Report CopyRight/DMCA Form For : Distributed Control And Optimization For Autonomous Power
control and optimization with a focus on the analysis of multi agent decision scenarios and on the design of decentralized and networked control strategies We cover the topics of decentral ized control of power converters in low inertia power systems real time control of distribution grids optimal and distributed
The remainder of this article is organized as follows M ir L is. Section II focuses on decentralized power converter control vg. Section III covers real time distribution grid control Section m. IV discusses distributed frequency control Section V ad. dresses supply and demand coordination Finally Section VI. concludes the paper and presents open research directions. Fig 1 Sketch of a simple permanent magnet synchronous machine SM. II D ECENTRALIZED C ONTROL OF P OWER C ONVERTERS,IN L OW I NERTIA P OWER S YSTEMS. coupling the currents is R2 and ir R of the SM stator. At the heart of the energy transition is the change in. and rotor respectively The electromagnetic field is driven by. generation technology from conventional rotational power. the generator rotor with angle S1 and frequency R, generation based on synchronous machines SM towards. The mechanical rotor swing and stator flux dynamics are. power converter interfaced generation CIG as in the case. of renewable energy sources battery storage or high voltage d. DC HVDC links interconnecting different synchronous 1a. areas This transition poses major challenges to the operation d sin. control and stability of the power system due to M D m Lm ir cos. i the loss of rotational kinetic energy in the SM whose dis. Rs is v g Lm ir cos,inertia acts a safeguard against disturbances dt. ii the loss of the stable and robust nonlinear synchroniza. tion mechanism which is physically inherent to SMs where all parameters are as depicted in Figure 1 v g R2 is. iii the loss of the robust frequency and voltage control the AC grid voltage at the SM terminals D and Rs model. since SMs are the main points of actuation and thus the mechanical and electrical dissipation m is the mechani. iv the loss a stable global frequency signal altogether cal control input relatively slow actuation typically used for. which is the basis for many services see Section IV speed droop and automatic generation control AGC and. In future low inertia power systems all of these functional ir is the excitation rotor current control input fast actuation. ities have to be provided by control of power converters used for automatic voltage regulation AVR and damping. which in absence of control lack all of the above control More detailed higher order models consider addi. features These so called low inertia challenges are currently tionally the mechanical turbine governor actuation stage. regarded as one of the ultimate bottlenecks to massively the rotor flux dynamics and stabilizing damper windings. integrating renewables in power systems around the world. We refer to 17 23 surveying the challenges of low VSC model The VSC converts a DC voltage vdc R. inertia systems and possible solution approaches In what and current ix R to an AC voltage vx R2 and inductor. follows we give a tutorial introduction from the perspective current if R2 by modulating the converter switches as. of control theory present the models of SMs and CIGs their. control specifications and different control approaches ix m. if and v x m, A Models of Synchronous Machines and Power Converters where m 1 2 1 2 1 2 1 2 are the normalized. We refer to 24 27 for control theoretic expositions and duty cycle ratios Here we consider an averaged model where. the text books 28 33 concerning the modeling of circuits m is a continuous variable typically parameterized as. voltage source converters VSCs and SMs We consider a sin. three phase AC power system and assume that all phases m umag cos. are balanced and all impedances are symmetric This allows. us to express all AC quantities as two dimensional variables where ufreq R and umag 1 2 1 2 are the controllable. e g in rectangular coordinates rotating with a reference switching frequency and magnitude The overall VSC is then. frequency dq coordinates polar complex valued phasor. coordinates or combinations thereof For simplicity we. choose coordinates in what follows While there are many idc ix. SM and VSC models we consider two prototypical models. shown in Figures 1 2 used among others in 34 35 vdc if vg. SM model The SM converts mechanical to electrical, energy via a rotational magnetic field with inductance matrix Cdc. Ls 0 Lm cos,L 0 Ls Lm sin, Lm cos Lm sin Lr Fig 2 Sketch of a two level voltage source converter VSC. modeled by the DC capacitor and AC inductor dynamics 50 Hz. power 51 49,d supply power,Cdc Gdc vdc idc umag cos. Rf if v g umag cossin, where all parameters are as in Figure 2 Gdc and Rf model control inertia. the lumped switching and conduction losses v g R2 is the. AC voltage at the VSC terminals and idc is the controllable Fig 4 Mechanical analogue of the power balance 4 adapted from 41. DC side current typically coming from an upstream DC con. verter or storage element Aside from an upstream converter balance equation 4 is illustrated in Figure 4 via a simpli. higher order CIG models consider additionally LC or LCL fied mechanical analogue where we also indicated control. filters at the AC terminals rather than a single inductor mechanisms discussed in Section II C While under similar. B Similarities Differences and Control Limitations assumptions an analogous power balance holds for the VSC. The SM and VSC models 1 and 2 can be abstracted d,dt vdc Cdc vdc. as a power preserving interconnection of energy supply. the equivalent inertia in the DC capacitor is rather small and. conversion and storage 36 from a controllable supply via. its effect as a safeguard against disturbances is negligible. m and idc over storage via M and Cdc a nearly lossless. However as an opportunity the equivalent actuation and. conversion or signal transformation via L and m that. energy supply idc is rather fast compared to m, is controllable via ir and m to the AC grid see Figure 3. Control limitations The above insights lead to the fol. controllable,controllable, lowing characteristics of SMs and VSCs as control systems. energy energy, storage system fast vs slow actuation of the energy supply the turbine. supply conversion, governor input m of a SM is rather slow whereas the. Fig 3 Conceptual abstraction of VSC and SM control systems VSC DC input idc can be controlled very fast though. idc is typically constrained in power and energy, significant vs negligible energy storage since M is or. Structural similarities On a less abstract level note the. ders of magnitudes larger than Cdc SMs are much more. similarity of the SM and VSC models when controlling the. robust to disturbances than VSCs c f low inertia, VSC frequency proportional to its DC voltage e g as. limited vs full AC actuation the SM s excitation, ufreq vdc 3 control via the single input ir is complex and often. limited in practice whereas the VSC s AC voltage, where ref vdc ref is chosen as the ratio of the nominal v x m vdc is fully controllable unless vdc 0. AC grid frequency ref and DC voltage vdc ref then the through the two modulation inputs ufreq and umag and. SM model 1 and closed loop VSC 2 3 are structurally state constraints whereas an SM can tolerate large fault. equivalent when identifying vdc is if and currents up to a factor 6 10 above nominal the VSC s. umag Lm ir Accordingly we can match the associated switches cannot withstand any over currents. parameters e g the equivalent inertia M as the normalized. In summary SMs and VSCs share lots of structural similar. DC capacitance Cdc 2 and the imbalance kinetic energy. ities but they have nearly antipodal control characteristics. signal as vdc These analogies and electro mechanical. dualities have recently given rise to a variety of control C Control Specifications and Classic Control Approaches. methods primarily based on the DC voltage as control signal We specify the control objectives