Time-varying Projected Dynamical Systems with Applications to Feedback Optimization of Power Systems

This paper is concerned with the study of continuous-time, non-smooth dynamical systems which arise in the context of time-varying non-convex optimization problems, as for example the feedback-based optimization of power systems. We generalize the notion of projected dynamical systems to time-varying, possibly non-regular, domains and derive conditions for the existence of so-called Krasovskii solutions. The key insight is that for trajectories to exist, informally, the time-varying domain can only contract at a bounded rate whereas it may expand discontinuously. This condition is met, in particular, by feasible sets delimited via piecewise differentiable functions under appropriate constraint qualifications. To illustrate the necessity and usefulness of such a general framework, we consider a simple yet insightful power system example, and we discuss the implications of the proposed conditions for the design of feedback optimization schemes

Experimental Validation of Feedback Optimization in Power Distribution Grids

We consider the problem of controlling the voltage of a distribution feeder using the reactive power capabilities of inverters. On a real distribution grid, we compare the local Volt/VAr droop control recommended in recent grid codes, a centralized dispatch based on optimal power flow (OPF) programming, and a feedback optimization (FO) controller that we propose. The local droop control yields suboptimal regulation, as predicted analytically. The OPF-based dispatch strategy requires an accurate grid model and measurement of all loads on the feeder in order to achieve proper voltage regulation. However, in the experiment, the OPF-based strategy violates voltage constraints due to inevitable model mismatch and uncertainties. Our proposed FO controller, on the other hand, satisfies the constraints and does not require load measurements or any grid state estimation. The only needed model knowledge is the sensitivity of the voltages with respect to reactive power, which can be obtained from data. As we show, an approximation of these sensitivities is also sufficient, which makes the approach essentially model-free, easy to tune, compatible with the current sensing and control infrastructure, and remarkably robust to measurement noise. We expect these properties to be fundamental features of FO for power systems and not specific to Volt/VAr regulation or to distribution grids

Ribozyme Knockdown of the ␥-Subunit of Rod cGMP Phosphodiesterase Alters the ERG and Retinal Morphology in Wild-Type Mice

PURPOSE. To generate an animal model of retinal degeneration by using AAV-mediated ribozyme knockdown of the ␥-subunit of the rod cGMP phosphodiesterase (PDE␥) mRNA in the retina of wild-type mice. METHODS. Two hammerhead ribozymes, HRz35 and HRz42, were designed to target the PDE␥ gene in wild-type C57BL/6 mice. The efficiency and specificity of the ribozyme cleavage was tested in vitro against three different types of target: short synthetic RNA oligomers, longer targets transcribed from clones, and full-length mRNA from total retinal RNA extracts. After in vitro validation, the ribozymes were cloned and packaged in a recombinant adenoassociated virus (rAAV) containing a proximal 472-bp murine rod opsin promoter (MOPS) to drive ribozyme synthesis. Three-week-old wild-type C57BL/6 mice were injected subretinally with the vectors. For treated versus partner control retinas, responses to light were measured by full-field electroretinography (ERG), and retinal tissues were examined by light microscopy. Messenger RNA and protein levels of PDE␥ gene were monitored by reverse transcription-polymerase chain reaction (RT-PCR) and Western immunoblot assay. RESULTS. The ribozymes had comparable in vitro kinetic properties in multiple turnover kinetic analyses. Ribozyme HRz35 exhibited a K cat of 0.48 minute Ϫ1 and a K m of 980 nM, and HRz42 showed a K cat of 0.17 minute Ϫ1 and a K m of 971 nM. Both ribozymes cleaved at accessible sites in the RNA, as they digested long transcripts transcribed from clones and fulllength mRNA from total retinal RNA extracts in vitro. At increasing intervals after subretinal injection with either AAV ribozyme, a 30% to 90% reduction in a-and b-wave amplitudes was observed compared with those in contralateral control eyes that were not injected. Retinal tissue analysis showed that loss of the photoreceptor cells and PDE␥ mRNA and protein paralleled the ERG results. CONCLUSIONS. Ribozyme-mediated somatic knockdown of wildtype PDE␥ mRNA in vivo can efficiently reduce the target RNA leading to a loss in rod photoreceptors and in rod-mediated ERG amplitudes, thus generating an animal model of retinal degeneration resembling human RP in an essentially normal adult retina. This vector ribozyme technique should be applicable to other genes associated with RP and perhaps also to mRNAs of phototransduction genes not yet associated with RP. Application of this approach may be age and species independent. (Invest Ophthalmol Vis Sci. 2005;46: 3836 -3844

Human blue-opsin promoter preferentially targets reporter gene expression to rat s-cone photoreceptors

PURPOSE. To develop a gene therapy system that specifically targets transgene expression to S-cones of the mammalian retina, the authors coupled recombinant AAV-mediated delivery with the use of a human blue-opsin (HB) promoter to drive expression. METHODS. Two regions of the HB promoter sequence, HB569 and HB996, were amplified from human DNA, cloned into an AAV vector cassette upstream of the green fluorescent protein (GFP) gene, and packaged into AAV2 and AAV5 capsids. Eyes of postnatal day (P) 40 to P48 Sprague-Dawley rats were subretinally injected with 2 L vector. Animals were humanely killed 2 to 3 weeks or 20 months after injection, and the pattern and persistence of GFP expression were analyzed in the treated retinas by immunohistochemistry, Western blotting, and RT-PCR. RESULTS. AAV5.HB.GFP vectors targeted photoreceptor transduction with an efficiency 20-fold higher than analogous serotype 2 vector. Both AAV5.HB.GFP vectors exhibited similar transduction efficiencies with patterns of GFP expression that did not vary depending on the size of the HB promoter used. Transgene expression was exclusively localized to photoreceptors of retinas treated with either vector. Furthermore, GFP expression was observed for at least 20 months. Dual GFP immunostaining with S-or M-opsin antibodies and GFP/PNA labeling revealed that cones coexpressing S-opsin/GFP or Mopsin/GFP constituted 37.5% Ϯ 8% and 13.5% Ϯ 3% of the GFP-positive photoreceptors, respectively, whereas rods constituted 49% Ϯ 5% of the GFP-positive photoreceptors. Because cones constitute approximately 1% of adult rat retinal photoreceptors, it was estimated that the relative transduction efficiency of AAV5.HB.GFP vectors was approximately 100:1 for cones versus rods. CONCLUSIONS. AAV5.HB.GFP vector injected into the subretinal space of Sprague-Dawley rats targeted gene expression to photoreceptor cells with an efficiency approximately 20-fold higher than that for AAV2.HB.GFP. Transgene expression regulated by the human blue cone-promoter persisted at least for 20 months. Cones coexpressing S-opsin and the GFP transgene appeared to prevail, confirming that in addition to having properties of the AAV serotype, the promoter choice is key to fine-tuning transgene delivery and expression in specific retinal cells. The system described here may be effective in a therapeutic setting in which strong S-cone transgene expression is required. (Invest Ophthalmol Vis Sci. 2006;47:3505-3513