , −1 if ij is an inter-area line with j∈         , and 0 otherwise. 
         [0084]    Finally, the thermal limit constraints are usually given by 
         [0000]          P ≦P≦  P     (21)
 
         [0000]    where  P :=(  P   e ) e∈ε  and  P :=( P   e ) e∈ε  represent the line flow limits, usually  P =−  P  so that |P|≦  P . 
       Efficient Load Control 
       [0085]    Suppose the system (17) is in equilibrium, i.e. {dot over (ω)} i =0 for all i and {dot over (P)} ij =0 for all ij, and at time 0, there is a disturbance, represented by a step change in the vector P m :=(P i   m , i∈         ), that produces a power imbalance. Then, in various embodiments, a distributed control mechanism can rebalance the system while preserving the frequency within its nominal value as well as maintaining the operational constraints described above. Furthermore, in some embodiments this mechanism can produce an efficient allocation among all the users (or loads) that are willing to adapt. 
         [0086]    c i (d i ) and 
         [0000]    
       
         
           
             
               
                 d 
                 ^ 
               
               2 
             
             
               2 
                
               
                   
               
                
               
                 D 
                 i 
               
             
           
         
       
     
         [0000]    can denote the cost or disutility of changing the load consumption by d i  and {circumflex over (d)} i  respectively, which describes efficiency in terms of the loads&#39; welfare. More precisely, a load control (d, {circumflex over (d)}) is efficient if it solves the following problem. 
         [0000]    
       
         
           
             
               
                 
                   Problem 
                    
                   
                       
                   
                    
                   1 
                    
                   
                       
                   
                    
                   
                     ( 
                     WELFARE 
                     ) 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       minimize 
                       
                         d 
                         , 
                         
                           d 
                           ^ 
                         
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                            
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           c 
                           i 
                         
                          
                         
                           ( 
                           
                             d 
                             i 
                           
                           ) 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           d 
                           ^ 
                         
                         i 
                         2 
                       
                       
                         2 
                          
                         
                             
                         
                          
                         
                           D 
                           i 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    subject to operational constraints. 
         [0087]    This can balance supply and demand, i.e. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         ∈ 
                          
                       
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         
                           d 
                           i 
                         
                         + 
                         
                           
                             d 
                             ^ 
                           
                           i 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         ∈ 
                          
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         P 
                         i 
                         m 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    It is shown that when 
         [0000]        d   i   =c   i ′ −1 (ω i ),  (24)
 
         [0000]    then (17) is a distributed primal-dual algorithm that solves (22) subject to (23). 
         [0088]    Therefore, problem (22)-(23) can be used to forward engineering the desired node controllers, by means of primal-dual decomposition, that can rebalance supply and demand. Like primary frequency control, the system (17) and (24) suffers from the disadvantage that the optimal solution of (22)-(23) may not recover the frequency to the nominal value and satisfy the additional operational constraints described above. 
         [0089]    A clever modification of (22)-(23) can restore the nominal frequency while maintaining the interpretation of (17) as a component of the primal-dual algorithm that solves the modified optimization problem. An additional byproduct of the formulation is that any type of linear equality and inequality constraint that the operator may require can be imposed. 
       Optimal Load-Side Control 
       [0090]    The crux of a solution in various embodiments comes from including additional constraints to Problem 1 that implicitly guarantee the desired operational constraints, yet still preserves the desired structure which allows the use of (17) as part of the optimization algorithm. 
         [0091]    Thus in several embodiments, the following modified version of Problem 1 can be used: 
         [0000]    
       
         
           
             
               
                 
                   Problem 
                    
                   
                       
                   
                    
                   2 
                    
                   
                       
                   
                    
                   
                     ( 
                     OLC 
                     ) 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       minimize 
                       
                         d 
                         , 
                         
                           d 
                           ^ 
                         
                         , 
                         P 
                         , 
                         v 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                            
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           c 
                           i 
                         
                          
                         
                           ( 
                           
                             d 
                             i 
                           
                           ) 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           d 
                           ^ 
                         
                         i 
                         2 
                       
                       
                         2 
                          
                         
                             
                         
                          
                         
                           D 
                           i 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     25 
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    subject to 
         [0000]        P   m −( d+{circumflex over (d)} )= CP   (25b)
 
         [0000]        P   m   −d=L   B ν  (25c)
 
         [0000]        ĈD   B   C   T   ν={circumflex over (P)}   (25d)
 
         [0000]          P ≦D   B   C   T   ν≦  P     (25e)
 
         [0000]    where L B :=CD B C T  is the B ij -weighted Laplacian matrix. 
         [0092]    Although not clear at first sight, the constraint (25c) implicitly enforces that any optimal solution of OLC (d*, {circumflex over (d)}*, P*, ν*) will restore the frequency to its nominal value, i.e. {circumflex over (d)}* i =D i ω*=0. Similarly, constraint (25d) can be used to impose (18) (or equivalently (20)) and (25e) to impose (21). 
         [0093]    In various embodiments, the following assumptions can be utilized: 
         [0000]    Assumption 1 (Cost function). The cost function c i (d i ) is α-strongly convex and second order continuously differentiable (c i ∈C 2  with c″ i (d i )≧α&gt;0) in the interior of its domain            i :=[ d   i ,  d   i ] ⊂           ,
 
such that c i (d i )→+∞ whenever d i →∂         .
 
Assumption 2 (Slater Condition). The OLC prnblemn (25) is feasible and there is at least one feasible (d, {circumflex over (d)}, P, ν) such that d∈Int         :=         .
 
