Guillaume Ganet-Lepage
email: guillaume.ganet-lepage@centralesupelec.fr

Alessio Iovine
email: alessio.iovine@centralesupelec.fr

Sorin Olaru
email: sorin.olaru@centralesupelec.fr

Estimation of Power Transmission Distribution Factors for Predictive Control in Sub-transmission Areas

Keywords: Predictive control, congestion management, PTDF

Transmission system operators target to maximize renewable power injection into the grids while preserving power lines' safety. Recently, decentralised approaches are developed using Power Transmission Distribution Factors (PTDFs), and consider model-based control strategies for local sub-transmission area. However, the computation of the models is complex due to the time-varying nature of the parameters. In order to answer these shortcomings, the present paper proposes a realtime parameter estimation using past observations of the power flow on a finite time window. Simulations results validate the proposed approach.

I. INTRODUCTION

The energy transition of power grids faces an increase of generations from renewable energies connected to the power grids. The transmission system operators' (TSO) objective is to maximize the injections on the grid for such generations while preserving the safety of the power lines. This objective is handled by power congestion management and the related computational methods are important for the guarantees they can offer regarding the equipment damage and cascading failures implications [START_REF] Monforti-Ferrario | The impact of power network congestion, its consequences and mitigation measures on air pollutants and greenhouse gases emissions. a case from germany[END_REF].

Due to the large size of power grids, controlling the grid as a whole is both complex and sensitive. The recent decentralized approaches [START_REF] Iovine | Modeling the partial renewable power curtailment for transmission network management[END_REF] focus on controlling some predefined subtransmission areas, alternatively called zones, associated to a particular geographical area. These areas are known to face power congestion issues due to high generations from renewable sources. The local controls to tackle power congestion, acting on levers like curtailment/storage/topological changes, have been shown to exhibit the expected impact on the congestion despite the interconnection uncertainty with the rest of the grid. This work is building on a series of recent works [START_REF] Iovine | Modeling the partial renewable power curtailment for transmission network management[END_REF] [3] [START_REF] Ganet-Lepage | Towards a safe maximisation of renewable's flexibility in power transmission sub-grids: An mpc approach[END_REF], all based on a Model Predictive Control (MPC) [START_REF] Camacho | Model predictive control[END_REF] approach.

In the existing works, a discrete-time prediction model associated to each area uses a linearization of the power flow equations based on Power Transfer Distribution Factors [START_REF] Cheng | PTDF-based power system equivalents[END_REF] (PTDFs). These parameters for power flow equations evolve in time and can be represented by the percentage of increase of power flow a given line would receive due to an increase of generation at a given bus. The use of timeinvariant PTDFs is an advantage for the MPC's computation. As a drawback, the computation of these time-invariant PTDFs requires the topology of the whole power grid to get just the values concerning the sub-transmission areas, and thus a timeconsuming computation done at an off-line stage. In order to avoid time-invariant PTDFs computed off-line by means of optimal power flow analysis, exploiting tools like [START_REF] Zimmerman | MAT-POWER: Steady-state operations, planning, and analysis tools for power systems research and education[END_REF], the current work aims at estimating the parameters of the power flow equations dynamically at each time step and with the ultimate goal of including them in the prediction-based control design.

As a first attempt in this endeavor, the method we propose is based on parameter estimation using past observations on a finite time window. More precisely, the power flows and the power generations recorded over time will be used to fit the parameters in a classical least squares framework [8] [9]. A discussion on the impact of the observation window and the computation will be done. A biased estimation is expected to be retrieved whenever the past horizon considered is too long as the distant past data might not be relevant for the current mode of functioning. At the same time, if the past horizon is too short, inaccuracy can occur and complementary tools need to be found for regularization.

