Wide Area Control of IEEE 39 New England Power Grid Model

New Enland model 

Contributed by Florian Dorfler and Mihailo R. Jovanovic

Matlab Files
neWAC.m  —  A Matlab function required to run lqrsp.m.
neData.mat  —  Data for the New England Power Grid example.
neDataPST.m  —  Power System Toolbox data file.

Presentations
LANL Talk
2013 ACC Talk

Papers
Journal (IEEE Trans. Power Syst. 2014; in press)
Conference (2013 ACC)

Problem description

The IEEE 39 New England Power Grid Model consists of 39 buses and 10 generators. Generator 10 is an equivalent aggregated model for the part of the network that we do not have control over; generators 1 to 9 are equipped with Power System Stabilizers (PSSs), which provide good damping of local modes and stabilize otherwise unstable open-loop system. We use a sparsity-promoting optimal control strategy to design a supplementary wide area control signal aimed at achieving a desirable trade-off between damping of the inter-area oscillations and communication requirements in the distributed controller. Inter-area oscillations are associated with the dynamics of power transfers and they are characterized by groups of coherent machines that swing relative to each other. These oscillations are caused by weakly damped modes of the linearized swing equations and they physically correspond to active power transfer between different generator groups.

In the absence of higher-order generator dynamics, for purely inductive lines and constant-current loads, the dynamics of each generator can be represented by the electromechanical swing equations

     M_i , ddot{theta}_i ,+, D_i , dot{theta}_i ,=, P_{i} ,-,     displaystyle{sum_{j , = , 1}^{N}}      ,     |, Y_{ij} ,|, E_i , E_j sin ( , theta_i ,-, theta_j , )

where N denotes the number of generators, P_i is the generator power injection in the network-reduced model, and Y_{ij} is the Kron-reduced admittance matrix describing the interactions among generators. After linearization at an operating point the swing equations reduce to

     M , ddot{theta} ,+, D , dot{theta} ,+, L , theta ,=, 0


where M and D are diagonal matrices of inertia and damping coefficients, and the coupling among generators is entirely described by the weighted graph induced by the Laplacian matrix L.

Let the state of the power network be partitioned as  x , = , [ ~~ theta^T  quad dot{theta}^T quad x_{rm rem}^T ~~ ]^T where theta and dot theta are the rotor angles and frequencies of 10 synchronous generators and x_{rm rem} are the state variables which correspond to fast electrical dynamics.

The sparsity-promoting minimum-variance optimal control problem can then be formulated as:

     begin{array}{rrcl}     mbox{bf linearized dynamics:}     &     dot{x}(t)     & = &      A , x(t) ,+, B_1 , d(t) ,+, B_2 , u(t)     [0.35cm]     mbox{bf objective function:}     &     J      & = &       displaystyle{lim_{t ; to ; infty}}     {cal E}     left(     theta^{T} (t) , Q_{theta} , theta(t)      ,+,      dot{theta}^{T} (t) , Q_{dot{theta}} , dot{theta} (t)      ,+,     u^{T} (t) , R , u(t)      ,+,      gamma , displaystyle{sum_{i, , j}} , w_{ij} , |F_{ij}|     right)     [0.6cm]     mbox{bf memoryless controller:}     &     u     & = &     -F , x(t)     end{array}


where we use the weighted l_1 norm to induce sparsity in the state feedback gain matrix F.

Computational results

We next present computations resulting from the use of the sparsity-promoting framework with 40 logarithmically-spaced points for  gamma in [ , 10^{-4},, 1 , ] . In the objective function we choose Q_{dot{theta}} = I, R = I, and set

 	Q_{theta}     ~ = ~     epsilon , I ~+; left( I ~ - ~ frac{1}{N} , {bf 1} {bf 1}^T right)

in order to penalize deviation from synchrony. This choice of Q_{theta} is inspired by slow coherency theory with epsilon > 0 denoting a small regularization parameter and {bf 1} denoting the vector of all ones.

Download Matlab code neWAC.m and data file neData.mat to reproduce these figures.

Performance vs Sparsity

sparsityVSrelativecost 

Relative to the optimal centralized controller, performance of the optimal sparse controller deteriorates gracefully with increased emphasis on sparsity.

sparsityVSsparsity 

For gamma ,=, 0, the optimal centralized feedback gain F_c is a dense matrix populated with nonzero elements.

Increased emphasis on sparsity induces feedback gains with smaller number of nonzero elements.

Signal exchange network

The sparsity pattern of the optimal feedback gain illustrates interactions between different generators. In the figures below, nine rows correspond to nine control inputs to controlled generators; the columns correspond to different states variables. Blue dots in each block represent local interactions within each generator, and red dots represent interactions between different generators.

K4block_color 

gamma , = , 0.0289, {bf card} , (F) , = , 90

Identified long range interactions are represented by red dots.

The optimal sparse controller promotes the use of angles and frequencies (i.e., the first and second states of each subsystem) as signals to be communicated.

K6block_color 

gamma , = , 1, {bf card} , (F) , = , 37

Only a single long range interaction is identified: the controller at generator 1 needs to access the angle of generator 9, theta_9(t).

Relative to the optimal centralized feedback gain, the resulting sparse controller degrades performance by only 1.6%.

singlelink 

As our emphasis on sparsity increases, most of the control burden is on generator 1, which has the largest inertia of all controlled generators. For gamma = 1, only the angle of loosely connected generator 9 needs to be measured and communicated to generator 1.

Dominant inter-area modes

mode1 

The polar plots show the generator frequency components of one of the poorly damped inter-area modes in the open-loop system (i.e., the system controlled with only local PSSs).

This inter-area mode is characterized by active power transfer between generator 10 and the rest of the grid.

mode1closedloop 

The least damped inter-area mode of the closed-loop system (i.e., the system controlled with both local PSSs and the optimal sparse wide-area controller obtained for  gamma = 1).

Relative to the open-loop system, the closed-loop system is characterized by much smaller active power transfer between generator 10 and the rest of the grid.

Presentations

Publications

Citation

If you are using information provided here in your research or teaching, please include explicit mention of this website

        www.umn.edu/~mihailo/software/lqrsp/

and of our papers.

@article{dorjovchebulTPS14,
   author  = {F. D\"orfler and M. R. Jovanovi\'c and M. Chertkov and F. Bullo},
   title   = {Sparsity-promoting optimal wide-area control of power networks},
   journal = {IEEE Trans. Power Syst.},
   year    = {2014},
   note    = {in press; also arXiv:1307.4342}
}

@inproceedings{dorjovchebulACC13,
   author       = {F. D\"orfler and M. R. Jovanovi\'c and M. Chertkov and F. Bullo},
   booktitle    = {Proceedings of the 2013 American Control Conference},
   title        = {Sparse and optimal wide-area damping control in power networks},
   year         = {2013},
   address      = {Washington, DC},
   pages        = {4295-4300}
}

@article{linfarjovADMM13,
   author  = {F. Lin and M. Fardad and M. R. Jovanovi\'c},
   title   = {Design of optimal sparse feedback gains via the alternating direction method of multipliers},
   journal = {IEEE Trans. Automat. Control},
   volume  = {58},
   number  = {9},
   pages   = {2426-2431},
   year    = {2013}
}