2020 International Graduate School of Control - www.eeci-igsc.eu

**M11 - STOCKHOLM - Control and Optimization of Autonomous Power Systems**

Florian Dörfler and Saverio Bolognani

28 September - 1 October 2020

**NCCR Collaborative projects**

- Dynamic population games for efficient autonomous mobility

- Urban Driving Games

2021 - 2023

**Swiss Federal Office of Energy - GRIDS program**

a UNIfied COntrol framework for Real-time power system operatioN

2019 - 2021

**EU H2020 Research Programme - ERIGrid project**

Networked feedback control of distributed energy resources for real-time voltage regulation

2017, 2019

- Optimization on manifolds for power system applications
- Projected dynamical systems on manifolds
- Distributed optimal power flow
- Power flow analysis and voltage stability
- Virtual inertia in power systems
- Chance-constrained real-time decision
- Control of deferrable power loads
- Identification of power distribution grid topology
- Clock synchronization algorithms
- Distributed computing and network congestion control
- Wireless sensor networks
- Control of quantum systems
- Model predictive control of electric drives

Download my BiBTeX biblography.

In [HBH+16] we propose an unconventional approach to the problem of real-time operation of power systems. This ideas was first proposed in [BCC+15] for the specific application of voltage regulation, and later extended to a unified feedback control architecture in which the full non-linear AC power system is actuated based on real-time measurements. This setup can be casted into the mathematical framework of projected dynamical systems on manifolds (see next section).

An illustration of the effectiveness of this approach on a realistic power system test bed has been presented in [HZB+17].

[HBH+16] | Projected gradient descent on Riemannian manifolds with applications to online power system optimization. In Proc. 54th Annual Allerton Conference on Communication, Control, and Computing, 2016. [doi: 10.1109/ALLERTON.2016.7852234] |

[BCC+15] | Distributed reactive power feedback control for voltage regulation and loss minimization. IEEE Transactions on Automatic Control, 60(4):966-981, 2015. [doi: 10.1109/TAC.2014.2363931] |

[HZB+17] | Online Optimization in Closed Loop on the Power Flow Manifold. In Proc. 12th IEEE PES PowerTech, 2017 (Best Student Paper Award). [doi: 10.1109/PTC.2017.7980998] |

Inspired by the power system application presented in the previous section, we are investigating the more abstract topic of projected dynamical systems on manifold, with a particular interest towards applications to feedback (autonomous) optimization on manifolds.

A mathematical formulation of this setup, together with conditions for the existence of solutions for these flows, is presented in [HBD18]. In [HSB+18] we consider the effect of time-varying constraints on the existence of solutions for these systems.

The stability of the interconnection of continous-time optimization flows with the underlying controlled system are discussed in [?].

[HBD18] | Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization. arXiv:1809.04831 [math.OC], 2018. |

[HSB+18] | Time-varying Projected Dynamical Systems with Applications to Feedback Optimization of Power Systems. In Proc. 57th IEEE Conference on Decision and Control, 2018. [doi: 10.1109/CDC.2018.8618660] |

We investigated the problem of designing distributed control algorithms for the solution of the optimal reactive power flow problem in a power distribution network, i.e. to exploit the reactive power compensation capabilities of compensators and microgenerators in order to minimize power losses and regulate the grid voltage.

One possible approach is based on a gossip (peer-to-peer) coordination among compensators [BZ11]. Interestingly, there is an interaction between the communication topology (who talks to who) and the electric grid topology. It is possible to show that neighbor-to-neighbor communication achieves the best performance [BZ13], possibly with a multi-hop approach to guarantee convergence for any initial condition [BCD+12].

Another approach consists in designing a distributed (networked) feedback law that drives the grid to a configuration where all voltages are between bounds, and power losses are minimized [BCC+15]. The feedback strategy is obtained from the dual decomposition of the original optimization problem, and requires no measurements from the loads.

In [BCC+19] we showed how there is a fundamental suboptimality gap when communication among agents is not allowed, and only purely local strategies can be employed (the proposed benchmark is available in [BCC+19]).

In [BD15], we show how the manifold of the power flow equation solutions can be locally approximated via a tangent plane, obtaining a sparse linear implicit model that can be efficiently used for fast decision and optimization problems in (possibly unbalanced) power systems. The resulting approximation is sparse, structure preserving, and computationally extremely efficient.

The source code for the approximation is available online [B15].

In [BZ16] we derived a linear approximation for the power flow analysis of a distribution grid, in order to enable the design of feedback control laws, identification algorithms, and state estimation procedures. We also provide explicit bounds on the quality of the approximation, and sufficient conditions for the existence of a solution, based on an implicit function theorem argument.

The source code for the approximation, together with a test feeder inspired by the IEEE123 distribution test feeder, is available online [B14].