       Properties of Optimal Solutions of OLC 
       [0094]    The optimal solutions of OLC can have various properties. ν i , λ i  and π k  can be used as Lagrange multipliers of constraints (25b), (25c) and (25d), and ρ ij   +  and ρ ij   −  can be used as multipliers of the right and left constraints of (25e), respectively. In order to make the presentation more compact sometimes x=(P, ν)∈          and σ=(ν, λ, π, ρ + , ρ − )∈         , as well as σ i =(ν i , λ i ), σ k =(π k ) and σ ij =(ρ ij   + , ρ ij   − ) can be used. ρ:=(ρ + , ρ − ) will also be used. 
         [0095]    Next, the dual function D(σ) of the OLC problem can be considered. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       D 
                        
                       
                         ( 
                         σ 
                         ) 
                       
                     
                     = 
                     
                       
                         inf 
                         
                           d 
                           , 
                           
                             d 
                             ^ 
                           
                           , 
                           x 
                         
                       
                        
                       
                         L 
                          
                         
                           ( 
                           
                             d 
                             , 
                             
                               d 
                               ^ 
                             
                             , 
                             x 
                             , 
                             σ 
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           L 
                            
                           
                             ( 
                             
                               d 
                               , 
                               
                                 d 
                                 ^ 
                               
                               , 
                               x 
                               , 
                               σ 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 ∈ 
                                  
                               
                             
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     c 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       d 
                                       i 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     
                                       d 
                                       ^ 
                                     
                                     i 
                                     2 
                                   
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     
                                       D 
                                       i 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               v 
                               T 
                             
                              
                             
                               ( 
                               
                                 
                                   P 
                                   m 
                                 
                                 - 
                                 
                                   ( 
                                   
                                     d 
                                     + 
                                     
                                       d 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                                 - 
                                 CP 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               λ 
                               T 
                             
                              
                             
                               ( 
                               
                                 
                                   P 
                                   m 
                                 
                                 - 
                                 d 
                                 - 
                                 
                                   
                                     L 
                                     B 
                                   
                                    
                                   v 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               π 
                               T 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     C 
                                     ^ 
                                   
                                    
                                   
                                     D 
                                     B 
                                   
                                    
                                   
                                     C 
                                     T 
                                   
                                    
                                   v 
                                 
                                 - 
                                 
                                   P 
                                   ^ 
                                 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             
                               ρ 
                               
                                 + 
                                 T 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     D 
                                     B 
                                   
                                    
                                   
                                     C 
                                     T 
                                   
                                    
                                   v 
                                 
                                 - 
                                 
                                   P 
                                   _ 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ρ 
                               
                                 - 
                                 T 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   P 
                                   _ 
                                 
                                 - 
                                 
                                   
                                     D 
                                     B 
                                   
                                    
                                   
                                     C 
                                     T 
                                   
                                    
                                   v 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               i 
                               ∈ 
                                
                             
                           
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               
                                 
                                   c 
                                   i 
                                 
                                  
                                 
                                   ( 
                                   
                                     d 
                                     i 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       λ 
                                       i 
                                     
                                     + 
                                     
                                       v 
                                       i 
                                     
                                   
                                   ) 
                                 
                                  
                                 
                                   d 
                                   i 
                                 
                               
                               + 
                               
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   i 
                                   2 
                                 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     D 
                                     i 
                                   
                                 
                               
                               - 
                               
                                 
                                   v 
                                   i 
                                 
                                  
                                 
                                   
                                     d 
                                     ^ 
                                   
                                   i 
                                 
                               
                               + 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                             
                            
                           
                             
                               ( 
                               
                                 
                                   v 
                                   i 
                                 
                                 + 
                                 
                                   λ 
                                   i 
                                 
                               
                               ) 
                             
                              
                             
                               P 
                               i 
                               m 
                             
                           
                           ) 
                         
                         - 
                         
                           
                             P 
                             T 
                           
                            
                           
                             C 
                             T 
                           
                            
                           v 
                         
                         - 
                         
                           
                             v 
                             T 
                           
                           ( 
                           
                             
                               
                                 L 
                                 B 
                               
                                
                               λ 
                             
                             - 
                             
                               
                                 CD 
                                 B 
                               
                                
                               
                                 
                                   C 
                                   ^ 
                                 
                                 T 
                               
                                
                               π 
                             
                             - 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                             
                            
                           
                             
                               CD 
                               B 
                             
                              
                             
                               ( 
                               
                                 
                                   ρ 
                                   + 
                                 
                                 - 
                                 
                                   ρ 
                                   - 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                         - 
                         
                           
                             π 
                             T 
                           
                            
                           
                             P 
                             ^ 
                           
                         
                         - 
                         
                           
                             ρ 
                             
                               + 
                               T 
                             
                           
                            
                           
                             P 
                             _ 
                           
                         
                         + 
                         
                           
                             ρ 
                             
                               - 
                               T 
                             
                           
                            
                           
                             P 
                             _ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0096]    Since c i (d i ) and 
         [0000]    
       
         
           
             
               
                 d 
                 ^ 
               
               i 
               2 
             
             
               2 
                
               
                   
               
                
               
                 D 
                 i 
               
             
           
         
       
     
         [0000]    are radially unbounded, the minimization over d and {circumflex over (d)} in (26) is always finite for given x and σ. However, whenever C T ν≠0 or L B λ−CD B Ĉ T  π−CD B (ρ + −ρ − )≠0, P or ν can be modified to arbitrarily decrease (27). Thus, the infimum is attained if and only if 
         [0000]        C   T ν=0 and  (28a)
 
         [0000]        L   B   λ−CD   B   Ĉ   T   π−CD   B (ρ + −ρ − )=0.  (28b)
 
         [0097]    Moreover, the minimum value must satisfy 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         c 
                         i 
                         ′ 
                       
                        
                       
                         ( 
                         
                           d 
                           i 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           v 
                           i 
                         