separately as nominal. and DC capacitor as equivalent inertia 17 34 37 40 steady state perturbed steady state and transient objectives. We will return to these control strategies in Section II D since the conventional power system stability classification. Deceiving similarities The above similarities can be mis and associated controllers also make such distinctions 30. leading as highlighted in the following key observation The We additionally discuss slower time scale objectives such. power balance of the SM neglecting the comparatively small as secondary regulation and tertiary set point re scheduling. magnetic energy and dissipation amounts to 26 35 that will be addressed in the forthcoming sections. M m i Nominal steady state specifications During nominal sys. dt z z z tem operation at frequency ref the requirements on a. change kinetic energy mechanical power supply power demanded by grid synchronous steady state are as follows 42 43. Thus any power imbalance e g caused by a grid fault S1 all DC states need to be constant that is v dc 0. will be absorbed in the SM s kinetic energy before any S2 all AC states need to be synchronous at 0 that is. is 0 ref 0ref is and so on and, restoring control mechanism via m even acts The power ref dt. S3 for the VSC and analogously for the SM the active Transient specifications Finally the grid should be ro. power P i 0 1, s v g reactive power Q is 1 0 vg bust to perturbations Especially on the fast so called tran. and terminal voltage magnitude kv g k take pre specif sient time scale up to five seconds where. ied values Pref and Qref and v ref respectively S7 transient disturbances and faults need to be rejected. These set points are scheduled offline see specification S6 For SMs disturbance rejection is achieved passively via their. Perturbed steady state specifications The power system large inertia M and actively via Power System Stabilizers. typically fluctuates around a nominal stationary operating PSSs inducing damping by closing a control loop between. point due to variable generation loads and disturbances and ir 28 This classic SM control problem has been. Aside from high frequency perturbations the system typi approached from many angles 47 49 However the rejec. cally operates at a synchronous frequency 0 different from tion of large disturbances is still an unresolved and contested. ref i e all signals satisfy the specifications S1 and S2 problem for CIGs which is at the heart of making low. above with ref replaced by 0 Additionally grid codes grid inertia systems reliable In the next section we will give an. interaction protocols demand pre specified sensitivities so introduction and overview of the proposed control solutions. called droop slopes P resp Q kv g k prescribing We conclude this section with Figure 5 which summarizes. a linear trade off between power injection frequency and the different control actions that need to be undertaken to. voltage 28 30 For example S3 changes to meet the specifications S1 S7 The next subsection will. S4 droop specification P Pref K ref focus on the fastest time scale and the control of CIG. where K 0 is the droop slope Similar droop specifications Sections III IV and V will address the slower secondary. exist for Q and kv g k 44 45 and they are typically and tertiary time scales though noting that these nominally. enforced via proportional also called primary or droop slower control actions get ever closer to real time operation. control of the VSC inputs idc or also ufreq or the SM. input m resp the modulation amplitude umag or excitation control of converter tertiary control. interfaced generation, current ir as function of the frequency deviation resp the. voltage deviation see 28 29 and also Sections III IV primary control. Since droop control provides physical energy to the system. and results in generation costs the actual values of the droop. slopes P resp Q kv g k are negotiated in ancillary. service markets see Section V or determined by grid codes. e g inversely proportional to the generation capacity 5s 30 s 15 min 75 min. inertial response secondary control, Regulation specification In a perturbed steady state the. synchronous frequency 0 is a global signal that reflects. the load generation imbalance which can be seen by eval Fig 5 Time scales of power system control operation adapted from 18. uating the power balance 4 for a constant t 0 The, non trivial insight is that the global system imbalance can D Decentralized Control of Power Converters. be inferred locally by measuring see also Figure 4 for Forming and following classifications Since power con. an illustration Hence the synchronous frequency is used verters are fast modular and nearly fully actuated control. as a floating variable that indicates imbalance serves as systems a vast number of articles discuss different control. a control signal and is regulated only longer time scales architectures and tuning criteria For the considered grid. on the order of minutes by means of secondary integral connected converters an important classification is that of. control approaches see Sections III IV for details grid following and grid forming control 18 While there is. S5 secondary regulation regulate 0 to ref no universally accepted definition of these concepts whose. Similar arguments apply to voltage magnitudes though they exact distinctions are often vigorously debated we provide. do not carry global information and are regulated tighter in the following a control theoretic characterization in terms. of system behavior 50 causality and reachability, Set point specifications The pre specified power and. 1 steady state behavior grid following converters mea. voltage set points for the nominal steady state are computed. sure the grid frequency e g by passing v g to a phase locked. offline e g through an OPF or via various markets see. loop PLL command their modulation frequency ufreq to. Sections III IV for details This set point scheduling must be. track this measured frequency and regulate their terminal. consistent i e the physical power balance equations must be. current to meet a specification e g a P Q set point or a. feasible satisfy operational constraints e g voltage limits. droop For these reasons grid following converters are re. and typically minimize an economic dispatch criterion 46. ferred to as frequency following and current controlled. S6 the set points Pref Qref and v ref need to be consistent. In comparison grid forming converters are frequency. satisfy operational limits and be economically efficient. forming voltage controlled 1 and invert inputs and out. Often these set points have to be updated during operation. time and generators have to be re dispatched This set point 1 A voltage controlled converter is interfaced to the grid through an LC. re scheduling is typically referred to as tertiary control filter that is an AC side capacitor is added to the VSC in Figure 2. puts the terminal voltage frequency and magnitude are con from a nominal steady state which leads to a poor transient. trolled e g based on active power measurement as in droop performance of droop and a limited basin of attraction. 