[BD15] | Fast power system analysis via implicit linearization of the power flow manifold. In Proc. 53rd Annual Allerton Conference on Communication, Control, and Computing, 2015. [doi: 10.1109/ALLERTON.2015.7447032] |

[B15] | 1ACPF - First-order AC power flow model. GitHub repository, 2015. |

[BZ16] | On the existence and linear approximation of the power flow solution in power distribution networks. IEEE Transactions on Power Systems, 31(1):163-172, 2016. [doi: 10.1109/TPWRS.2015.2395452] |

[B14] | approx-pf - Approximate linear solution of power flow equations in power distribution networks. GitHub repository, 2014. |

In [HBH+18] we study contraint qualifications (and therefore the existence of unique Lagrange multipliers) for AC OPF programs, showing that they generically hold, i.e. they always hold, except for a zero-measure set of problem parameters.

[HBH+18] | Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow. IEEE Control Systems Letters, 2(4):791-796, 2018. [doi: 10.1109/LCSYS.2018.2849598] |

In [ABD18] we show how the determinant of the power flow Jacobian, which is used to determine wheter a given power system state is voltage-stable, can be approximated extremely well via a voltage stability index which can be computed in a scalable and distributed fashion, both in a leader-less communication architecture and in a hierarchical one.

[ABD18] | Hierarchical and Distributed Monitoring of Voltage Stability in Distribution Networks. IEEE Transactions on Power Systems, 33(6):6705-6714, 2018. [doi: 10.1109/TPWRS.2018.2850448] |

In [PBD17] we consider the problem of optimal allocation of virtual inertia (via storage, flywheels, spinning reserves, etc.) in a power grid, in order to mitigate the effect of disturbances (including fluctuating renewable power generation) on the system frequency. We develop a framework for the mathematical analysis of the perfomance of different allocation choices and we derive an efficient numerical method to tackle large scale optimal allocation problems. Different metrics towards this goal are contrasted in [GBP+17] and [PGB+18].

Based on this framework, in [PBN+20] we propose a market mechanism for the efficient provision of virtual inertia in a liberalized market.

[PBD17] | Optimal Placement of Virtual Inertia in Power Grids. IEEE Transactions on Automatic Control, 62(12):6209-6220, 2017. [doi: 10.1109/TAC.2017.2703302] |

[GBP+17] | Increasing the Resilience of Low-inertia Power Systems by Virtual Inertia and Damping. In IREP Bulk Power System Dynamics \& Control Symposium, 2017. |

[PGB+18] | Virtual Inertia Placement in Electric Power Grids. In Energy Markets and Responsive Grids, 162, Springer, New York, NY, 2018. [doi: 10.1007/978-1-4939-7822-9] |

[PBN+20] | A Market Mechanism for Virtual Inertia. IEEE Transactions on Smart Grid, 11(4):3570-3579, 2020. [doi: 10.1109/TSG.2020.2969518] |

In [BAD17] we studied the problem of real-time chance-constrained decision making, with a particular focus on the operation of distribution network in presence of stochastic uncertainties. This decision problem has a specific structure: on one hand, we assume that some a-priori information about the unknown parameters is known, in the form of samples; on the other hand, we assume that it is possible to gather further information regarding the true value of these parameters via real-time measurements. We specialized the scenario-based approach towards this task, obtaining a two-phase algorithm composed of an offline processing of the samples, and an online part to be executed in real-time. This online part is extremely lightweight and is suited for implementation in embedded systems. A tutorial (with source code) on the proposed method is available on [B19].

[BAD17] | A Fast Method for Real-Time Chance-Constrained Decision with Application to Power Systems. IEEE Control Systems Letters, 1(1):152-157, 2017 (Presented at the 56th IEEE Conference on Decision and Control). [doi: 10.1109/LCSYS.2017.2711140] |

[B19] | A Fast Method for Real-Time Chance-Constrained Decision with Application to Power Systems [Source Code]. CodeOcean, 2019. [doi: 10.24433/CO.0494755.v1] |

We considered the problem of deferrable loads (power loads that can be postponed up to a deadline) in a scenario where consumers face time varying energy prices. We showed how to compute the optimal consumption strategy of the individual users that observes the stochastic price signal [MBR+15].

We then consider a scenario where multiple flexible loads are subject to a coupling constraint on the power that they can request (e.g. car batteries in a charging station). Given their deadlines, we characterized the set of feasible power consumption decisions that guarantee the satisfaction of all the deadlines, while allowing full participation of each consumer to the energy market via a constrained auction [MBR+14].