                         + 
                         
                           
                             λ 
                             i 
                           
                            
                           
                               
                           
                            
                           and 
                            
                           
                               
                           
                            
                           
                             
                               
                                 d 
                                 ^ 
                               
                               i 
                             
                             
                               D 
                               i 
                             
                           
                         
                       
                       = 
                       
                         v 
                         i 
                       
                     
                   
                   , 
                   
                       
                   
                    
                   
                     ∀ 
                     
                       i 
                       ∈ 
                       
                          
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0098]    Using (28) and (29) dual function can be computed 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                      
                     
                       ( 
                       σ 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               Φ 
                                
                               
                                 ( 
                                 σ 
                                 ) 
                               
                             
                           
                           
                             
                               σ 
                               ∈ 
                               
                                 N 
                                 ~ 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               ∞ 
                             
                           
                           
                             otherwise 
                           
                         
                       
                       , 
                       
                         
 
                       
                        
                       
                         
                           where 
                            
                           
                             
 
                           
                            
                           
                             N 
                             ~ 
                           
                         
                         := 
                         
                           { 
                           
                             σ 
                             ∈ 
                             
                               
                                 ℝ 
                                 
                                   
                                     2 
                                      
                                     
                                        
                                        
                                        
                                     
                                   
                                   + 
                                   
                                      
                                      
                                      
                                   
                                   + 
                                   
                                     2 
                                      
                                     
                                        
                                       ℰ 
                                        
                                     
                                   
                                 
                               
                                
                               
                                 : 
                               
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   28 
                                    
                                   a 
                                 
                                 ) 
                               
                                
                               
                                   
                               
                                
                               and 
                                
                               
                                   
                               
                                
                               
                                 ( 
                                 
                                   28 
                                    
                                   b 
                                 
                                 ) 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the function Φ(σ) is decoupled in σ i =(ν i , λ i ), σ k =(π k ) and σ ij =(ρ ij   + , ρ ij   − ). That is, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Φ 
                        
                       
                         ( 
                         σ 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                              
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             Φ 
                             i 
                           
                            
                           
                             ( 
                             
                               σ 
                               i 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             k 
                             ∈ 
                              
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             Φ 
                             k 
                           
                            
                           
                             ( 
                             
                               σ 
                               k 
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             ij 
                             ∈ 
                             ℰ 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             Φ 
                             ij 
                           
                            
                           
                             ( 
                             
                               σ 
                               ij 
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         where 
                          
                         
                             
                         
                          
                         
                           
                             Φ 
                             k 
                           
                            
                           
                             ( 
                             
                               σ 
                               k 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           - 
                           
                             π 
                             k 
                           
                         
                          
                         
                           
                             P 
                             ^ 
                           
                           k 
                         
                       
                     
                     , 
                     
                       
                         
                           Φ 
                           ij 
                         
                          
                         
                           ( 
                           
                             σ 
                             ij 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             ρ 
                             ij 
                             - 
                           
                            
                           
                             
                               P 
                               _ 
                             
                             ij 
                           
                         
                         - 
                         
                           
                             ρ 
                             ij 
                             + 
                           
                            
                           
                             
                               P 
                               _ 
                             
                             ij 
                           
                            
                           
                               
                           
                            
                           and 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         Φ 
                         i 
                       
                        
                       
                         ( 
                         
                           σ 
                           i 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           c 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               d 
                               i 
                             
                              
                             
                               ( 
                               
                                 σ 
                                 i 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               v 
                               i 
                             
                             + 
                             
                               λ 
                               i 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               P 
                               i 
                               m 
                             
                             - 
                             
                               
                                 d 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   σ 
                                   i 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           
                             D 
                             i 
                           
                           2 
                         
                          
                         
                           v 
                           i 
                           2 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   with 
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       d 
                       i 
                     
                      
                     
                       ( 
                       
                         σ 
                         i 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         c 
                         i 
                         
                           ′ 
                           - 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             v 
                             i 
                           
                           + 
                           
                             λ 
                             i 
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0099]    The dual problem of the OLC (DOLC) is then given by 
       DOLC: 
       [0100]    
       
         
           
             
               
                 
                   
                     
                       
                         maximize 
                         
                           v 
                           , 
                           λ 
                           , 
                           π 
                           , 
                           ρ 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                              
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             Φ 
                             i 
                           
                            
                           
                             ( 
                             
                               
                                 v 
                                 i 
                               
                               , 
                               
                                 λ 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                            
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           Φ 
                           k 
                         
                          
                         
                           ( 
                           
                             π 
                             k 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           ij 
                           ∈ 
                           ℰ 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           Φ 
                           ij 
                         
                          
                         
                           ( 
                           
                             ρ 
                             ij 
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       
                         
                           subject 
                            
                           
                               
                           
                            
                           to 
                         
                       
                       
                         
                           
                             ( 
                             
                               28 
                                
                               a 
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           and 
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               28 
                                
                               
                                   
                               
                                
                               b 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                           
                             
                               ρ 
                               ij 
                               + 
                             
                             ≥ 
                             0 
                           
                           , 
                           
                               
                           
                            
                           
                             
                               ρ 
                               ij 
                               - 
                             
                             ≥ 
                             0 
                           
                           , 
                           
                               
                           
                            
                           
                             ij 
                             ∈ 
                             ℰ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0101]    Clearly, DOLC is feasible (e.g. take σ=0). Then, Assumption 2 implies dual optimal is attained. 
         [0102]    Although D(σ) is only finite on Ñ, Φ i (σ i ), Φ k (σ k ) and Φ ij (σ ij ) are finite everywhere. Thus sometimes the extended version of the dual function can be used (31) instead of D(σ), knowing that D(σ)=Φ(σ) for σ∈Ñ. Given any S⊂         , K⊂          or U⊂ε, define 
         [0000]    
       