1 A second approach is based on emulation of virtual SMs. ufreq ref K P Pref 5, making sure that the terminal behavior of the VSC 2 equals. with droop slope K 0 or as in the matching control 3 that of a SM 1 see e g 17 32 56 59 The core of. 2 reachability grid following converters require a strong this approach is a reference system e g a software model of. grid with stiff frequency which they can track In absence of the SM 1 on a micro controller driven by measurements. a grid they lack their driving input and will not output a si of the terminal signals if v g and the output of which. nusoidal wave form In comparison grid forming converters serves as reference signal for controlling the modulation. can reach the desired behavior i e form the grid frequency ufreq umag typically via cascaded PI tracking controllers. and voltage and can thus function in islanded mode e g The appeal of the virtual SM approach is backward com. when connected only to a resistor By this classification patibility to the legacy system but it suffers from various. a constant feedforward modulation e g ufreq ref and practical drawbacks such as time delays in control loopst. umag 1 p u is forming a grid but it cannot adapt to ambi overshoots and current violations Since SMs and VSCs have. ent conditions such as a shortage or surplus of generation antipodal characteristics as argued Section II B it seems. cannot produce a floating frequency indicating the global shortsighted to control a VSC without storage though fast. imbalance and finally such a feedforward scheme is not to mimic a SM with large storage and slow actuation 18. robust to clock drifts 51 52 For this reason grid forming A third approach is virtual oscillator control VOC 60. converters are controlled in feedback e g by droop 5 64 which uses a nonlinear oscillator as reference system. Other desirable features of droop control are that it induces reminiscent of the classic van der Pol model Based on recent. a frequency consensus thus grid forming converters can theoretic developments in the synchronization of coupled. synchronize with another or with a stiff grid and it achieves oscillator models it can be shown that such oscillator. fair power sharing that is any load generation imbalance is controlled inverters robustly and almost globally synchronize. picked up equally according to the droop slopes 53 55 60 61 even when pre specified set points are assigned. 63 64 and it locally near steady state reduces to droop. Limitations of grid following control Simplified models. control 62 Thus VOC is a global and multivariable taking. of grid following and grid forming converters give rise to. cross coupling into account implementation of droop control. the same input output behavior e g a droop specification. which generally appears to be faster and more robust 58. but their controllers have different inputs and outputs which. Fourth and finally the matching control approaches 34. can make a significant difference e g when measuring a. 37 40 reviewed in Section II B rely on the VSC and SM. brittle frequency different performance characteristics e g. dualities and match the SM s power conversion mechanism. grid following converters are limited by the accuracy and the. the rotating electro magnetic field by the control 3 A. bandwidth of their PLLs and they can function with either. notable feature compared to all other approaches is that the. only strong or also with weak grids Thus far grid following. converter AC control makes use of a DC signal whereas. converters were dominating in conventional grids where CIG. the previous three approaches decouple AC and DC sides. was not required to provide any ancillary control services. and stabilize the latter through SISO P I controls Thus. other than injecting free renewable energy However future. unsurprisingly due to its multivariable nature there are. low inertia power systems require grid forming converters. scenarios when the matching control 3 is resilient to a. that take over the role of SMs form the frequency and. disturbance whereas the other three controllers fail 57. voltage reject disturbances and provide ancillary services. We close here by stating that quest for the best grid. such as frequency droop reactive power support and fast. forming control is still vigorously debated accross communi. inertial frequency response to stabilize the grid, ties and the field is enjoying tremendous interest right now. Overview of grid forming strategies There are manifold Finally all the presented device level controllers can give. proposals how to control grid forming converters see the excellent system level performance when their gains tuned. reviews and comparison studies 17 18 56 59 The well 65 and their set points are regularly updated which. bulk of which can be classified into four distinct groups that will be one of the subjects of the forthcoming sections. all have to cope with the limitations reviewed in Section II B. III R EAL T IME C ONTROL OF D ISTRIBUTION G RIDS, The first and most developed approach is droop control as. in 5 and regulation of the terminal voltage kv g k through A Challenges in Future Power Distribution Grids. umag While droop control is well studied and understood One of the consequences of the integration of renewable. stability characterizations exist only for simplified models energy sources in the generation mix is the additional stress. 53 55 and practical implementation usually demands faced by power distribution grids In fact most renewable. various extra heuristics such as low pass filters or virtual sources are interfaced to the grid through power converters. impedances inducing damping in the control loop Finally and are deployed as small scale installations microgenera. droop control closes two independent SISO loops separately tors in the low and medium voltage power network rather. for P and as well as for Q and kv g k However these than in the high voltage transmission grid Power distribution. quantities are actually coupled especially when far away lines have non neglibile resistance and their power transfer. Feedforward approach Feedback approach, capacity is limited Because of these reasons congestion. phenomena are expected to occur increasingly often voltage w estimate w w. collapse instability violation of voltage limits and line and y y. Optimization System Controller System,transformer overloading 66. In order to guarantee a safe reliable and efficient opera. tion of these grids microgenerators must be provided with Fig 6 A schematic illustration of the difference between the standard. appropriate set points as already discussed in Section II C optimization approach to the generation of set points and the proposed. feedback solution The signal u y and w denote decision variables. specification S6 In transmission grids generator set points measurements and exogenous parameters of the optimization problem. are typically scheduled via offline programming by solving respectively. an Optimal Power Flow problem in which economic cost is. power voltage, minimized based on relatively accurate models of the grid. and of the power demand profiles The same approach is. generally unfeasible for distribution grids the power demand. of each distribution bus is often unmonitored and highly PCC. uncertain and network parameters are often known with sig. nificant error Moreover the flexibility and the fast dynamic. response of microgenerators allow to update set points at a. much faster rate than the traditional time scale of tertiary. control depicted in Figure 5 This increased responsiveness. is expected to be critical for the efficient accomodation of u. fluctuating and intermittent renewable sources yk line. In this section we will therefore present a recent research. trend consisting in the design of responsive and automated. strategies to update the generator set points in real time microgenerator ui set point. based on measurements collected from the grid either by yi. load measurement, the same microgenerators or by dedicated sensors As. represented schematically in Figure 6 this approach can be Fig 7 An example of the control architecture that we assume in a power. interpreted as a feedback approach to set point generation distribution grid highlighting the available measurements and the controlled. in contrast to the feedforward nature of optimization based variables set points. schemes The two main advantages of a feedback architecture. are its robustness to model mismatch and its capability to. reject exogenous disturbances without measuring them in B Distribution Grid Modeling and Control Architecture. this application disturbances represent for example unmoni We consider the control architecture that is schematically. tored load demands Both these points will be elaborated represented in Figure 7 The diagram represent a power dis. in the rest of this section Another important advantage tribution grid that hosts multiple loads and microgenerators. of a feedback optimization approach is the availability of We denote. rigorous methods to design the controller so