II. THE PRINCIPLES OF CONGESTION MANAGEMENT

BASED ON MODEL PREDICTIVE CONTROL In this section, we recall in a concise manner all the elements needed in order to construct a mathematical model for the evolution of the power flows within a selected subtransmission area, described by a subset of buses, generators, and batteries along with the power lines in the selected zone to be controlled. For more details on the construction of the state space model, the interested reader is referred to [START_REF] Ganet-Lepage | Towards a safe maximisation of renewable's flexibility in power transmission sub-grids: An mpc approach[END_REF]. Main sets of topological information: Z: set of buses in the zone; Z G ⊂ Z: set of buses with a generator in the zone; Z B ⊂ Z: set of buses with a battery in the zone; L: set of power lines in the zone. Parameter: The system dynamics of the zone model is defined, ∀l ∈ Z L , ∀g ∈ Z C , ∀b ∈ Z B , ∀n ∈ Z N as:

n N , n G , n B , n L : cardinality of Z, Z G , Z B ,
F l (t + 1) = F l (t) + b∈Z B ψ B l,b (t) • δP B b (t -τ B )+ g∈Z G ψ G l,g (t)• z G g (t) -P G g (t) + n∈Z N ψ N l,n (t) • δP T n (t) P C g (t + 1) = P C g (t) + δP C g (t -τ C ) P B b (t + 1) = P B b (t) + δP B b (t -τ B ) E B b (t + 1) = E B b (t) -T • c B b • [P B b (t) + δP B b (t -τ B )] P G g (t + 1) = z G g (t) P A g (t + 1) = P A g (t) + δP A g (t) (I) z G g (t) = min ( P A g (t) + δP A g (t), P G g -P C g (t) -δP C g (t -τ C ) (II)
In practice, the curtailment control delay τ C is 45 seconds, while the battery control delay τ B is 5 seconds. To determine the appropriate curtailment control at present, the controller must anticipate the future impact of the control. Model Predictive Control (MPC) is chosen for this purpose, as it can foresee the outcomes over a prediction horizon, enabling informed decision-making and control actions. More details about the control strategy are in [START_REF] Iovine | Modeling the partial renewable power curtailment for transmission network management[END_REF] [START_REF] Ganet-Lepage | Towards a safe maximisation of renewable's flexibility in power transmission sub-grids: An mpc approach[END_REF], where the model-based predictive control for congestion management was presented.

III. REAL-TIME PTDF ESTIMATION

The previous works of [START_REF] Pham | Nonlinearity handling in MPC for Power Congestion management in sub-transmission areas[END_REF] and [START_REF] Ganet-Lepage | Towards a safe maximisation of renewable's flexibility in power transmission sub-grids: An mpc approach[END_REF] builds on the working hypothesis that the power flow equation of (I) exhibits PTDFs with constant values computed offline based on MATPOWER library [START_REF] Zimmerman | MAT-POWER: Steady-state operations, planning, and analysis tools for power systems research and education[END_REF], i.e.,

ψ G l,g (t) = ψ G,mat l,g , ψ B l,b (t) = ψ B,mat l,b , ψ N l,n (t) = ψ N,mat l,n
(III) In the present work, we revisit this hypothesis. The PTDF are now time-varying, and the estimation of these PTDF is databased: a collection of data estimates the values of the PTDF using regression analysis. The ultimate goal for the control design point of view is to use these time-varying PTDFs for the prediction mechanism. Ideally (from the robustness of the controller), the reconstructed PTDFs are equal to the values computed offline. If this ideal situation differs from the real system operation conditions, then one can represent more accurately the current state of functioning using the estimated PTDFs values, thus leading to better controls, which exploits most of the information available.

Indeed, a possible cause of errors in the [START_REF] Pham | Nonlinearity handling in MPC for Power Congestion management in sub-transmission areas[END_REF] and [START_REF] Ganet-Lepage | Towards a safe maximisation of renewable's flexibility in power transmission sub-grids: An mpc approach[END_REF] is that the offline computation of PTDF from [START_REF] Zimmerman | MAT-POWER: Steady-state operations, planning, and analysis tools for power systems research and education[END_REF] assumes the generation of only one generator is changing at a time, but the superposition principle might not be applicable for several generators simultaneously. This work aims at estimating the time-varying PTDFs : ψ G l,g (t), ψ B l,b (t), ψ N l,n (t) from (I).

