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\section{State of the Art and Limits of Current Practice}
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The principal aim of this research is to create autonomous reactor control
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systems that are tractably safe. To understand what is being automated, we must
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first understand how nuclear reactors are operated today. This section examines
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reactor operators and the operating procedures we aim to leverage, then
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investigates limitations of human-based operation, and concludes with current
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formal methods approaches to reactor control systems.
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\subsection{Current Reactor Procedures and Operation}
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Nuclear plant procedures exist in a hierarchy: normal operating procedures for
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routine operations, abnormal operating procedures for off-normal conditions,
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Emergency Operating Procedures (EOPs) for design-basis accidents, Severe
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Accident Management Guidelines (SAMGs) for beyond-design-basis events, and
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Extensive Damage Mitigation Guidelines (EDMGs) for catastrophic damage
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scenarios. These procedures must comply with 10 CFR 50.34(b)(6)(ii) and are
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developed using guidance from NUREG-0900~\cite{NUREG-0899, 10CFR50.34}, but their
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development relies fundamentally on expert judgment and simulator validation
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rather than formal verification. Procedures undergo technical evaluation,
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simulator validation testing, and biennial review as part of operator
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requalification under 10 CFR 55.59~\cite{10CFR55.59}. Despite this rigor,
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procedures fundamentally lack formal verification of key safety properties. No
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mathematical proof exists that procedures cover all possible plant states, that
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required actions can be completed within available timeframes, or that
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transitions between procedure sets maintain safety invariants.
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\textbf{LIMITATION:} \textit{Procedures lack formal verification of correctness
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and completeness.} Current procedure development relies on expert judgment and
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simulator validation. No mathematical proof exists that procedures cover all
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possible plant states, that required actions can be completed within available
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timeframes, or that transitions between procedure sets maintain safety
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invariants. Paper-based procedures cannot ensure correct application, and even
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computer-based procedure systems lack the formal guarantees that automated
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reasoning could provide.
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Nuclear plants operate with multiple control modes: automatic control, where the
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reactor control system maintains target parameters through continuous reactivity
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adjustment; manual control, where operators directly manipulate the reactor; and
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various intermediate modes. In typical pressurized water reactor operation, the
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reactor control system automatically maintains a floating average temperature
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and compensates for power demand changes through reactivity feedback loops
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alone. Safety systems, by contrast, operate with implemented automation. Reactor
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Protection Systems trip automatically on safety signals with millisecond
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response times, and engineered safety features actuate automatically on accident
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signals without operator action required.
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The division between automated and human-controlled functions reveals the
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fundamental challenge of hybrid control. Highly automated systems handle reactor
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protection---automatic trips on safety parameters, emergency core cooling
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actuation, containment isolation, and basic process
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control~\cite{WRPS.Description, gentillon_westinghouse_1999}. Human operators,
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however, retain control of strategic decision-making: power level changes,
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startup/shutdown sequences, mode transitions, and procedure implementation.
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\subsection{Human Factors in Nuclear Accidents}
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Current-generation nuclear power plants employ over 3,600 active NRC-licensed
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reactor operators in the United States~\cite{operator_statistics}. These
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operators divide into Reactor Operators (ROs), who manipulate reactor controls,
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and Senior Reactor Operators (SROs), who direct plant operations and serve as
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shift supervisors~\cite{10CFR55}. Staffing typically requires at least two ROs
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and one SRO for current-generation units~\cite{10CFR50.54}. Becoming a reactor
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operator requires several years of training.
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The persistent role of human error in nuclear safety incidents---despite decades
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of improvements in training and procedures---provides the most compelling
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motivation for formal automated control with mathematical safety guarantees.
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Operators hold legal authority under 10 CFR Part 55 to make critical decisions,
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including departing from normal regulations during emergencies. The Three Mile
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Island (TMI) accident demonstrated how a combination of personnel error, design
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deficiencies, and component failures led to partial meltdown when operators
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misread confusing and contradictory readings and shut off the emergency water
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system~\cite{Kemeny1979}. The President's Commission on TMI identified a
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fundamental ambiguity: placing responsibility for safe power plant operations on
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the licensee without formal verification that operators can fulfill this
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responsibility does not guarantee safety. This tension between operational
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flexibility and safety assurance remains unresolved: the person responsible for
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reactor safety is often the root cause of failures.