that the dynamic by u the vector of set points for the microgenerators as. performance of the closed loop system is improved i e explained in Section II these can be set points for the. time varying disturbances are tracked while ensuring that the active and reactive power injection or for the voltage. interconnection with the system dynamics does not introduce magnitude. instabilities by y the vector of measurements that we perform on. Instead of reviewing the relevant pieces of work from the the grid these outputs can include for example voltage. literature here we will first present a unified formulation of magnitudes line currents and power flows. this feedback strategy and we will then discuss the main When the grid is at steady state including electrical. design challenges one at a time referring the reader to transients load dynamics and the local controllers reviewed. different solutions available in the recent literature in Section II the output y converges to a steady state value. We will not cover the dynamic part of the control design. in this section which is a thriving topic on its own We can y h u w 6. refer the reader to some very recent works on this matter that is function both of the set points u and of a vector w. 67 73 of exogenous inputs e g power demand of the loads at all. Finally it is important to notice that while distribution buses substation voltage. grids constitute a very compelling motivation and benchmark The steady state map h descends directly from Kirchhoff. for this strategy most of the proposed methods can be laws from the tracking characteristics of local controllers. applied to transmission grids as well In fact some of the It will become clear later that an explicit knowledge of h. contributions that we will review have a broader scope than is not needed for the control design procedure presented in. the distribution systems this section It is however necessary to guarantee its existence. and well posedness To illustrate this point with an example In the following we assume that the constraint 9b is. consider a distribution grid in which v p and q denote replaced by an opportune penalty function yielding the. the bus voltages angles active power injection and reactive following optimization program. power injection respectively We can partition these vectors. min J u h u w h u w 10,h v0 i 0 h p0 i h q0 i,v vG G p ppG q qqG. where y is a scalar valued transformation of g y, Note that the optimization program 10 cannot be solved. where the first element of each vector refers to the substation offline because the exogenous input w which we assume. the subvectors with subscript G refer to the buses that constant for the time scale of interest is unknown This. host a microgenerator and no loads while the remaining prevents the application of optimal power flow solvers even. subvectors with subscript L refer to load buses By Kirchhoff distributed one e g 76 77 for this task Strictly speak. laws the electrical state of the grid needs to satisfy at steady ing the feedback optimization methods proposed in 78. state would also be unsuited for this task as they assume full grid. diag Y p jq 7 state measurement although the underlying methodology. is similar to the one presented in this section Moreover. where denotes the vector of complex bus voltages i uncertainty in the grid parameters translates into uncertainty. vi ej i and where Y is the bus admittance matrix Equation on the nonlinear map h and consequently on the evaluation. 7 can be interpreted as an implicit relation between the of the penalty function for the constraint 9b. quantities that we introduced before In the rest of this section we will consider a specific line of. attack to this challenge inspired by gradient descent methods. f v p q 0 8, for nonlinear optimization It is not however the only possible. Let us assume that we can control the power set points strategy we will review a few other options at the end of. for the generator buses with no tracking error u pqG G. the section Nevertheless the formulation that we present. and we can measure their voltage y vG while all other hereafter is sufficient to highlight the key features and more. power demands and the substation voltage are uncontrolled importantly the main challenges that characterize this control. w v0 0 p design problem,L qL The vectors u y and w are therefore. a simple rearrangement of the coordinates v p q The D Gradient Flow Design. existence of an explicit nonlinear function y h u w. We consider the key problem of designing a continuous. that solves 8 can be guaranteed for example in the. time gradient descent flow that converges to the local. neighborhood of the no load solution 74 75 although. minimizer of 10 Note that the gradient u can be, a closed form expression is in general not available. computed explicitly as,C Real Time Control Specifications u u J u h u w. We now elaborate the set point specifications that have u h u w y J u h u w 11. been introduced as S6 in Section II Set points for the gen. u h u w h u w, erators need to be compatible with the power flow equations. and with the power demand on the system they need to Once u is available 10 can be tackled via standard. satisfy some operational limits of the grid and they need iterative solvers e g to drive a projected gradient descent. to be economically efficient We unify these specifications method of the form. via the following general non convex Optimal Power Flow u Tu U u arg min kv u k 12. program v Tu U, where Tu U denotes the tangent cone of U at u and cor. min J u y 9a responds to the set of all velocities in u that maintain the. subject to g y 0 9b trajectory in U see the formal definition in 79 Chapter. 6 In practical terms this projection operation ensures that. the gradient descent flow never leaves the set U In the. y h u w 9d special but common case in which U is an hyper cube i e. each component ui of u is subject to independent bounds. where g y 0 describes operational constraints of the grid. then the projection operator amounts to an coordinate wise. e g voltage limits and U is the set of feasible set points. saturation of the flow,which we assume to be compact. Multiple technical issues need to be taken care of We. In the example introduced before where u pqG G, and review the most important ones hereafter pointing at some. y vG the set U would describe the power capability limits of the latest efforts in those directions with a preference. of microgenerators while g y could beh used toi describe for those results that have been specialized to the context of. under and over voltage bounds g y vGmin vG 0 control of power distribution networks. feedback optimizer computed and the only model knowledge that is needed is. c y the sensitivity matrix u h u w All the remaining terms in. c y are design parameters and are therefore known, grid generator The sensitivity matrix u h u w describes the first order. y u set points,measurements, approximation of the effect of changes in the set points u on. power system w the grid measurements y A few comments are due. steady state y h u w In some cases such as the example presented before. an analytic expression for u h u w is available as a. Fig 8 Feedback implementation of the proposed gradient descent opti function of both u and w Namely u h u w can be. mization scheme recovered via the implicit function theorem applied to. 8 as both u f u y w and y f u y w are avail, able in analytic form 85 Under some standard approx. Convergence analysis The vector flow 12 is disconti imations e g DC power flow and V decoupling. nouous and the projection operator may be set valued The the resulting u h u w becomes even independent on. existence and uniqueness of solutions of 12 together with u and w. its convergence to solutions on 10 depend on opportune Numerical experiments show that approximations of. regularity assumptions on both and U Some of these u h u w yield very good results when used in the. conditions are derived in 80 proposed feedback optimization scheme see for exam. Second order methods The gradient can be used to ple 83 A mathematical