A. Receding-horizon parameter estimation and constrained least squares formulation

The quality of the PTDFs' estimation has an important impact on the model-based prediction capabilities and consequently on the optimal control. For this reason, at each time step t of the simulation, the estimation of the PTDFs will be updated. The estimation is based on a receding horizon strategy and two approaches are considered and detailed next:

• a variable-length horizon starting from the beginning of the simulation and including all the available historical data; • a fixed horizon, the data considered for the estimation corresponds to a fixed number of previous steps of the simulation, 10 and 30 steps are used in the numerical examples.

The estimation of the time-varying PTDFs is formulated in terms of an optimisation problem. Let us introduce the Power flow variations δF l and generation power variations δP G g :

δF l (t) = F l (t + 1) -F l (t), δP G g (t) = z G g (t) -P G g (t) (IV)
δF l (t) = b∈Z B δP B b (t -d) • ψ B l,b (t) + g∈Z G δP G g (t) • ψ G l,g (t)
+

n∈Z N δP T n (t) • ψ N l,n (t) (V)
The aim is to formulate Eq. (V) in matrix form, and the following notations of column vectors are instrumental:

δF (t) =    δF 1 (t) . . . l δF n L (t)    ∈ R n L , δP B (t) =    δP B 1 (t) . . . b δP B n B (t)    ∈ R n B , δP G (t) =    δP G 1 (t) . . . g δP G n G (t)    ∈ R n G , δP T (t) =    δP T 1 (t) . . . n δP T n N (t)    ∈ R n N . Let Ψ B (t) = ψ B l,b (t) ∈ R n L ×n B , Ψ G (t) = ψ G l,g (t) ∈ R n L ×n G , Ψ N (t) = ψ N l,n (t) ∈ R n L ×n N
, and the block matrix

Ψ(t) = Ψ B (t) Ψ G (t) Ψ N (t) ∈ R n L ×(n B +n G +n N )
. Each block corresponds to some PTDF values (batteries, generators or buses) at time t.

Consequently, the power flow equation (V) can be formulated in a compact form, by considering all lines l:

δF (t) = Ψ(t) •   δP B (t -τ B ) δP G (t) δP T (t)   (VI)
Considering the available information on the past H p -step horizon from t -H p to t -1 as illustrated in Fig. 1, lead to the definition of the matrices of observation:

∆F (t) = δF (t -H p ) t • • •δF (t -1) ∈ R n L ×Hp , ∆P B (t -τ B ) = δP B (t -H p -τ B ) t • • •δP B (t -1 -τ B ) ∈ R n B ×Hp , ∆P G (t) = δP G (t -H p ) t • • •δP G (t -1) ∈ R n G ×Hp , ∆P T (t) = δP T (t -H p ) t • • •δP T (t -1) ∈ R n N ×Hp
, and the block matrix

∆P (t) =   ∆P B (t -τ B ) ∆P G (t) ∆P T (t)  
collecting the past observations from t -H p up to t -1.

A working assumption of pseudo-constant PTDFs over the observation horizon will be made in order to exploit the structure of the dependencies at the computation stage. It should be noticed that the receding-horizon principle employed here, re-inforces the time-varying PTDFs nature of the estimation. Practically, at each sampling instant a linear regression will be employed to estimate point-wise (at time t) the PTDFs Ψ(t):

∆F (t) = Ψ(t) • ∆P (t) + S(t) (VII)
where S(t) ∈ R H×n L corresponds to the slack matrix of the residuals between the left-hand side and the remaining terms on the right-hand side. ∆F (t) and ∆P (t) are collections of available data, namely past observations. We want to select the PTDFs in Ψ(t), such that the Frobenius-norm of the slack matrix S(t) is the smallest, noted ∥.∥ F .