[MBR+15] | Optimal consumption policies for power-constrained flexible loads under dynamic pricing. IEEE Transactions on Smart Grid, 6(4):1884-1892, 2015. [doi: 10.1109/TSG.2015.2393053] |

[MBR+14] | Deferrable loads in an energy market: coordination under congestion constraints. In Proc. 22nd Mediterranean Conference on Control and Automation, 2014. [doi: 10.1109/MED.2014.6961443] |

We considered the problem of identifying the topology of the power distribution grid from nodal voltage measurements, in order to enable a plug-and-play deployment of control and automation in these grids. We showed how the topology can be inferred from the statistical properties of the voltage measurements, in particular from their conditional correlation [BBM+13], and we also derive a statistical test that can be performed in a distributed way [B18].

[BBM+13] | Identification of power distribution network topology via voltage correlation analysis. In Proc. 52nd Conference on Decision and Control, 2013. [doi: 10.1109/CDC.2013.6760120] |

[B18] | Grid Topology Identification Via Distributed Statistical Hypothesis Testing. In Big Data Application in Power Systems, Elsevier Science, 2018. [doi: 10.1016/B978-0-12-811968-6.00013-9] |

In [BCL+16] we have shown how a very large family of clock synchronization algorithms (implemented on wireless or wired networks, and based on peer-to-peer or broadcast communications) can be cast into the same framework (that we called RandSync), for which sufficient conditions for convergence can be provided. We implemented RandSync on a wireless testbed [B14], showing how an accuracy of less than 0.1 ms can be achieved with just about 2 broadcasted messages from each node evey hour.

[BCL+16] | A randomized linear algorithm for clock synchronization in multi-agent systems. IEEE Transactions on Automatic Control, 61(7):1711-1726, 2016. [doi: 10.1109/TAC.2015.2479136] |

[B14] | RandSync source code. GitHub repository, 2014. |

In [KBD16], we consider the problem of optimal allocation of computing power for IaaS (Infrastructure as a Service) cloud computing applications. We formulate a bilevel optimization problem where bandwidth and computing capacity constraints for each involved node are taken into account, showing that the optimal allocation strategy is independent by the congestion control mechanism employed by the data network infrastructure.

[KBD16] | A separation principle for optimal IaaS cloud computing distribution. In Proc. EUSIPCO, 2016. [doi: 10.1109/EUSIPCO.2016.7760477] |

We condidered a problem that arises when trying to infer agent-to-agent distances in a wireless sensor networks based on received power strength. We showed how it is possible to identify the parameters of the wireless channel and to compensate the measurement offsets via a distributed algorithm that require no central coordinator [BDS+10].

[BDS+10] | Consensus-based distributed sensor calibration and least-square parameter estimation in WSNs. International Journal of Robust and Nonlinear Control, 20(2):176-193, 2010. [doi: 10.1002/rnc.1452] |

We considered the problem of achieving attractivity and invariance of a quantum subspace by designing an appropriate feedback, discrete-time, control law. The control scheme consists in a quantum measurement, which is given, and a coherent control action, which has to be designed. An algorithm has been proposed to check feasibility of the control problem and to return a stabilizing unitary control [BT10].

[BT10] | Engineering stable discrete-time quantum dynamics via a canonical QR decomposition. IEEE Transactions on Automatic Control, 55(12):2721-2734, 2010. [doi: 10.1109/TAC.2010.2049291] |

We designed and implemented a model predictive controller for PMSM electric drives, where we included both the electrical and the mechanical dynamics in a single aggregate model [BBP+09].

[BBP+09] | Design and implementation of model predictive control for electrical motor drives. IEEE Transactions on Industrial Electronics, 56(6):1925-1936, 2009. [doi: 10.1109/TIE.2008.2007547] |

For the material of the Game Theory and Control course, refer to the official Moodle page.

Final reports and presentation are available upon request.

- Ashwin Venkatraman,
**"Real-time control of power systems for voltage collapse prevention",**2018. - Sandeep Menta,
**"Feedback optimisation of linear time invariant systems",**2018. - Christos Konstantinopoulos,
**"Integration of PV units in the primary and secondary control ancillary services in Switzerland",**2018. - David Rodriguez Flores,
**"Time domain performance metrics in optimal inertia placement",**2017. - Felix Böwing,
**"Optimal nonlinear frequency control in power systems",**2017. - Liviu Aolaritei,
**"A decentralized voltage collapse distance for power distribution networks,"**2017. - József Pázmány,
**"Constrained optimization over manifolds for power system application,"**2017. - Elena Arcari,
**"Fast chance-constrained optimization using real-time measurements with applications to power systems,"**2017 - Philipp Lütolf,
**"Optimal placement of virtual damping and inertia,"**2017 - Alessandro Zanardi,
**"Constrained optimization over manifolds for power system application,"**2016 - Jan Schulze,
**"Peer-to-peer clock synchronization in wireless sensor networks,"**2016 - Panagiotis Kyriakis,
**"Formation of robust networks for secure exchange of cryptocurrencies,"**2016 - Felix Kottmann,
**"Computational load and congestion control in cloud environments,"**2015 - Dalibor Drzajic,
**"Energy theft detection using compressive sensing methods,"**2015