analysis of the performance. drive iterative optimization schemes that make use of or of the closed loop scheme under this source of uncer. estimate the second derivative of as well e g quasi tainty is however still missing. As we are assuming that the performance outputs y, Newton methods This has been done for example in 81. Note that once we adopt a descent direction which is not the are measured the sensitivity matrix u h u w can be. steepest descent then the projection on the feasible set U also inferred from past data or through ad hoc system. can become non trivial compared to the simpler coordinate identification techniques. wise saturation that we mentioned before In second order F Distributed Implementation. descent methods the projection needs to be computed via. According to 13 the real time computation of a descent. a nested optimization problem a quadratic program in the. direction requires in general a centralized processing of all. case of polytopic U The analysis of convergence when this. the performance outputs y and all the set points u This. inner optimization cannot be solved exactly because of the. reduces scalability of the approach when many microgener. finite time available requires special care see for example. ators are connected to the same grid and a natural question. is whether the proposed feedback optimization law can be. Time varying problem parameters The parameters of computed in a distributed manner. the optimization problem 10 can vary over time resulting in Let us assume with minimal loss of generality that for. some interesting questions on both the existence of solutions each i the set point ui and the measurement yi are co located. to 12 and on its tracking performance with respect to and managed by an agent i The adoption of separable cost. the true minimizer The case of time varying exogenous and penalty functions i e. quantities w has been studied for example in 67 83 while X X. a time varying feasible region U has been considered in 84 J u y Ji ui yi y i yi. E Feedback Optimization and Robustness would partly distribute the computation of each component. The dynamic flow 12 can be conveniently implemented. c u y of the gradient However from 13 coupling still. in a feedback fashion based on measurements of the output exists via the matrices u h u w As mentioned before. y that are collected from the system see Figure 8 In fact analytic expressions for u h u w exist and these expres. we can replace u by sions preserve the structure of the system i e they exhibit a. sparsity pattern which directly descends from the sparsity of. c y the electrical interconnection between agents Depending on. u J u y u h u w y J u y u h u w y the choice of set points and of performance measurements. this sparsity can appear directly in the term u h u w. In general however we can aim at constructing a sparse. where the last equality is valid under the assumption that the matrix Q such that S Q u h u w is also sparse or. output y is at steady state and therefore satisfies the model even the identity matrix If Q is positive definite then the. y h u w direction Q u,c y can be used in the feedback optimiza. By adopting u, c y as a measurement based evaluation tion scheme instead of u. c y The resulting direction will, of the gradient we drastically reduce the model dependence still be a descent direction but not the steepest one While. of the optimization scheme as h u w does not need to be in general this may affect performance it is also possible. that the matrix Q can be chosen to approximate the Hessian f. restoration time, of the cost function u therefore yielding a second order. descent flow This is what happens for example in 86 nominal frequency. Note that as discussed before special care and most likely effort. extra communication between the agents will have to be max deviation. used to compute the projection on the feasible set U. G Further extensions f, Fig 9 Response of power system with primary and secondary. nominal frequency, We focused on the gradient descent flow to drive the control to an increase in load figure adapted from 18. system to the solution of the optimization problem 9 or. more precisely of the optimization problem in which 9b. is replaced by a penalty function This is clearly not the is inversely proportional to the inertia of the system a. only solution We refer the reader to 83 87 88 for larger inertia yields a shallower slope Within a few seconds. an alternative approach in which the inequality constraints speed governors interfaced with SGs through their turbines. of the problem have been dualized and a saddle point flow become active and SGs inject additional power proportional. on an augmented Lagrangian has been employed to drive to the frequency deviation this slows and eventually stops. the system to a solution of the corresponding KKT equa the drop in frequency This so called primary droop control. tions Alternatively in 89 the real time specifications are mechanism is entirely decentralized. expressed using the formalism of semidefinite programming Primary droop control through the governor system results. and a dual subgradient method is proposed The spirit in a non zero frequency deviation the secondary control. of these approaches remain the same as in the problem layer restores the frequency to its nominal value on a time. formulation that we presented to design an output feedback scale of a few minutes The industry standard secondary. law i e without full state disturbance measurements that control methodology in interconnected bulk power systems. ensure ultimate convergence to set of minimizer of a given is a semi decentralized control system called automatic gen. optimization program while guaranteed satisfaction of a eration control AGC Finally on a time scale of roughly. subset of hard constraints at all times Any advance towards 15 minutes the nominal generation setpoints of SGs are re. the solution of this challenge has the potential to translate computed via a centralized constrained optimal dispatch this. immediately into effective solutions for the core goal that we tertiary control layer sits atop the temporal and architectural. reviewed in this section which is the autonomous generation control hierarchy. of feasible and efficient microgenerator set points To summarize in control theoretic terms primary fre. IV O PTIMAL AND D ISTRIBUTED F REQUENCY C ONTROL quency control is concerned with disturbance attenuation. OF T RANSMISSION G RIDS via decentralized proportional control secondary frequency. control is a problem of asymptotic disturbance rejection. A Frequency Control Background and Fundamentals, via integral control and tertiary control deals with setpoint. All AC power systems are designed to operate only in design implemented via centralized feedforward control. a very narrow range 1 around a nominal frequency, value ref The importance of frequency as the key control B The Need for Modernized Frequency Control. variable in large scale power systems arises from the physics. of SGs As mentioned in Section II SGs equipped with For the reasons discussed in Section II low inertia power. standard control systems and interconnected into a large systems are more susceptible to frequency deviations and. power system will synchronize with one another converging experience more severe disturbances than classical SG based. to a common network wide frequency in steady state The systems Moreover the standard AGC architecture for sec. difference between this common frequency and the nominal ondary control was devised in the 1950 s and has not been. frequency ref is directly proportional to the imbalance substantially updated since then to utilize improvements. between demand and scheduled generation in the system in communication and computation technologies With this. For instance if demand exceeds scheduled generation the context recent research on advanced secondary frequency. frequency will settle to a value below ref as illustrated in control has broadly focused on the following questions. Figure 4 This relationship forms the basis for the real time i to what extent