Thus, the optimization problem O 1 is defined as:

O 1 (t) = min Ψ(t) ∥∆F (t) -Ψ(t) • ∆P (t)∥ F (VIII)
Additionally, the PTDFs represent a percentage of generation distribution over a line, which is directed but flow is allowed in both directions, so the percentage can also be negative. To impose a percentage-like solution for the coefficients, a constraint is added by defining the all-ones matrix J ∈ R (n B +n G +n N )×n L :

-J ≤ Ψ ≤ J (IX)
With these main principles, several versions of estimations can be obtained according to the length of the observation window and the weight on the information within the respective window. One can enumerate some intuitive choices: use all the available data (with obvious storage complexity); use the data from a finite window; impose a forgetting factor on the data coming from a long observation window, that last point is not investigated here.

B. Topological aspect of the solution

Based on the physical balance, the PTDFs are defined in relationship with the following property: for any given bus, the generation must be entirely distributed on the lines connected with the bus. This topological property can be translated through constraints on the PTDFs estimates: the sum of the PTDFs of all lines connected to this bus is equal to 100%.

The above property should be valid for any type of PTDF coefficient, either ψ B , ψ G or ψ N :

l∈L s.t. n=fromBus of l ψ l,n - l∈L s.t. n=toBus of l ψ l,n = 100% (X)
As lines are directed edges, the direction partitions the sums into two categories: incoming with a positive sign, and outgoing with an opposite sign. The respective constraints (X) need to be accounted for in the optimization problem constructing the PTDF estimates at each sampling time.

C. Numerical results with all past data collected

A comparison of results based on the methods used for estimation is presented next. It should be noted that for the simulations, no information regarding the transit power disturbances ∆P N is available and thus the hypothesis ∆P N = 0 is adopted. In real-world scenarios, such information may be available and thus enable the estimation of the PTDF Ψ N , which is not possible here.

Four methods were considered for illustration, the first one, the baseline using the grid properties and three others are based on data observation:

• MATPOWER's computation of the PTDF, based on the topology and the properties of the power network • Use of the explicit unconstrained optimum via Matlab's built-in pseudoinverse function • Solving a Least Squares optimization problem, Eq.(VIII)-(IX) encoded with Yalmip [START_REF] Lofberg | Yalmip: A toolbox for modeling and optimization in matlab[END_REF] and solved with Gurobi [START_REF]Gurobi optimizer reference manual[END_REF].

• The optimization framework with additionally the topological constraints, i.e. Eq. (VIII)-(IX)-(X). Along the simulations, the estimation at each step can use all the past observations. Gradually the number of past observations grows and structurally helps in estimating accurately the PTDFs using the optimisation-based procedure. It can be observed that results based on pseudoinverse and least squares are equal. Three types of behavior are apparent. Firstly, MATPOWER's method and the two Least Squares methods result in the same PTDFs. A clear example in this sense is the PTDF estimation between branch 1 and the battery, cf. Fig. 2. The battery remains inactive for the first 400 seconds, offering no observations for PTDF estimation. It becomes operational afterward, contributing to the estimation process. The PTDF converge rapidly towards MATPOWER's PTDFs. A second interesting behavior can be observed between the two Least Squares methods: with and without topology constraint. Convergence to to the same PTDF is obtained, but the limit values are different from MATPOWER's value. For example, the PTDF estimation of branch 5 with generator 3, Fig. 3.

Third, a case where each method converges towards a different PTDF value. For example, the PTDF estimate of branch 7 with generator 4, cf. Fig. 4, where the Least Squares method without topology constraint converges to -43%, MAT-POWER's PTDF to -45% and the Least Squares method with topology constraint converges to -46%.

In summary, when the two Least Squares methods converge to distinct values, the one incorporating topology constraints is usually closer to MATPOWER's PTDF. Previous tests assumed that all past observations are collected and stored during the simulation. In the context of a receding-horizon policy, this equates to a dynamically expanding receding horizon estimation, where the past observation window gradually grows. In practical scenarios with limited storage, we relax this structural property and adjust the estimation to a fixed window for the receding horizon.