         
           
             
               
                 
                   Φ 
                   S 
                 
                  
                 
                   ( 
                   
                     σ 
                     S 
                   
                   ) 
                 
               
               := 
               
                 
                   ∑ 
                   
                     i 
                     ∈ 
                     S 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     Φ 
                     i 
                   
                    
                   
                     ( 
                     
                       σ 
                       i 
                     
                     ) 
                   
                 
               
             
             , 
             
                 
             
              
             
               
                 
                   Φ 
                   K 
                 
                  
                 
                   ( 
                   
                     σ 
                     K 
                   
                   ) 
                 
               
               := 
               
                 
                   ∑ 
                   
                     k 
                     ∈ 
                     K 
                   
                 
                  
                 
                     
                 
                  
                 
                   
                     Φ 
                     k 
                   
                    
                   
                     ( 
                     
                       σ 
                       k 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               and 
                
               
                   
               
                
               
                 
                   Φ 
                   U 
                 
                  
                 
                   ( 
                   
                     σ 
                     U 
                   
                   ) 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   ij 
                   ∈ 
                   U 
                 
               
                
               
                   
               
                
               
                 
                   Φ 
                   ij 
                 
                  
                 
                   ( 
                   
                     σ 
                     ij 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    such that Φ(σ)=         +         +Φ ε (σ ε ). 
         [0103]    The optimization problem has the following property: Given a connected graph G(         , ε), then there exists a scalar ν* such that (d*, {circumflex over (d)}*, x*, σ*) is a primal-dual optimal solution of OLC and DOLC if and only if (d*, {circumflex over (d)}*, x*) is primal feasible (satisfies (25b)-(25e)), σ* is dual feasible (satisfies (28) and (34)), 
         [0000]        {circumflex over (d)}*   i   =D   i ν* i   ,d*   i   =c   i ′ −1 (ν* i +λ* i ),ν* i   =ν*, i∈             ,   (35)
 
         [0000]      and 
         [0000]      ρ ij   + *( B   ij (ν i *−ν j *)−   P     ij )=0,  ij∈ε,   (36a)
 
         [0000]      ρ ij   − *(   P     ij   −B   ij (ν* i −ν* j ))=0,  ij∈ε   (36b)
 
         [0000]    Moreover, d*, {circumflex over (d)}*, ν* and λ* are unique with ν*=0. 
       Distributed Optimal Load Control 
       [0104]    In several embodiments, power network dynamics can be leveraged to solve the OLC problem in a distributed way. In many embodiments, the solution is based on the classical primal dual optimization algorithm that has been of great use to design congestion control mechanisms in communication networks. 
         [0105]    Let 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           L 
                            
                           
                             ( 
                             
                               x 
                               , 
                               σ 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             minimize 
                             
                               d 
                               , 
                               
                                 d 
                                 ^ 
                               
                             
                           
                            
                           
                               
                           
                            
                           
                             L 
                              
                             
                               ( 
                               
                                 d 
                                 , 
                                 
                                   d 
                                   ^ 
                                 
                                 , 
                                 x 
                                 , 
                                 σ 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           L 
                            
                           
                             ( 
                             
                               
                                 d 
                                  
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   d 
                                   ^ 
                                 
                                  
                                 
                                   ( 
                                   σ 
                                   ) 
                                 
                               
                               , 
                               x 
                               , 
                               σ 
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             Φ 
                              
                             
                               ( 
                               σ 
                               ) 
                             
                           
                           - 
                           
                             
                               P 
                               T 
                             
                              
                             
                               C 
                               T 
                             
                              
                             v 
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             v 
                             T 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   L 
                                   B 
                                 
                                  
                                 λ 
                               
                               - 
                               
                                 
                                   CD 
                                   B 
                                 
                                  
                                 
                                   
                                     C 
                                     ^ 
                                   
                                   T 
                                 
                                  
                                 π 
                               
                               - 
                               
                                 
                                   CD 
                                   B 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       ρ 
                                       + 
                                     
                                     - 
                                     
                                       ρ 
                                       - 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where L(d, {circumflex over (d)}, x, σ) is defined as in (27), d(σ):=(d i (σ i )) and {circumflex over (d)}(σ):=({circumflex over (d)} i (σ i )) according to (35). 
         [0106]    The following partial primal-dual law can be proposed 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       v 
                       . 
                     
                      
                   
                   = 
                   
                     
                       ζ 
                        
                       v 
                     
                      
                     
                       ( 
                       
                         
                           P 
                            
                           m 
                         
                         - 
                         
                           ( 
                           
                             
                               
                                 d 
                                  
                               
                                
                               
                                 ( 
                                 
                                   σ 
                                    
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 D 
                                  
                               
                                
                               
                                 v 
                                  
                               
                             
                           
                           ) 
                         
                         - 
                         
                           
                             C 
                              
                           
                            
                           P 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     38 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   0 
                   = 
                   
                     
                       P 
                       ℒ 
                       m 
                     
                     - 
                     
                       ( 
                       
                         
                           
                             d 
                             ℒ 
                           
                            
                           
                             ( 
                             
                               σ 
                               ℒ 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             D 
                             ℒ 
                           
                            
                           
                             v 
                             ℒ 
                           
                         
                       
                       ) 
                     
                     - 
                     
                       
                         C 
                         ℒ 
                       
                        
                       P 
                     
                   
                 
               
               
                 
                   ( 
                   
                     38 
                      
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     λ 
                     . 
                   