should the primary secondary tertiary. control of supply demand balance in the system control layers be co designed and under what circum. Frequency control is performed in power systems on a stances can or should they be merged into a single. hierarchy of time scales 30 Chapter 11 1 ranging from fast time scale control architecture. approximately one second to tens of minutes 28 Chapter ii what are the appropriate control architectures for inte. 9 as shown in Figure 5 Figure 9 shows the response of grating large numbers of low capacity heterogeneous. a typical power system to a sudden and persistent load step control resources into secondary frequency control and. or equivalently a loss of generation how should feedback signals be designed to achieve. From 1 the initial slope of the frequency decrease economic objectives and while maintaining or improv. the so called rate of change of frequency RoCoF ing dynamic performance. iii how can power converter interfaced resources be most immediate control conclusions can be drawn from 15 by ap. effectively controlled and utilized for frequency con plying results from multivariable servoregulator theory 90. trol Firstly since range G0 span 1n the only admissible. A full exploration of these issues is far beyond our scope steady state frequency vectors are those with equal entries. here our goal is to introduce the reader to the secondary i e a common steady state. Pn frequency Moreover since, frequency control problem identify some of its key charac ker G0 y Rn i 1 y i 0 only the total sum of all. teristics and note some recent developments power references and all uncontrolled disturbances contribute. to the common steady state frequency value For constant. C Models for Frequency Control and Key Insights, exogenous inputs um and Pu the resulting steady state. As secondary frequency control is concerned with asymp frequency deviation ss is. totically correcting small deviations in frequency around the. nominal value small signal models are most appropriate ss 1 1. with reactive power and voltage magnitude dynamics typ Since G0 has row rank equal to one a second conclusion. ically neglected As a representative model for discussion which can be drawn is the following it is necessary that. we consider a network reduced linearized model of n SGs there be only one frequency integrator in any internally stable. secondary control system see Appendix I for a short proof. We conclude that in steady state or by continuity on. Mi i Di i m i Pu i Pe i 14b sufficiently long time scales a power system acts much like. Pe i Tij i j 14c a single input single output control system with total power. imbalance as input and synchronous frequency as output. Tch i m i m i g i 14d We conclude that there is incredible flexibility in how to. Tg i g i g i um i Rd i i 14e achieve steady state frequency regulation both in terms of. how frequency measurements can be spatially collected and. The point of linearization for the vectorized state x. used for control purposes and in terms of how steady state. m g and vectorized inputs u Pu um is, control actions are allocated among actuators in the system. x ref 1n m, This flexibility permits centralized distributed and semi. decentralized approaches to the frequency control problem. where 1n 1 1 1 Rn Equations 14a 14c, describe the linearized mechanical dynamics of a network D Formulation of Optimal Frequency Control Problem. of SGs Pu i R is the uncontrolled injection generation Given the noted flexibility in terms of how steady state. minus demand at the same bus as the ith SG and Pe i control actions are allocated to actuators in the system. is the electrical power injected by SG i and Tij 0 is a steady state optimization problem can be posed by the. the synchronizing torque coefficient for line i j 30 Eq system operator to minimize the cost of control provision. 11 10 subject to system wide balance of power2 or equivalently. The equation 14d models a non reheat steam turbine and subject to frequency regulation 3. 14e models the governor where Tch i 0 Tg i 0 Xn, is the turbine governor time constant Rd i 0 is the min Ji um i 17a. droop slope control gain for primary control and um i Xn. subject to um i Pu i 0 17b, is the external reference for secondary control the load i 1. reference setpoint The reference values um i are therefore um i Ui m i m i 17c. set points to lower level controllers Section II C The where m i and m i are upper and lower setpoint limits. equations 14a 14b also describe reduced order models for unit i cf 9 and Ji m i m i R is the cost rate. of power converters 24 In any case the conclusions and function of unit i we assume Ji is convex and differentiable. development that follow are insensitive to these modeling The formulation 17 assumes that resistive network losses. assumptions and extend to complex scenarios involving are negligible standard for transmission system control and. nonlinear models CIG load side participation and so forth that the grid is sufficiently far from congested scenarios. To begin our analysis we consider 14a 14e as together standard for frequency control. defining a vectorized input output LTI system Going forward we will assume that the box constraints. s G1 s Pu s G2 s um s 17c have been incorporated into the objective functions. Ji via a differentiable penalty function see 91 for a. and examine the response if the disturbances and power related approach With this assumption the gradient KKT. references are constant Analysis shows that the open loop optimality condition for 17 is. system is input output stable and that the DC gain matrices. satisfy Ji um i i 1 n 18, G1 0 G2 0 G0 1 1n 1 n 15 2 This balancing problem will be further studied explicitly in Section V. Pn Pn 1 in the context of competitive markets, The constant i 1 i i 1 Di Ri is called the 3 We omit additional equality constraints tie line flow schedules for. stiffness constant of the power system 30 Eq 11 9 Two simplicity see 28 Sec 9 5 2 for a discussion of tie line flow control. where R is the dual variable associated with the as a local estimate of the global marginal cost for power. power balance constraint 17b Equation 18 is the famous imbalance and evolves according to. economic dispatch criteria which states that at optimality Xn. the marginal cost for each generation unit should be equal ki i i aij i j 21a. um i Ji 1 i 21b,E Control Architectures and Recent Solutions. The problem 17 cannot be solved in a feedforward cen Asymptotically the consensus feedback in 21a ensures that. tralized fashion as the disturbances Pu i are unmeasured all i converge towards the optimal price variable while. spatially distributed and time varying The control problem eliminating the local frequency deviations the control actions. is to develop feedback controllers to ensure that in closed 21b are computed just as in 20b Finally we note that. loop the power system converges to an optimizer of 17 despite having distributed integral action this control scheme. this is a problem of real time economic balancing is entirely consistent with the statement directly below 16. if w Rn is the left eigenvector associated with the zero. Automatic Generation Control The AGC methodology eigenvalue of Laplacian matrix of the communication graph. takes a single frequency deviation measurement AGC 98 Theorem 7 4 then 21a implies. integrates and broadcasts the resulting control signal4 Xn Xn. dt w i k i i wi i,KI b AGC i 1 i 1, um i m i i which is the integral mode of the controller all other modes. have a stable low pass characteristic, where f 0 is the filter time constant KI 0 is the integral There is now an extensive literature on 21 which was. gain and b 0 is the frequency bias constant typically initially proposed independently in 53 and 99 Stability. chosen equal to The quantity i 0 is the participation analyses for nonlinear dynamic models can be found in. factor of unit i at the operating point defined as 100 103 with communication network design and delay. 