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Multiple independent analyses converge on a striking statistic: 70--80\% of
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nuclear power plant events are attributed to human error, versus approximately
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20\% to equipment failures~\cite{WNA2020}. More significantly, the root cause of
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all severe accidents at nuclear power plants---Three Mile Island, Chernobyl, and
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Fukushima Daiichi---has been identified as poor safety management and safety
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culture: primarily human factors~\cite{hogberg_root_2013}. A detailed analysis
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of 190 events at Chinese nuclear power plants from
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2007--2020~\cite{zhang_analysis_2025} found that 53\% of events involved active
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errors, while 92\% were associated with latent errors---organizational and
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systemic weaknesses that create conditions for failure.
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\textbf{LIMITATION:} \textit{Human factors impose fundamental reliability limits
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that cannot be overcome through training alone.} The persistent human
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error contribution despite four decades of improvements demonstrates that these
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limitations are fundamental rather than a remediable part of human-driven control.
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\subsection{Formal Methods}
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\subsubsection{HARDENS}
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The High Assurance Rigorous Digital Engineering for Nuclear Safety (HARDENS)
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project represents the most advanced application of formal methods to nuclear
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reactor control systems to date~\cite{Kiniry2024}.
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HARDENS aimed to address a fundamental dilemma: existing U.S. nuclear control
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rooms rely on analog technologies from the 1950s--60s. This technology is
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obsolete compared to modern control systems and incurs significant risk and
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cost. The NRC contracted Galois, a formal methods firm, to demonstrate that
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Model-Based Systems Engineering and formal methods could design, verify, and
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implement a complex protection system meeting regulatory criteria at a fraction
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of typical cost. The project delivered a Reactor Trip System (RTS)
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implementation with full traceability from NRC Request for Proposals and IEEE
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standards through formal architecture specifications to verified software.
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HARDENS employed formal methods tools and techniques across the verification
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hierarchy. High-level specifications used Lando, SysMLv2, and FRET (NASA Formal
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Requirements Elicitation Tool) to capture stakeholder requirements, domain
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engineering, certification requirements, and safety requirements. Requirements
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were analyzed for consistency, completeness, and realizability using SAT and SMT
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solvers. Executable formal models used Cryptol to create a behavioral model of
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the entire RTS, including all subsystems, components, and limited digital twin
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models of sensors, actuators, and compute infrastructure. Automatic code
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synthesis generated verifiable C implementations and SystemVerilog hardware
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implementations directly from Cryptol models---eliminating the traditional gap
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between specification and implementation where errors commonly arise.
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Despite its accomplishments, HARDENS has a fundamental limitation directly
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relevant to hybrid control synthesis: the project addressed only discrete
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digital control logic without modeling or verifying continuous reactor dynamics.
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The Reactor Trip System specification and verification covered discrete state
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transitions (trip/no-trip decisions), digital sensor input processing through
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discrete logic, and discrete actuation outputs (reactor trip commands). The
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project did not address continuous dynamics of nuclear reactor physics. Real
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reactor safety depends on the interaction between continuous
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processes---temperature, pressure, neutron flux---evolving in response to
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discrete control decisions. HARDENS verified the discrete controller in
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isolation but not the closed-loop hybrid system behavior.
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\textbf{LIMITATION:} \textit{HARDENS addressed discrete control logic without
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continuous dynamics or hybrid system verification.} Verifying discrete control
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logic alone provides no guarantee that the closed-loop system exhibits desired
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continuous behavior such as stability, convergence to setpoints, or maintained
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safety margins.
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HARDENS produced a demonstrator system at Technology Readiness Level 2--3
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(analytical proof of concept with laboratory breadboard validation) rather than
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a deployment-ready system validated through extended operational testing. The
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NRC Final Report explicitly notes~\cite{Kiniry2024} that all material is
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considered in development, not a finalized product, and that ``The demonstration
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of its technical soundness was to be at a level consistent with satisfaction of
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the current regulatory criteria, although with no explicit demonstration of how
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regulatory requirements are met.'' The project did not include deployment in
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actual nuclear facilities, testing with real reactor systems under operational
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conditions, side-by-side validation with operational analog RTS systems,
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systematic failure mode testing (radiation effects, electromagnetic
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interference, temperature extremes), NRC licensing review, or human factors
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validation with licensed operators in realistic control room scenarios.