                   = 
                   
                     
                       ζ 
                       λ 
                     
                      
                     
                       ( 
                       
                         
                           P 
                           m 
                         
                         - 
                         
                           d 
                            
                           
                             ( 
                             σ 
                             ) 
                           
                         
                         - 
                         
                           
                             L 
                             B 
                           
                            
                           v 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     38 
                      
                     c 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     π 
                     . 
                   
                   = 
                   
                     
                       ζ 
                       π 
                     
                      
                     
                       ( 
                       
                         
                           
                             C 
                             ^ 
                           
                            
                           
                             D 
                             B 
                           
                            
                           
                             C 
                             T 
                           
                            
                           v 
                         
                         - 
                         
                           P 
                           ^ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     38 
                      
                     d 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ρ 
                       . 
                     
                     + 
                   
                   = 
                   
                     
                       
                         ζ 
                         
                           ρ 
                           + 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               D 
                               B 
                             
                              
                             
                               C 
                               T 
                             
                              
                             v 
                           
                           - 
                           
                             P 
                             _ 
                           
                         
                         ] 
                       
                     
                     
                       ρ 
                       + 
                     
                     + 
                   
                 
               
               
                 
                   ( 
                   
                     38 
                      
                     e 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ρ 
                       . 
                     
                     - 
                   
                   = 
                   
                     
                       
                         ζ 
                         
                           ρ 
                           - 
                         
                       
                        
                       
                         [ 
                         
                           
                             P 
                             _ 
                           
                           - 
                           
                             
                               D 
                               B 
                             
                              
                             
                               C 
                               T 
                             
                              
                             v 
                           
                         
                         ] 
                       
                     
                     
                       ρ 
                       - 
                     
                     + 
                   
                 
               
               
                 
                   ( 
                   
                     38 
                      
                     f 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     P 
                     . 
                   
                   = 
                   
                     
                       χ 
                       P 
                     
                      
                     
                       ( 
                       
                         
                           C 
                           T 
                         
                          
                         v 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     38 
                      
                     g 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     v 
                     . 
                   
                   = 
                   
                     
                       χ 
                       v 
                     
                      
                     
                       ( 
                       
                         
                           
                             L 
                             B 
                           
                            
                           λ 
                         
                         - 
                         
                           
                             CD 
                             B 
                           
                            
                           
                             
                               C 
                               ^ 
                             
                             T 
                           
                            
                           π 
                         
                         - 
                         
                           
                             CD 
                             B 
                           
                            
                           
                             ( 
                             
                               
                                 ρ 
                                 + 
                               
                               - 
                               
                                 ρ 
                                 - 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     38 
                      
                     h 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 ζ 
                  
                 v 
               
               = 
               
                 
                   diag 
                    
                   
                     ( 
                     
                       ζ 
                       i 
                       v 
                     
                     ) 
                   
                 
                 
                   i 
                   ∈ 
                    
                 
               
             
             , 
             
               
 
             
              
             
               
                 ζ 
                 λ 
               
               = 
               
                 
                   diag 
                    
                   
                     ( 
                     
                       ζ 
                       i 
                       λ 
                     
                     ) 
                   
                 
                 
                   i 
                   ∈ 
                    
                 
               
             
             , 
             
               
 
             
              
             
               
                 ζ 
                 π 
               
               = 
               
                 
                   diag 
                    
                   
                     ( 
                     
                       ζ 
                       k 
                       π 
                     
                     ) 
                   
                 
                 
                   k 
                   ∈ 
                    
                 
               
             
             , 
             
               
 
             
              
             
               
                 ζ 
                 
                   ρ 
                   + 
                 
               
               = 
               
                 
                   diag 
                    
                   
                     ( 
                     
                       ζ 
                       e 
                       
                         ρ 
                         + 
                       
                     
                     ) 
                   
                 
                 
                   e 
                   ∈ 
                   ℰ 
                 
               
             
             , 
             
               
 
             
              
             
               
                 ζ 
                 
                   ρ 
                   - 
                 
               
               = 
               
                 
                   diag 
                    
                   
                     ( 
                     
                       ζ 
                       e 
                       
                         ρ 
                         - 
                       
                     
                     ) 
                   
                 
                 
                   e 
                   ∈ 
                   ℰ 
                 
               
             
             , 
             
               
 
             
              
             
               
                 χ 
                 P 
               
               = 
               
                 
                   diag 
                    
                   
                     ( 
                     
                       χ 
                       e 
                       P 
                     
                     ) 
                   
                 
                 
                   e 
                   ∈ 
                   ℰ 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               χ 
               v 
             
             = 
             
               
                 
                   diag 
                    
                   
                     ( 
                     
                       χ 
                       i 
                       v 
                     
                     ) 
                   
                 
                 
                   i 
                   ∈ 
                    
                 
               
               . 
             
           
         
       
     
         [0107]    The operator [•] u   +  is an element-wise projection that maintains each element of the u(t) within the positive orthant when {dot over (u)}=[•] u   + , i.e. given any vector a with same dimension as u then [α] u   +  is defined element-wise by 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         a 
                         e 
                       
                       ] 
                     
                     
                       u 
                       e 
                     
                     + 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               a 
                               e 
                             
                           
                           
                             
                               
                                 
                                   if 
                                    
                                   
                                       
                                   
                                    
                                   
                                     a 
                                     e 
                                   
                                 
                                 &gt; 
                                 
                                   0 
                                    
                                   
                                       
                                   
                                    
                                   or 
                                    
                                   
                                       
                                   
                                    
                                   
                                     u 
                                     e 
                                   
                                 
                                 &gt; 
                                 0 
                               
                               , 
                             
                           
                         
                         
                           
                             0 
                           
                           
                             otherwise 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0108]    One property that can be used later is that given any constant vector u*≧0, then 
         [0000]      ( u−u *) T   [a]   u   + ≦( u−u *) T   a   (40)
 
         [0000]    since for any pair (u e , a e ) that makes the projection active, so u e =0 and a e &lt;0 must be by definition and therefore 
         [0000]      ( u   e   −u*   e ) a   e   =−u*   e   a   e ≧0=( u   e   −u*   e ) T   [a   e ] u     e     + .
 