The goal is to evaluate the impact of the size of horizon on the PTDF estimation. The PTDF estimation between branch 7 and generator 4 is chosen as example. Two cases are illustrated: a 30-step and a 10-step window, with Fig. 5 and 6 respectively; to be compared with the case of all past observations of Fig. 4. For the 30-step window case, the Least Squares method without the topology constraint behaves similarly to the estimation using all past observations depicted in Fig 4 . The convergence is similar, with a slight variation in the PTDF asymptotic value. The addition of the topology constraint leads to estimations with higher fluctuations.

For the 10-step window case, the shortness of the observation window implies locally a lack of excitation within the observation window and leads to singularities in the computation of the PTDF values.

To tackle the high variability of the estimates, we introduce a regularized version of the Least Squares method with topology constraint. The PTDF estimation variables are decomposed into two parts: the default time-invariant PTDF (XII) where δψ l,n (t) are the entries of ∆Ψ(t). Thus, the optimisation problem becomes:

O 2 (t) = min ∆Ψ(t) ∥∆F (t)-Ψ(t)•∆P (t)∥ F +c∥∆Ψ(t)∥ F (XIII)
The first term focuses on accurately determining the PTDFs through precise linear regression. The second term introduces flexibility in PTDF variations while preventing unrealistic values. The parameter c serves as a weighting factor and is set to 1 in these simulations.

In experiments, the case with a 30-step horizon and regulation exhibited no difference compared to previous methods, but significantly improved the behavior of the 10-step horizon case.The PTDF estimation of branch 7 with generator 4 is depicted on Fig. 7. One can observe that whenever the PTDF estimation was previously reaching extreme values, e.g., at 200 or 800 seconds of simulation, the regularization mechanism leads to a selection of the time-invariant PTDF

IV. CONCLUSION

The paper investigated different methods for the real-time estimation of the Power Transmission Distribution Factors (PTDFs). These factors represent the parameters of the dynamical model used by the controller for prediction purposes.

The numerical examples show a good convergence whenever the persistent excitation is ensured despite the fact that the controller used a time-invariant prediction model. As future research direction, the goal is to determine the efficiency of the controllers based on a time-varying dynamical model employing the different PTDF estimations.

Fig. 1 .

 1 Fig. 1. past horizon Hp, made of past observations usable at time t

1 Fig. 2 .

 12 Fig.2. PTDF estimate between branch 1 and the battery, located at bus 6

1 Fig. 3 . 1 Fig. 4 .

 1314 Fig. 3. PTDF estimate between branch 5 and generator 3 located on bus 5

Fig. 7 .

 7 Fig. 7. PTDF estimate: branch 7, gen 4, 10-step horizon, regularization


Youngest age groups depend (a) mostly on the intercensal fertility and the number of women in reproductive age groups in census 1 (surviving and present in the country during intercensal period), and (b) to a lesser extent on infant/child mortality and migration between censuses 1 and 2, and (c) potential errors in census 2.

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Population Division • Population projections by the United Nations are derived from models of future trends in the demographic components of change, in particular fertility and mortality • UN projection models have a strong basis in demographic theory; for each country, the models are calibrated using data from the country itself and, especially when data are sparse, from other countries as well • Uncertainty of the UN population projections is reflected in traditional deterministic/normative scenarios; the 2012 Revision introduced new methods based on formal models that yields probabilistic statements about plausible future trends ; the 2022 revision also takes into account uncertainty in past fertility estimates • Work on the probabilistic assessment of uncertainty is ongoing and could benefit from further efforts to incorporate additional sources of uncertainty, notably for international migrations Summary

•

  1950-2021: 6945 country/area x years with mortality crises out of 16,827 locations x years or about 41% (n=237 over 71 years) • Conflicts and Battle deaths: 2355 locations x years (n=171) • Mass killings (including genocide): 95 locations x years (n=18) • Floodings: 2323 locations x years (n=162) • Cyclones: 1605 locations x years (n=169) • Epidemics: 1988 locations x years (n=152) • Earthquakes: 1143 locations x years (n=124) • COVID-19: 431 locations x years (n=218) • Famines/Droughts: 160 locations x years (n=32) • Tsunami: 45 locations x years (n=25)