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\textbf{LIMITATION:} \textit{HARDENS achieved TRL 2--3 without experimental
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validation.} While formal verification provides mathematical correctness
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guarantees for the implemented discrete logic, the gap between formal
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verification and actual system deployment involves myriad practical
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considerations: integration with legacy systems, long-term reliability
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under harsh environments, human-system interaction in realistic
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operational contexts, and regulatory acceptance of formal methods as
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primary assurance evidence.
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\subsubsection{Sequent Calculus and Differential Dynamic Logic}
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There has been additional work to do verification of hybrid systems by extending
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the temporal logics directly. The result has been the field of differential
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dynamic logic (dL). dL introduces two additional operators
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into temporal logic: the box operator and the diamond operator. The box operator
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\([\alpha]\phi\) states that for some region \(\phi\), the hybrid system
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\(\alpha\) always remains within that region. In this way, it is a safety
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ivariant being enforced for the system. The second operator, the diamond
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operator \(<\alpha>\phi\) says that for the region \(\phi\), there is at least
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one trajectory of \(\alpha\) that enters that region. This is a declaration of a
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liveness property.
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%source: https://symbolaris.com/logic/dL.html
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While dL allows for the specification of these liveness and safety properties,
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actually proving them for a given hybrid system is quite difficult. Automated
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proof assistants such as KeYmaera X exist to help develop proofs of systems
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using dL, but so far have been insufficient for reasonably complex hybrid
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systems. The main issue behind creating system proofs using dL is state space
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explosion and non-terminating solutions.
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%Source: that one satellite tracking paper that has the problem with the
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%gyroscopes overloding and needing to dump speed all the time
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Approaches have been made to alleviate
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these issues for nuclear power contexts using contract and decomposition based
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methods, but are far from a complete methodology to design systems with.
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%source: Manyu's thesis.
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Instead, these approaches have been used on systems that have been designed a
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priori, and require expert knowledge to create the system proofs.
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%Maybe a limitation here? Something about dL doesn't scale or help in design
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%very much, so the limitation is that logic based hybrid system approaches have
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%not been used in the DESIGN of autonomous controllers, only in the analysis of
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%systems that already exist.
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{/usr/local/texlive/2025/texmf-var/fonts/map/pdftex/updmap/pdftex.map}{/usr/local/texlive/2025/texmf-dist/fonts/enc/dvips/base/8r.enc}] (./1-goals-and-outcomes/research_statement_v1.tex)
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...
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[6]
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[7]
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[9] (./4-metrics-of-success/v1.tex
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[14] (./6-broader-impacts/v1.tex)
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[17]
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\OT1/cmtt/m/n/12 nuclear . org / information -[] library / safety -[] and -[] security / safety -[] of -[]
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[]