         [0000]    The name of the dynamic law (38) comes from the fact that 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         v 
                       
                     
                      
                     
                       
                         L 
                          
                         
                           ( 
                           
                             x 
                             , 
                             σ 
                           
                           ) 
                         
                       
                       T 
                     
                   
                   = 
                   
                     
                       P 
                       m 
                     
                     - 
                     
                       ( 
                       
                         
                           d 
                            
                           
                             ( 
                             
                               σ 
                               i 
                             
                             ) 
                           
                         
                         + 
                         Dv 
                       
                       ) 
                     
                     - 
                     CP 
                   
                 
               
               
                 
                   ( 
                   
                     41 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         λ 
                       
                     
                      
                     
                       
                         L 
                          
                         
                           ( 
                           
                             x 
                             , 
                             σ 
                           
                           ) 
                         
                       
                       T 
                     
                   
                   = 
                   
                     
                       P 
                       m 
                     
                     - 
                     
                       d 
                        
                       
                         ( 
                         σ 
                         ) 
                       
                     
                     - 
                     
                       
                         L 
                         B 
                       
                        
                       v 
                     
                   
                 
               
               
                 
                   ( 
                   
                     41 
                      
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         π 
                       
                     
                      
                     
                       
                         L 
                          
                         
                           ( 
                           
                             x 
                             , 
                             σ 
                           
                           ) 
                         
                       
                       T 
                     
                   
                   = 
                   
                     
                       
                         C 
                         ^ 
                       
                        
                       
                         D 
                         B 
                       
                        
                       
                         C 
                         T 
                       
                        
                       v 
                     
                     - 
                     
                       P 
                       ^ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     41 
                      
                     c 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           ρ 
                           + 
                         
                       
                     
                      
                     
                       
                         L 
                          
                         
                           ( 
                           
                             x 
                             , 
                             σ 
                           
                           ) 
                         
                       
                       T 
                     
                   
                   = 
                   
                     
                       
                         D 
                         B 
                       
                        
                       
                         C 
                         T 
                       
                        
                       v 
                     
                     - 
                     
                       P 
                       _ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     41 
                      
                     d 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           ρ 
                           - 
                         
                       
                     
                      
                     
                       
                         L 
                          
                         
                           ( 
                           
                             x 
                             , 
                             σ 
                           
                           ) 
                         
                       
                       T 
                     
                   
                   = 
                   
                     
                       P 
                       _ 
                     
                     - 
                     
                       
                         D 
                         B 
                       
                        
                       
                         C 
                         T 
                       
                        
                       v 
                     
                   
                 
               
               
                 
                   ( 
                   
                     41 
                      
                     e 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         P 
                       
                     
                      
                     
                       
                         L 
                          
                         
                           ( 
                           
                             x 
                             , 
                             σ 
                           
                           ) 
                         
                       
                       T 
                     
                   
                   = 
                   
                     - 
                     
                       ( 
                       
                         
                           C 
                           T 
                         
                          
                         v 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     41 
                      
                     f 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         v 
                       
                     
                      
                     
                       
                         L 
                          
                         
                           ( 
                           
                             x 
                             , 
                             σ 
                           
                           ) 
                         
                       
                       T 
                     
                   
                   = 
                   
                     - 
                     
                       ( 
                       
                         
                           
                             L 
                             B 
                           
                            
                           λ 
                         
                         - 
                         
                           
                             CD 
                             B 
                           
                            
                           
                             
                               C 
                               ^ 
                             
                             T 
                           
                            
                           π 
                         
                         - 
                         
                           
                             CD 
                             B 
                           
                            
                           
                             ( 
                             
                               
                                 ρ 
                                 + 
                               
                               - 
                               
                                 ρ 
                                 - 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     41 
                      
                     g 
                   
                   ) 
                 
               
             
           
         
       
     
         [0109]    Equations (38a), (38b) and (38g) show that dynamics (17) can be interpreted as a subset of the primal-dual dynamics described in (38) for the special case when ζ i   ν =M i   −1  and χ ij   p =B ij . Therefore, the frequency ω i  can be interpreted as the Lagrange multiplier ν i . 
       Optimal Load Control Processes 
       [0110]    An overview of a process for calculating optimal load control is illustrated in  FIG. 9 . 
         [0111]    Process  900  includes calculating  902  power network dynamics parameters. Power network dynamics parameters will be discussed in greater detail below but can include (but are not limited to) frequency, line flows, and/or change in load consumption. Dynamic load control parameters are calculated  904 . Dynamic load control parameters are also discussed in greater detail below and can include (but are not limited to) Lagrange multipliers, and/or load consumption. It can readily be appreciated that primal network dynamics parameters and dynamic load control parameters can be a result of using a primal-dual law as described above. Aggregate controllable load can be calculated  906  using power network dynamics parameters and dynamic load control parameters. Optimal load control can be achieved  908  by adjusting the aggregate controllable load. In many embodiments, this process can occur in a distributed manner. Although a number of processes are discussed above with respect to  FIG. 9 , any of a variety of different processes for controlling nodes for achieving optimal load control as appropriate to the requirements of specific applications in accordance with many embodiments of the system. Processes for adjusting aggregate controllable load in accordance with various embodiments of the invention are discussed further below. 
         [0112]    A process for a node to adjust an aggregate controllable load is illustrated in  FIG. 10 . Referring back to  FIG. 8  and  FIG. 9 , the process can be similar to parts of process  900  and performed by node controllers similar to node controller  800 . Process  1000  can include calculating  1002  dynamic load control parameters. Dynamic load control parameters are discussed in greater detail below and can include (but are not limited to) Lagrange multipliers, and/or load consumption. Aggregate controllable load is calculated  1004  using dynamic load control parameters and known power network dynamics parameters. In many embodiments, a shared static feedback loop can be used for parameters that must be calculated by both dynamic load control parameters and power network dynamics parameters. A shared static feedback loop is discussed further below. In many other embodiments, approximations can be made to simplify calculations of dynamic load control parameters, which may eliminate the need for a shared static feedback loop and generate a more distributed process. The aggregate controllable load is updated  1006  based on these calculations. Updating this load drives the network towards optimal load control. Although a number of processes are discussed above with respect to  FIG. 10 , any of a variety of different processes for achieving distributed optimal load control as appropriate to the requirements of specific applications in accordance with various embodiments of the system. 
         [0113]    A distributed load control scheme can be proposed that is naturally decomposed into Power Network Dynamics: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ω 
                       . 
                     