2 Ji m i 1 Xn issues treated in 104 105 and higher order dynamic. i 1 models in 106 107 see 108 109 for further microgrid. k 1 Jk m k, applications Tuning and fundamental performance limita. assuming Ji is twice differentiable The AGC scheme is tions have been examined in 110 113 Stability proofs. centralized and typically treated with quadratic cost functions for distributed controller have been restricted to the case of. 31 See 92 95 for overviews and surveys of AGC and quadratic cost functions Ji a more general stability proof. 96 for a more recent treatment e g for strictly convex Ji remains an open problem. Gather and Broadcast Control A generalization of the Primal Dual Control and Other Methods Another pop. AGC scheme collects frequency measurements from across ular frequency control approach which we do not cover. the system integrates a weighted average and broadcasts out in detail here is based on continuous time saddle point or. the control signals primal dual optimization methods representative references. Xn are 114 120 The key idea is to encode system equi. KI ci i 20a librium and steady state specifications in an optimization. um i Ji 1 20b problem and then apply augmented Lagrangian methods. P to derive an equilibrium seeking controller this results in a. where KI 0 ci 0 and i ci 1 The equation peer to peer distributed control architecture in the same spirit. 20b sets the power references of all units based on a as 21 Centralized and distributed model predictive control. common variable t this common variable converges approaches have been proposed in 121 124 which allow. asymptotically to the marginal cost of system wide power for transient constraint satisfaction at the expense of im. imbalance cf 17 18 This scheme was proposed in 97 plementation complexity Price based control strategies have. which contains a nonlinear Lyapunov stability analysis under been proposed in 114 125 127 and an approximate. a homogeneity assumption on the cost functions decentralized approach was studied in 128 Finally we. note that there is an emerging line of theoretical research on. Distributed Averaging Integral Control Methods from. steady state optimizing feedback control which encompasses. consensus can be leveraged to distribute the integral control. some or all of the control schemes discussed in this section. computation and thereby remove the centralized point of. see 70 73 for different formulations,computation from the previous two methods Let A A. To conclude we note that important aspects we have not. aij Rn n denote the adjacency matrix of a connected. discussed in this section include i market based provision. weighted graph containing a globally reachable node 98. of frequency control services in deregulated power systems. this models peer to peer communication links between SGs. ii heterogeneity of resources within the grid which can. Each unit is assigned an integral variable i which serves. allow for control authority to be spread out both spatially. 4 This is a stylized model of AGC with considerable variation seen in and temporally iii balancing authority based secondary. practical implementations see 30 Section 11 1 6 for further discussion frequency control where inter area tie line flows must be. regulated to scheduled setpoints and iv practical imple control i e the reference input of node i for the secondary. mentation aspects including delay tolerance and required control and u t u1 t uN t the vectorized. sampling rates While advanced control strategies for im input as in model 14 Section IV C For balancing energy. proving classical frequency control architectures show much supply ui t 0 and demand ui t 0 at all times the. promise practically relevant strategies must in the end be power schedules shall be chosen such that. robust reliable and simple P,i I ui 1N In u 0n 22,V C OORDINATION OF E NERGY S UPPLY AND D EMAND. where u col u1 uN Thus we can define the local, A Market Mechanisms for Balancing Supply and Demand. feasible set as U U1 UN and the coupling, In modern autonomous power grids supply and demand constraint set as C u RnN 1 N In u 0n. side management distributed energy generation and storage so that the overall feasible set reads as K U C Let us. are seen as among the main facilitators for the integration assume that all the sets are nonempty compact and convex. of renewable energy sources These technologies have in fact and that the set K satisfies Slater s constraint qualification. shown the potential to increase energy efficiency via control 136 5 2 3 5 26. lable generation consumption and storage in local balancing. markets 129 130 In general active supply and demand Cooperative balancing If the autonomous nodes belong. management can be referred as programs implemented by to the same energy company or agree to cooperatively. system operators or utility companies to efficiently use balance energy supply and demand as in Section IV D. the available energy without installing new generation and we can assume that the nodes aim at solving an economic. transmission infrastructure These include demand response balancing optimization problem 17. programs residential and or commercial load management P. programs e g for reducing or shifting consumption 131. nN i I Ji ui, The latter aims at shifting the high power loads to off peak s t ui Ui i I 23. hours to reduce the peak to average ratio in energy demand u. For example appropriate load shifting is foreseen to become. crucial for charging discharging control in areas with high Problem 23 has separable cost function and separable. local penetration of plug in electric vehicles 132 constraints It is also known as optimal exchange problem. For load shifting and in general for balancing supply and and under suitable technical assumptions be solved effi. demand in electricity markets efficiently semi decentralized ciently via dual decomposition Douglas Rachford ADMM. or distributed bidding mechanisms paired with aggregative and proximal algorithms 137 7 3 2 138 5 3 1 From. market clearing algorithms have been proposed 133 134 an economics perspective these methods rely on a price. Namely a market coordinator controls an incentive variable adjustment process so called tatonnement see the theory of. based on an aggregative measure of energy supply and general Walras equilibrium 139 140 141 Namely the. demand The latter is determined as an aggregation of in market coordinator acts via price adjustments by increasing. dividual bids e g computed via local optimization routines or decreasing the price of electricity depending on whether. Such coordination mechanism is traditionally in place in the there is an excess demand or excess supply respectively. slowest time scale of power system control and is referred Competitive balancing More realistically whenever the. as tertiary control see Figure 5 autonomous nodes belong to different energy parties or. While in general the optimal balancing problem is a hard companies it is natural to assume that each node aims at. problem namely a mathematical problem with equilibrium minimizing its local cost function Ji ui u i which usually. constraints MPEC 135 simplified yet representative depends on both the local variable ui first argument due to. models have been proposed In this section we model active the local energy generation or storage cost and also on the. supply and demand management via convex optimization decision variables of the other agents u i col uj j6 i. and monotone game theory and review simple incentive second argument This is because the electricity price set by. mechanisms coupled with iterative bidding for balancing the market is typically an aggregative function of all intended. energy supply and demand in a competitive setting power schedules u1 u2 uN Formally we obtain the. B Mathematics of Balanced Energy Supply and Demand following game which is a collection of inter dependent. optimization problems 5, We consider N autonomous nodes e g generators flexible. storage and loads indexed by i I 1 2 N P, where each shall decide on its power schedule ui umin. n i I Ji ui u i, ui 1 ui n over a multi period horizon of n time i I s t ui P. periods from the local decision set Ui Rn which rep. ui j I i uj 0n, resents