o▪

  Project fertility, mortality and migration and use with base population in cohortcomponent framework o Can do this through trend extrapolation, parametric modelling, assumptions (e.g., expert-based) o Can investigate implications of different scenarios • Fast vs slow pace of fertility decline • Instant-replacement fertility • Constant-mortality • No migration • Probabilistic o Employ statistical models to project likely trajectories of fertility, mortality and migration, then use results with base population in cohort-component framework How to project the components of population change? Click to edit Master subtitle style Click to edit Master title style Population Division▪ UN projections of fertility and mortality are guided by historical trends in those same variables ▪ Regularities in historical trends have led to theories of demographic change, which give structure to the projection model ▪ Model is calibrated for each country using an estimation procedure that relies primarily on data for that country ▪ Data for other countries influence estimates especially for countries in which the transition is less complete ▪ Bayesian hierarchical model of past trends, combined with time series model of future trends, yields probabilistic depiction of plausible Rate of TFR decline depends on level of TFR ▪ Peak rate of decline when total fertility rates is around 5 or 6 ▪ Slower decline for TFR > 6 or TFR < 5 ▪ Rate of decline in the TFR, as a function of its current level, is modeled using a double-logistic function, which has an inverted U-shape ▪ Bayesian hierarchical model used to estimate model for the world and for each country ▪ In addition, standard time series methods are used to project future trends once countries reach below replacement levelPhase II: fertility decline model by definition, after the first

B t,t+n and D t,t+n are

  Click to edit Master subtitle styleClick to edit Master title style t+n = P t + B t,t+n -D t,t+n + I t,t+n -E t,t+n P t+n -P t = B t,t+n -D t,t+n + I t,t+n -E t,t+n P t is the population at time t number of births and deaths occurring between t and t + n.I t,t+n and E t,t+n are number of immigrants and emigrants from the country during the same period

	Population balance (demographic accounting)
	Population Division

P

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Population Division

•

  For each of the 237 countries or areas:

	o Base population by sex and single year of age (or graduated
	5-year age group) in 1950
	o For every year from 1950-2021, time series of:
	• fertility rates for women aged 10-54 years by single year of
	age group (or graduated 5-year age groups)
	• sex ratio at birth (males/females)
	• sex and age-specific mortality rates (life tables) for single
	ages (or graduated abridged age groups)
	• net international migration by sex and single ages

Input data requirements for WPP

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Population Division

•

  Estimates and projections are revised every 2 years, incorporating all newly available data from censuses, administrative registers, surveys, etc.

	• Data collected through:
	IPUMS International, DHS, MICS, etc.)
	o Published reports and NSO web sites
	o Ad-hoc requests to national institutions
	• Latest revision in 2022 considered:
	o 1,758 population censuses conducted between 1950 and 2021
	o Information on births and deaths from vital registration systems for 169 countries
	o Demographic indicators from 2,890 surveys

o UN Statistics Division (Demographic Yearbook System) o UN regional commissions & agencies (UNICEF, UNFPA, UNHCR, UNAIDS, etc.) o Global or regional databases (WHO, Eurostat, HMD, HFD, etc.) o Public-use micro-datasets (

Procedures/frequency to incorporate newly available data ?

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	Population Division

• To fill-in gaps in missing data: most

	information often available only for
	some countries and/or dates, or not sufficiently disaggregated by age
	• To reconcile differences between (a) data sources and/or estimation
	method(s) for a specific date and (b) within sources over time
	• To ensure international comparability using similar
	definitions/concepts, methodology and assumptions across countries
	UN population estimates and projections are used in the calculation of many
	development indicators used by the United Nations system, including about one-
	quarter of the indicators used to monitor global progress towards the achievement of
	the SDGs.