                      
                   
                   = 
                   
                     
                       M 
                        
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           P 
                            
                           m 
                         
                         - 
                         
                           ( 
                           
                             
                               d 
                                
                             
                             + 
                             
                               
                                 d 
                                  
                               
                               ^ 
                             
                           
                           ) 
                         
                         - 
                         
                           
                             C 
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                            
                           P 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     42 
                      
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   0 
                   = 
                   
                     
                       P 
                       ℒ 
                       m 
                     
                     - 
                     
                       ( 
                       
                         
                           d 
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                         + 
                         
                           
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                             ^ 
                           
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                     - 
                     
                       
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                   ( 
                   
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                      
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                     P 
                     . 
                   
                   = 
                   
                     
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                       B 
                     
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                   ( 
                   
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                      
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                   ( 
                   
                     42 
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                   ) 
                 
               
             
           
         
       
     
         [0000]    and 
       Dynamic Load Control: 
       [0114]      {dot over (λ)}=ζ λ ( P   m   −d−L   B ν)  (43a)
 
         [0000]      {dot over (π)}=ζ π ( ĈD   B   C   T   ν−{circumflex over (P)} )  (43b)
 
         [0000]      {dot over (ρ)} + =ζ π     +     [D   B   C   T   ν−  P ]   π     +     +   (43c)
 
         [0000]      {dot over (ρ)} − =ζ π     −     [ P −D   B   C   T ν] ρ     −     +   (43d)
 
         [0000]      {dot over (ν)}=χ ν ( L   B   λ−CD   B   Ĉ   T   π−CD   B (ρ + −ρ − ))  (43e)
 
         [0000]        d=c′   −1 (ω+λ)  (43f)
 
         [0115]    Equations (42) and (43) show how the network dynamics can be complemented with dynamic load control such that the whole system amounts to a distributed primal-dual algorithm that tries to find a saddle point on L(x, σ). It will be shown below that this system does achieve optimality as intended. 
         [0116]    In various embodiments, the control architecture can be derived from the OLC problem. Unlike traditional observer-based controller design architecture, the dynamic load control block does not try to estimate state of the network. Instead, it drives the network towards the desired state using a shared static feedback loop, i.e. d i (λ i +ω i ). 
         [0000]    Remark 3. One of the limitations of (43) is that in order to generate the Lagrange multipliers λ i , P i   m −d i  must be estimated which is not easy since P i   m  cannot easily be separated from P i   m −D i ω i  when power injection is measured at a given node (bus) without knowing D i . This will be addressed further in other embodiments below where a modified control scheme that can achieve the same equilibrium without needing to know D i  exactly. 
       Convergence Under Uncertainty 
       [0117]    An important aspect of the implementation of the control law (43). A modified control law can be provided that solves the problem raised in Remark 3 will be described below, i.e. that does not require knowledge of D i . It will be shown that the new control law still converges to the same equilibrium provided the estimation error of D i  is small enough. 
         [0118]    An alternative mechanism can be proposed to compute λ i . Instead of (43), consider the following control law: 
       Dynamic Load Control (2): 
       [0119]      {dot over (λ)}=ζ λ ( M{dot over (ω)}+Bω+CP−L   B ν)  (44a)
 
         [0000]      {dot over (π)}=ζ π ( ĈD   B   C   T   ν−{circumflex over (P)} )  (44b)
 
         [0000]      {dot over (π)} + =ζ ρ     +     [D   B   C   T   ν   −{tilde over (P)}]   ρ     +     +   (44c)
 
         [0000]      {dot over (ρ)} − =ζ ρ     −     [ P −D   B   C   T ν] ρ     −     +   (44d)
 
         [0000]      {dot over (ν)}=χ ν ( L   B   λ−CD   B   Ĉ   T   π−CD   B (ρ + −ρ − ))  (44e)
 
         [0000]        d=c′   −1 (ω+λ)  (44f)
 
         [0000]    where M=diag          with M i =0 for i∈         , and B=diag           
         [0120]    Notice that the only difference between (43) and (44) is that (43a) can be substituted with (44a) where now M i  only needs to be estimated for the generators, which is usually known. 
         [0121]    The parameter b i  plays the role of D i . In fact, whenever b i =D i  then one can use (42a) and (42b) to show that (44a) is the same as (43a). In other words, if b i =D i +δb i  and δB=diag(δb i ) i∈           , then using (42a) and (42b): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           λ 
                           . 
                         