local operational constraints such as limits ui. ui t ui rate constraints ui t 1 P ui t i and 5 For. n each i I with ui j I i uj 0n we differentiate between. inter temporal constraints e g i t 1 ui t i At the local decision variable the power schedule ui and the given decision. each time period t 1 n ui t represents the tertiary variables of the other nodes the power schedules uj s. We note that for each node i I both the cost function comparing 29 and 30 we have that u is a v GNE of. Ji and the feasible set the competitive balancing game if and only if for some. n P o Rn 0 T u i e u is a zero of T, Ki u i ui Ui ui j I i uj 0n 25 which is maximally monotone by construction 144 Def. depend on the power schedules of the other nodes u i 20 20 Therefore the competitive power balancing problem. In this setting the nodes would competitively bid tentative 24 26 is a monotone inclusion problem 144 26. power schedules for ultimately balancing energy supply and. demand In game theory the balancing problem can be seen D Coordination Algorithms for Asymptotic Balancing. as that of computing a generalized Nash equilibrium GNE Monotone inclusion problems which include convex opti. of the game in 24 142 i e a set of power schedules mization problems as a special problem class can be solved. u col u 1 u N K such that for all i I efficiently via operator splitting methods In the following. u i argmin Ji v u i v Ki u i, 26 subsections we present two mechanisms for computing a. zero of T in 30 i e a pair u such that 0, In plain words a GNE is such that no node can improve T u In both mechanisms since the balance constraint. its revenue by unilaterally changing its power schedule to is dualized 28 energy supply and demand are balanced. another feasible one To ensure existence and uniqueness of asymptotically This is tolerable since balancing mechanisms. a v GNE 143 Prop 12 7 12 9 let us assume that the cost run off line to solve the day ahead market problem not in. functions Ji are continuously differentiable convex in their real time during the actual power grid operation. first argument and that the so called game mapping. Dual Decomposition DD Algorithm One of the sim, F u col u1 J1 u uN JN u 27 plest mechanisms to exploit the dualization of the balance. which collects the local partial derivatives is strongly mono constraint via the Lagrangian functions in 28 is the so. tone i e F u F v u v ku vk for some called dual decomposition algorithm. 0 for all u v and Lipschitz continuous, C Operator Theoretic Characterization i I ui k 1 argmin Ji v u i k k v. In this section we show the main steps to recast the prob. lem to compute a GNE u 26 into a monotone inclusion. problem i e the problem to find a zero of a monotone. k 1 k j I uj k 1 31b, operator First in order to decouple the balance constraint. 22 present in 24 we introduce the KKT conditions of In essence the DD algorithm consists of a first order price. each optimization problem in 24 For each node i I let adjustment mechanism together with local full optimization. us define its Lagrangian function steps In 31b the market coordinator adjusts the pricing. variable which based on the most recent tentative P power. Li u i Ji ui u i Ui ui, i 1N In u 28 schedules is increased if supply exceeds demand j uj. where Ui denotes the indicator function6 and Rn is 0 and decreased viceversa We note that if the cost func. the dual multiplier associated with the balance constraint tions J i have an aggregative structure i e Ji ui u i. J i ui j6 i uj for some function J i which is typical of. in 22 Thanks to convexity and regularity it follows from. 143 12 2 3 that u is a GNE of the game in 24 if and energy markets then the algorithm consists of fully decen. only if the following coupled KKT systems hold true tralized computations 31a by the autonomous nodes and of. a simple aggregative computation and broadcast communi. 0 ui Ji u i u i NUi u i i, i I cation 31b by the market coordinator The main advantage. 29 of this optimize gather and broadcast scheme is that the. where NUi Ui denotes the normal cone operator 144 competing nodes do not need to exchange information among. Def 6 38 Following the lines in 145 142 3 2 for each other. computational convenience let us look for a variational Under the postulated technical assumptions for small. GNE v GNE a solution to the KKT systems in 29 with enough step size 0 the DD algorithm converges to some. i Rn for all i I 146 3 2 Then to cast the u such that 0 T u with T as in 30 where. balancing problem in compact form let us introduce the set u is a v GNE of the balancing game in 24. valued mapping,Projected Pseudo Gradient PPG Algorithm One. F u NU u 1 In, N practical requirement of the DD algorithm is that at each. 1 N In u iteration each node has to solve an optimization problem. where F is the game mapping in 27 and NU NU1 31a To substantially reduce this computational burden a. NUN is the collective normal cone operator By viable alternative is to let each node take a single step along. the direction of its local pseudo gradient and then project it. 6 u 0 if ui Ui otherwise onto its local constraint set. challenges already emphasized in the previous sections we. highlight the following fruitful avenues for future research. i I ui k 1 projUi ui k 32a At the device level power converters can be efficiently. ui Ji ui k u i k k designed as lossless transformers of signals which means. P that the impactful control decisions injecting power are. k 1 k j I 2uj k 1 uj k 32b being taken at the DC terminal The extent to which the. dynamics of converter interfaced sources can be designed for. In 32b we note that the market coordinator performs grid stability appears expansive with no present consensus. a second order price adjustment while the nodes perform on the best strategy Aggregating many converter interfaced. local projected pseudo gradient steps in 32a Also for the sources into virtual power plants and controlling the ensem. PPG algorithm if the cost functions Ji have an aggregative ble will also be key for effective integration of renewables. structure then we recover a semi decentralized computation The flexibility of these devices suggests that their set. and communication scheme with the market coordinator in point could be regulated in a coordinated way There seems. the loop and where the competing nodes shall not exchange to be an untapped potential in the real time control of the. information among each other steady state of power converters together with traditional. The PPG algorithm in 32 also known as asymmetric generators in order to provide faster fine grained and more. projection algorithm 147 12 5 1 is the outcome of a efficient ancillary services to the grid Feedback optimization. preconditioned forward backward splitting applied to the can provide a unified MIMO approach for many real time. operator T in 30 splitted as control problems that are currently tackled via parallel and. possibly inefficient SISO controllers,T u N In F u 33. 1 Finally at the aggregate system level merging secondary. control and competitive tertiary balancing markets i e. We refer to 148 149 for the explicit derivation of the market in the loop is a key direction requiring careful prob. algorithm in 32 via operator splitting for aggregative and lem formulations and innovative solutions the extent to. network games respectively Under the postulated technical which the dynamic response of the system can be decoupled. assumptions for small enough step size 0 the algorithm from or co designed with market mechanisms is a crucial. converges to some zero of T in 30 u where u is question for the viability of such architectures. the unique v GNE of the game in 24 indeed,ACKNOWLEDGEMENTS. E Some Recent Semi Decentralized and Distributed Coor The authors want to thank their research groups collabo. dination Algorithms rators as well as the anonymous reviewers for many fruitful. The literature on coordination algorithms for solving discussions and comments which improved this paper. monotone games with general non differentiable convex. 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