With so many data available, why estimates are necessary…

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Population Division

• For many countries, data available vary greatly in quantity, frequency, quality, reliability and consistency. • Not all data points are as informative and can be trusted equally… • Estimates can vary based on the type of data sources (census, surveys, vital registration), the type of survey itself (national survey vs. international survey programs), the estimation methods (direct or indirect estimates) and by various

biases affecting reporting of retrospective data (e.g., birth histories or lifetime fertility, parental or siblings' survival). Estimation of robust time series for demographic rates (e.g., fertility and mortality)

Click to edit Master subtitle style Click to edit Master title style Population Division India TFR: 1950-2021 time trend Click to edit Master subtitle style Click to edit Master title style Population Division India TFR: 1950-2100 time trend Click to edit Master subtitle style Click to edit Master title style Population Division India fertility rates 5-year age groups: 1950-2021 Click to edit Master subtitle style Click to edit Master title style

Population Division India fertility rates graduated by age: 1950-2021

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Population Division India fertility rates graduated by age: 1950-2021

	Population Division

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•

  Compile and compute direct and indirect estimates from as many empirical data sources as possible for each country since 1950 • Review and assess the various series • Generate an initial robust time trend for overall summary indicators (e.g., TFR, 5q0, 45q15) and age-specific rates (e.g., ASFR, life tables) • Use this initial set of estimates within the full cohort-component population reconstruction by age and sex since 1950 • Compare and assess the reconstructed population cohorts with those enumerated across the various censuses • Revise and adjust the set of WPP estimates to reconcile the various demographic components (e.g., TFR) that satisfy the demographic balancing relationships over time, age and cohorts

WPP workflow process with vital rates

	• 113 locations with empirical life tables based on different types of
	inputs and data gaps
	Population Division

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Age-specific type of mortality input data

	Type of age pattern	Input data	Locations (n)
	Empirical	Complete	44
	Empirical	Abridged	69
	Model-based	Abridged	124
		Total	237
			21

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Population Division

Annual time series of excess mortality due to crises 22 About 61 million excess deaths: • 19.8 million due to conflicts (32%) • 19.9 million due to famines (33%) • 4.7 million due to natural disasters (8%) • 15.0 million due to COVID-19 (25%) About 61 million excess deaths: • 19.8 million due to conflicts (32%) • 19.9 million due to famines (33%) • 4.7 million due to natural disasters (8%) • 15.0 million due to COVID-19 (25%)

	UN cohort-component projections to cross-validate estimations
	Remember… over the last 70 years:
	• 87% of countries have ≥ 3 censuses, 77% have ≥ 4 and about
	half of the countries have ≥ 5.
	• Each census can be evaluated by itself, and in the context of the
	previous ones by following birth cohorts over time.
	With at least 3-5 censuses in the last decades…
	1. Start cohort-component projections with base year in 1950 for
	all 237 countries/areas.
	2. Insure full internal consistency by age and sex between all
	demographic components and enumerated populations.
	Population Division

See WPP 2022 methodology and structured metadata for further details Click to edit Master subtitle style Click to edit Master title style Population Division Click to edit Master subtitle style Click to edit Master title style Click to edit Master subtitle style Click to edit Master title style

Population Division Sri Lanka: 1963-2001 censuses

	1,
	1,600,000
	1,400,000
	1,200,000
	1,000,000
	800,000
	600,000
	400,000
	200,000
	0
	25

800,000 Sri Lanka -Comparison of the projected and observed Both Sexes population: WPP vs 1963 Census

	Both Sexes observed, Sri Lanka 1963	Both Sexes projected, Sri Lanka 1963
		2,
		1,800,000
		1,600,000
		1,400,000
		1,200,000
		1,000,000
		800,000
		600,000
		400,000
		200,000
		0

000,000 Sri Lanka -Comparison of the projected and observed Both Sexes population: WPP vs 1971 Census

	Both Sexes observed, Sri Lanka 1971	Both Sexes projected, Sri Lanka 1971
	1,500
	1,000,000
	500,000
	0

,000 2,000,000 2,500,000 Sri Lanka -Comparison of the projected and observed Both Sexes population: WPP vs 1981 Census

	Both Sexes observed, Sri Lanka 1981	Both Sexes projected, Sri Lanka 1981
		1,600
		1,400,000
		1,200,000
		1,000,000
		800,000
		600,000
		400,000
		200,000
		0

,000 1,800,000 2,000,000 Sri Lanka -Comparison of the projected and observed Both Sexes population: WPP vs 2001 Census

	1963 census	1971 census
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	Both Sexes observed, Sri Lanka 2001	Both Sexes projected, Sri Lanka 2001
		2001 census
	1981 census

Population Division 1. Survive the base population in t by age and sex to t+n in projection period

	Multiply each age group population by life table survival ratios
	(open-ended interval requires special handling)
	2.