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[18])
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[19] (./main.aux)
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***********
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LaTeX2e <2024-11-01> patch level 2
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L3 programming layer <2025-01-18>
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***********
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2
main.tex
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main.tex
@ -20,7 +20,7 @@
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\input{1-goals-and-outcomes/v1}
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\newpage
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\input{2-state-of-the-art/v1}
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\input{2-state-of-the-art/v2}
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\newpage
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\input{3-research-approach/v3}
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44
main.toc
44
main.toc
@ -3,24 +3,26 @@
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||||
\contentsline {section}{\numberline {2}State of the Art and Limits of Current Practice}{3}{}%
|
||||
\contentsline {subsection}{\numberline {2.1}Current Reactor Procedures and Operation}{3}{}%
|
||||
\contentsline {subsection}{\numberline {2.2}Human Factors in Nuclear Accidents}{3}{}%
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||||
\contentsline {subsection}{\numberline {2.3}HARDENS and Formal Methods}{4}{}%
|
||||
\contentsline {section}{\numberline {3}Research Approach}{6}{}%
|
||||
\contentsline {subsection}{\numberline {3.1}System Requirements and Specifications}{7}{}%
|
||||
\contentsline {subsection}{\numberline {3.2}Discrete Controller Synthesis}{10}{}%
|
||||
\contentsline {subsection}{\numberline {3.3}Continuous Controllers}{10}{}%
|
||||
\contentsline {subsubsection}{\numberline {3.3.1}Transitory Modes}{11}{}%
|
||||
\contentsline {subsubsection}{\numberline {3.3.2}Stabilizing Modes}{12}{}%
|
||||
\contentsline {subsubsection}{\numberline {3.3.3}Expulsory Modes}{13}{}%
|
||||
\contentsline {subsection}{\numberline {3.4}Industrial Implementation}{14}{}%
|
||||
\contentsline {section}{\numberline {4}Metrics for Success}{15}{}%
|
||||
\contentsline {paragraph}{TRL 3 \textit {Critical Function and Proof of Concept}}{15}{}%
|
||||
\contentsline {paragraph}{TRL 4 \textit {Laboratory Testing of Integrated Components}}{15}{}%
|
||||
\contentsline {paragraph}{TRL 5 \textit {Laboratory Testing in Relevant Environment}}{15}{}%
|
||||
\contentsline {section}{\numberline {5}Risks and Contingencies}{17}{}%
|
||||
\contentsline {subsection}{\numberline {5.1}Computational Tractability of Synthesis}{17}{}%
|
||||
\contentsline {subsection}{\numberline {5.2}Discrete-Continuous Interface Formalization}{17}{}%
|
||||
\contentsline {subsection}{\numberline {5.3}Procedure Formalization Completeness}{18}{}%
|
||||
\contentsline {section}{\numberline {6}Broader Impacts}{20}{}%
|
||||
\contentsline {section}{\numberline {7}Schedule, Milestones, and Deliverables}{22}{}%
|
||||
\contentsline {subsection}{\numberline {7.1}Milestones and Deliverables}{22}{}%
|
||||
\contentsline {section}{References}{23}{}%
|
||||
\contentsline {subsection}{\numberline {2.3}Formal Methods}{4}{}%
|
||||
\contentsline {subsubsection}{\numberline {2.3.1}HARDENS}{4}{}%
|
||||
\contentsline {subsubsection}{\numberline {2.3.2}Sequent Calculus and Differential Dynamic Logic}{5}{}%
|
||||
\contentsline {section}{\numberline {3}Research Approach}{7}{}%
|
||||
\contentsline {subsection}{\numberline {3.1}System Requirements and Specifications}{8}{}%
|
||||
\contentsline {subsection}{\numberline {3.2}Discrete Controller Synthesis}{11}{}%
|
||||
\contentsline {subsection}{\numberline {3.3}Continuous Controllers}{11}{}%
|
||||
\contentsline {subsubsection}{\numberline {3.3.1}Transitory Modes}{12}{}%
|
||||
\contentsline {subsubsection}{\numberline {3.3.2}Stabilizing Modes}{13}{}%
|
||||
\contentsline {subsubsection}{\numberline {3.3.3}Expulsory Modes}{14}{}%
|
||||
\contentsline {subsection}{\numberline {3.4}Industrial Implementation}{15}{}%
|
||||
\contentsline {section}{\numberline {4}Metrics for Success}{16}{}%
|
||||
\contentsline {paragraph}{TRL 3 \textit {Critical Function and Proof of Concept}}{16}{}%
|
||||
\contentsline {paragraph}{TRL 4 \textit {Laboratory Testing of Integrated Components}}{16}{}%
|
||||
\contentsline {paragraph}{TRL 5 \textit {Laboratory Testing in Relevant Environment}}{16}{}%
|
||||
\contentsline {section}{\numberline {5}Risks and Contingencies}{18}{}%
|
||||
\contentsline {subsection}{\numberline {5.1}Computational Tractability of Synthesis}{18}{}%
|
||||
\contentsline {subsection}{\numberline {5.2}Discrete-Continuous Interface Formalization}{18}{}%
|
||||
\contentsline {subsection}{\numberline {5.3}Procedure Formalization Completeness}{19}{}%
|
||||
\contentsline {section}{\numberline {6}Broader Impacts}{21}{}%
|
||||
\contentsline {section}{\numberline {7}Schedule, Milestones, and Deliverables}{23}{}%
|
||||
\contentsline {subsection}{\numberline {7.1}Milestones and Deliverables}{23}{}%
|
||||
\contentsline {section}{References}{24}{}%
|
||||
|
||||
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