                         = 
                           
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                               + 
                               
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                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which is equal to (43a) when δb i =0. 
         [0122]    Using (45), the system (42) and (44) can be expressed by 
         [0000]    
       
         
           
             
               
                 
                   
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                           ρ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with matrix δB S :=diag(δb i ) i∈S . 
         [0123]    In many embodiments, under certain conditions on b i  it can be proven that convergence to the optimal solution is preserved despite the fact that (42) with (44) is no longer a primal dual algorithm. The basic intuition behind this result is that when one uses b i  instead of D i , the system dynamics are no longer a primal-dual law, yet provided b i  does not distant too much form D i , the convergence properties are preserved. 
       Simulations of OLC 
       [0124]    Simulations of OLC using the control scheme are illustrated. The IEEE 39 node (bus) system is illustrated in  FIG. 11  to simulate the scheme. The simulation assumes that the system has two independent control areas that are connected through lines (1,2), (2,3) and (26,27). The simulated power network parameters as well as the initial stationary point (pre fault state) were obtained from the Power System Toolbox data set. 
         [0125]    Each simulated node (bus) is assumed to have a controllable load with            i =[−d max , d max ], with d max =1 p.u. on a 100 MVA base and disutility function 
         [0000]    
       
         
           
             
               
                 c 
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                 ( 
                 
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             = 
             
               
                 
                   ∫ 
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                     d 
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                      
                     
                       ( 
                       
                         
                           π 
                           
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                               d 
                               max 
                             
                           
                         
                          
                         s 
                       
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                                 i 
                               
                             
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                          
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
       Thus, 
       [0126]    
       
         
           
             
               
                 d 
                 i 
               
                
               
                 ( 
                 
                   σ 
                   i 
                 
                 ) 
               
             
             = 
             
               
                 
                   c 
                   i 
                   
                     ′ 
                     - 
                     1 
                   
                 
                  
                 
                   ( 
                   
                     
                       ω 
                       i 
                     
                     + 
                     
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                       i 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
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                      
                     
                       d 
                       max 
                     
                   
                   π 
                 
                  
                 
                   
                     arctan 
                      
                     
                       ( 
                       
                         
                           ω 
                           i 
                         
                         + 
                         
                           λ 
                           i 
                         
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0000]    See  FIGS. 12A-12B  for an illustration of both c i (d i ) and d i (σ i ). 
         [0127]    Throughout the simulations it can be assumed that the aggregate generator damping and load frequency sensitivity parameter D i =0.2 ∀i∈          and χ i   ν =ζ i   λ =ζ k   π =ζ e   ρ     +   =ζ e   ρ     −   =1, for all i∈         , k∈          and e∈ε. These parameter values do not affect convergence, but in general they will affect the convergence rate. The values of P m  are corrected so that they initially add up to zero by evenly distributing the mismatch among the load nodes (buses). {circumflex over (P)} is obtained from the starting stationary condition.  P  and  P  can initially be set so that they are not binding. 
         [0128]    The OLC-system can be simulated as well as the swing dynamics (42) without load control (d i =0), after introducing a perturbation at node (bus) 29 of P 29   m =−2 p.u.  FIGS. 13A-13C  and  FIGS. 14A-C  illustrate the evolution of the node (bus) frequencies for the uncontrolled swing dynamics (a), the OLC system without inter-area constraints (b), and the OLC with area constraints (c).  FIGS. 13A-13C  illustrate frequency evolution in area 1.  FIGS. 14A-14C  illustrate frequency evolution in area 2. 
         [0129]    It can be seen that while the swing dynamics alone fail to recover the nominal frequency, the OLC controllers can jointly rebalance the power as well as recovering the nominal frequency. The convergence of OLC seems to be similar or even better than the swing dynamics, as shown in  FIGS. 13A-13C  and  FIGS. 14A-14C . 
         [0130]    Next, the action of the thermal constraints can be illustrated by adding a constraint of  P   e =2.6 p.u. and  P   e =−2.6 p.u. to the tie lines between areas.  FIGS. 15A-15B  show the values of the multipliers λ i , that correspond to the Locational Marginal Prices (LMPs), and the line flows of the tie lines for the same scenario displayed in  FIGS. 13C and 14C  i.e. without thermal limits. It can be seen that neither the initial condition, nor the new steady state can satisfy the thermal limit (shown by a dashed line). However, once thermal limits are added to the OLC scheme,  FIGS. 16A-16B  show that the system converges to a new operating point that satisfies the constraints. 
         [0131]    Finally, the conservativeness of the bound can be shown. Simulate the system (42) and (44) under the same conditions as in  FIGS. 15A-15B . B i  can be set such that the corresponding δb i s are homogeneous for every node (bus) i. 
         [0132]      FIGS. 17A-17E  and  FIGS. 18A-18E  show the evolution of the frequency ω i  and LMPs λ i  for different values of δb i  belonging to {−0.4, −0.21, −0.2, −0.19, 0.0}. Since D i =0.2 at all the nodes (buses), then δb i =−0.2 is the threshold that makes B i  go from positive to negative as δb i  decreases. 
         [0133]    The simulations show that the system converges whenever B i ≧0 (δb i ≧−0.2). The case when δb i =−0.2 is of special interest. Here, the system converges, yet the nominal frequency is not restored. This is because the terms δb i ω i  (45) are equal to the terms D i ω i  in (42a)-(42b). Thus {dot over (ω)} i  and {dot over (λ)} i  can be made simultaneously zero with nonzero w* i . Fortunately, this can only happen when B i =0 which can be avoided since B i  is a designed parameter. 
         [0134]    Although the present invention has been described in certain specific aspects, many additional modifications and variations would be apparent to those skilled in the art. It is therefore to be understood that the present invention can be practiced otherwise than specifically described without departing from the scope and spirit of the present invention. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive. Accordingly, the scope of the invention should be determined not by the embodiments illustrated, but by the appended claims and their equivalents.