Make any necessary adjustments for migration 3. Calculate the total number of births (total, by sex) during t, t+n

48 countries/areas in Phase III by 2021

	• Europe		• Eastern & South-Eastern Asia / Pacific
	o Armenia	o Lithuania	o Australia
	o Austria	o Luxembourg	o China
	o Belarus	o Malta	o China -Hong Kong SAR
	o Belgium	o Netherlands	o China -Macao SAR
	o Bulgaria	o North Macedonia	o China -Taiwan Province of China
	o Czechia	o Norway	o Japan
	o Denmark	o Poland	o Singapore
	o Estonia	o Republic of Moldova	o Viet Nam
	o Finland	o Romania
	o France o Germany o Greece o Guernsey	o Russian Federation o Serbia o Slovakia o Slovenia	• Latin America and the Caribbean o Barbados o Cuba
	o Hungary	o Spain
	o Ireland	o Sweden	• Northern America
	o Italy	o Switzerland	o Canada
	o Jersey	o Ukraine	o United States of America
	o Latvia	o United Kingdom
			35

Sources of error vary in importance with the length of the projection horizon: 1950-1995.

	Population in 2015:
	• +3.1% (233 mio)
	to -3.7% (-272 mio)
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	Population Division

UN population projections: Past and Present 1980

  to 2022 revisions of the World Population Prospects (WPP)

		7.6	UN 1980 UN 1982
	(billions) population	5.6 6.0 6.8 7.2 6.4	UN 1984 UN 1988 UN 1990 UN 1992 UN 1994 UN 1996 UN 1998 UN 2000 UN 2002 UN 2004
	Total	5.2	UN 2006 UN 2008
			UN 2010
		4.8	UN 2012
			UN 2015
		4.4	UN 2017
		1980 1985 1990 1995 2000 2005 2010 2015	UN 2019
		Year	UN 2022
			53

* MARE is the mean absolute relative error. Coverage is the proportion of the 1990-2010 observations that fell within their prediction interval, in percent. Source: Raftery, A.E., N. Li, H. Ševčíková, P. Gerland, and G.K. Heilig (2012). "Bayesian probabilistic population projections for all countries." Proceedings of the National Academy of Sciences. 109 (35):13915-13921. doi: 10.1073/pnas.1211452109

• Differences in medium trajectories are driven mostly by differences in fertility assumptions • Differences in uncertainty ranges are closely linked to differences in assumptions about the strength of correlation across countries in changes taking place over time • UN population projections have a good track record and continue to be guided by the same core principle in choosing assumptions -that historical trends in fertility, mortality and migration are the best available predictor of future trends*.

Three important considerations about differences in long-term population projections 48 * More precisely, that historical trends in fertility, mortality and migration provide the most reliable basis for calibrating theoretically informed models of future trends

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Population Division

• National Research Council, & Committee on Population (2000). Beyond six billion: Forecasting the world's population. National Academies Press.

• Keyfitz, N. ( 1981). The limits of population forecasting. Population and development review.

• Lee, R. D. ( 1990). Long-run global population forecasts: A critical appraisal. PDR. • DataCatalog: comprehensive inventory for each country of primary data sources (censuses, demographic surveys, etc.) providing data on demographic processes (fertility, mortality, population structure and dynamics, marital status and family planning) for all countries and areas, as of Dec. 2023, about 8,250 entries

• DataArchive: a virtual repository of documents, tabular datasets and reports (potentially) for each data source, as of Dec. 2023, more than 32,750 files

• DemoData: SQL database to store in a structured and standardized way empirical data and demographic estimates (with meta-information) on population, fertility, mortality and (net) international migration data from as many sources as possible: Online resources